Research ArticleComplex Convexity of Musielak-Orlicz Function SpacesEquipped with the 119901-Amemiya Norm

Lili Chen Yunan Cui and Yanfeng Zhao

Department of Mathematics Harbin University of Science and Technology Harbin 150080 China

Correspondence should be addressed to Lili Chen cll2119hotmailcom

Received 27 November 2013 Revised 5 April 2014 Accepted 13 April 2014 Published 8 May 2014

Academic Editor Angelo Favini

Copyright copy 2014 Lili Chen et alThis is an open access article distributed under the Creative CommonsAttribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

The complex convexity of Musielak-Orlicz function spaces equipped with the 119901-Amemiya norm is mainly discussed It is obtainedthat for any Musielak-Orlicz function space equipped with the 119901-Amemiya norm when 1 le 119901 lt infin complex strongly extremepoints of the unit ball coincide with complex extreme points of the unit ball Moreover criteria for them in above spaces are givenCriteria for complex strict convexity and complex midpoint locally uniform convexity of above spaces are also deduced

1 Introduction

Let (119883 sdot ) be a complex Banach space over the complexfield C let 119894 be the complex number satisfying 1198942 = minus1 andlet 119861(119883) and 119878(119883) be the closed unit ball and the unit sphereof 119883 respectively In the sequel N and R denote the set ofnatural numbers and the set of real numbers respectively

In the early 1980s a huge number of papers in the area ofthe geometry of Banach spaces were directed to the complexgeometry of complex Banach spaces It is well known that thecomplex geometric properties of complex Banach spaces haveapplications in various branches among others in HarmonicAnalysis Theory Operator Theory Banach Algebras 119862lowast-Algebras Differential EquationTheory QuantumMechanicsTheory and Hydrodynamics Theory It is also known thatextreme points which are connected with strict convexity ofthe whole spaces are the most basic and important geometricpoints in geometric theory of Banach spaces (see [1ndash6])

In [7] Thorp and Whitley first introduced the conceptsof complex extreme point and complex strict convexitywhen they studied the conditions under which the StrongMaximum Modulus Theorem for analytic functions alwaysholds in a complex Banach space

A point 119909 isin 119878(119883) is said to be a complex extreme point of119861(119883) if for every nonzero119910 isin 119883 there holds sup

|120582|le1119909+120582119910 gt

1 A complex Banach space 119883 is said to be complex strictlyconvex if every element of 119878(119883) is a complex extreme pointof 119861(119883)

In [8] we further studied the notions of complex stronglyextreme point and complex midpoint locally uniform con-vexity in general complex spaces

A point 119909 isin 119878(119883) is said to be a complex strongly extremepoint of 119861(119883) if for every 120576 gt 0 we have Δ

119888(119909 120576) gt 0 where

Δ119888(119909 120576) = inf 1 minus |120582| exist119910 isin 119883st 1003817100381710038171003817120582119909 plusmn 119910

1003817100381710038171003817 le 1

1003817100381710038171003817120582119909 plusmn 1198941199101003817100381710038171003817 le 1

10038171003817100381710038171199101003817100381710038171003817 ge 120576

(1)

A complex Banach space 119883 is said to be complex midpointlocally uniformly convex if every element of 119878(119883) is a complexstrongly extreme point of 119861(119883)

Let (119879 Σ 120583) be a nonatomic and complete measure spacewith 120583(119879) lt infin By Φ we denote a Musielak-Orlicz functionthat is Φ 119879 times [0 +infin) rarr [0 +infin] satisfies the following

(1) for each 119906 isin [0infin] Φ(119905 119906) is a 120583-measurablefunction of 119905 on 119879

(2) for 120583-ae 119905 isin 119879Φ(119905 0) = 0 lim119906rarrinfin

Φ(119905 119906) = infin andthere exists 1199061015840 gt 0 such thatΦ(119905 1199061015840) lt infin

(3) for 120583-ae 119905 isin 119879 Φ(119905 119906) is convex on the interval[0infin) with respect to 119906

Let 119871119888(120583) be the space of all 120583-equivalence classes ofcomplex and Σ-measurable functions defined on 119879 For each119909 isin 119871119888(120583) we define on 119871119888(120583) the convex modular of 119909 by

119868Φ(119909) = int

119879

Φ (119905 |119909 (119905)|) 119889119905 (2)

Hindawi Publishing CorporationAbstract and Applied AnalysisVolume 2014 Article ID 190203 6 pageshttpdxdoiorg1011552014190203

2 Abstract and Applied Analysis

We define supp 119909 = 119905 isin 119879 |119909(119905)| = 0 and the Musielak-Orlicz space 119871

Φgenerated by the formula

119871Φ= 119909 isin 119871

119888(120583) 119868

Φ(120582119909) lt infin for some 120582 gt 0 (3)

Set

119890 (119905) = sup 119906 ge 0 Φ (119905 119906) = 0

119864 (119905) = sup 119906 ge 0 Φ (119905 119906) lt infin (4)

then 119890(119905) and 119864(119905) are 120583-measurable (see [9])The notion of 119901-Amemiya norm has been introduced in

[10] for any 1 le 119901 le infin and 119906 gt 0 define

119904119901(119906) =

(1 + 119906119901)1119901

for 1 le 119901 lt infin

max 1 119906 for 119901 = infin(5)

Furthermore define 119904Φ119901(119909) = 119904

119901∘119868Φ(119909) for all 1 le 119901 le infin

and 119909 isin 119871119888(120583) Notice that the function 119904119901is increasing on 119877

+

for 1 le 119901 lt infin however the function 119904infinis increasing on the

interval [1infin) onlyFor 119909 isin 119871

119888(120583) define the 119901-Amemiya norm by the

formula

119909Φ119901 = inf119896gt0

1

119896119904Φ119901(119896119909) 1 le 119901 le infin (6)

For convenience from now on we write 119871Φ119901

=

(119871Φ sdot Φ119901) it is easy to see that 119871

Φ119901is a Banach space For

1 le 119901 lt infin 119909 isin 119871Φ119901

set

119870119901(119909) = 119896 gt 0 119909Φ119901 =

1

1198961 + 119868

119901

Φ(119896119909)1119901

(7)

2 Main Results

We begin this section from the following useful lemmas

Lemma 1 (see [9]) For any 120576 gt 0 there exists 120575 isin (0 12)such that if 119906 V isin C and

|V| ge120576

8max119890|119906 + 119890V| (8)

then

|119906| le1 minus 2120575

4Σ119890 |119906 + 119890V| (9)

where

max119890|119906 + 119890V| = max |119906 + V| |119906 minus V| |119906 + 119894V| |119906 minus 119894V|

Σ119890 |119906 + 119890V| = |119906 + V| + |119906 minus V| + |119906 + 119894V| + |119906 minus 119894V|

(10)

Lemma 2 If lim119906rarrinfin

(Φ(119905 119906)119906) = infin for 120583-ae 119905 isin 119879 then119870119901(119909) = 0 for any 119909 isin 119871

Φ119901 0 where 1 le 119901 lt infin

Proof For any 119909 isin 119871Φ119901

0 there exists 120572 gt 0 such that120583119905 isin 119879 |119909(119905)| ge 120572 gt 0 Let 119879

1= 119905 isin 119879 |119909(119905)| ge 120572 and

define the subsets

119865119899= 119905 isin 119879

1Φ (119905 119906)

119906ge4119909Φ119901

120572 sdot 1205831198791

119906 ge 119899 (11)

for each 119899 isin N It is easy to see that 1198651sube F2sube sdot sdot sdot sube 119865

119899sube sdot sdot sdot

and lim119899rarrinfin

120583119865119899= 1205831198791 Then there exists 119899

0isin N such that

1205831198651198990gt (12)120583119879

1

It follows from the definition of 119901-Amemiya norm thatthere is a sequence 119896

119899 satisfying

119909Φ119901 = lim119899rarrinfin

1

119896119899

1 + 119868119901

Φ(119896119899119909)1119901

(12)

For any 119896 gt 0 we have

1

1198961 + 119868

119901

Φ(119896119909)1119901

=1

1198961 + [int

119879

Φ (119905 119896 |119909 (119905)|) 119889119905]

119901

1119901

ge

int1198651198990

Φ (119905 119896 |119909 (119905)|) 119889119905

119896

ge int1198651198990

Φ (119905 119896120572)

119896119889119905

= 120572int1198651198990

Φ (119905 119896120572)

119896120572119889119905

(13)

If 119896 gt 1198990120572 notice that

1

1198961 + 119868

119901

Φ(119896119909)1119901

ge 120572 sdot4119909Φ119901

120572 sdot 1205831198791

sdot 1205831198651198990gt 2119909Φ119901 (14)

which means the sequence 119896119899 is bounded Hence without

loss of generality we assume that 119896119899rarr 1198960as 119899 rarr infin

We can also choose the monotonic increasing or decreasingsubsequence of 119896

119899 that converges to the number 119896

0 Apply-

ing Levi Theorem and Lebesgue Dominated ConvergenceTheorem we obtain

119909Φ119901 = lim119899rarrinfin

1

119896119899

1 + 119868119901

Φ(119896119899119909)1119901

=1

1198960

1 + 119868119901

Φ(1198960119909)1119901

(15)

which implies 1198960isin 119870119901(119909)

In order to exclude the case when such sets 119870119901(119909) are

empty for 119909 isin 119871Φ1199010 in the sequel we will assume without

any special mention that lim119906rarrinfin

(Φ(119905 119906)119906) = infin for 120583-ae119905 isin 119879

Theorem 3 Assume 1 le 119901 lt infin 119909 isin 119878(119871Φ119901) Then the

following assertions are equivalent

(1) 119909 is a complex strongly extreme point of the unit ball119861(119871Φ119901)

(2) 119909 is a complex extreme point of the unit ball 119861(119871Φ119901)

(3) for any 119896 isin 119870119901(119909) 120583119905 isin 119879 119896|119909(119905)| lt 119890(119905) = 0

Proof The implication (1) rArr (2) is trivial Let 119909 be a complexextreme point of the unit ball 119861(119871

Φ119901) and there exists 119896

0isin

119870119901(119909) such that 120583119905 isin 119879 119896

0|119909(119905)| lt 119890(119905) gt 0 Then we can

Abstract and Applied Analysis 3

find 119889 gt 0 such that 120583119905 isin 119879 1198960|119909(119905)| + 119889 lt 119890(119905) gt 0

Indeed if 120583119905 isin 119879 1198960|119909(119905)| + 119889 lt 119890(119905) = 0 for any 119889 gt 0

then we set 119889119899= 1119899 and define the subsets

119879119899= 119905 isin 119879 119896

0 |119909 (119905)| + 119889119899 lt 119890 (119905) (16)

for each 119899 isin N Notice that 1198791sube 1198792sube sdot sdot sdot sube 119879

119899sube sdot sdot sdot and

119905 isin 119879 1198960|119909(119905)| le 119890(119905) = ⋃

infin

119899=1119879119899 we can find that

120583 119905 isin 119879 1198960 |119909 (119905)| le 119890 (119905) = 0 (17)

which is a contradictionLet 1198790= 119905 isin 119879 119896

0|119909(119905)| + 119889 lt 119890(119905) and define 119910 =

(1198891198960)1205941198790 we have 119910 = 0 and for any 120582 isin C with |120582| le 1

1003817100381710038171003817119909 + 1205821199101003817100381710038171003817Φ119901 le

1

1198960

1 + 119868119901

Φ(1198960(119909 + 120582119910))

1119901

=1

1198960

1 + [119868Φ(11989601199091205941198791198790

)

+119868Φ(11989601199091205941198790+ 120582119889120594

1198790)]119901

1119901

le1

1198960

1 + [119868Φ(11989601199091205941198791198790

)

+119868Φ((1198960 |119909| + 119889) 1205941198790

)]119901

1119901

=1

1198960

1 + [119868Φ(11989601199091205941198791198790

)]119901

1119901

=1

1198960

(1 + 119868119901

Φ(1198960119909))1119901

= 1

(18)

which shows that 119909 is not a complex extreme point of the unitball 119861(119871

Φ119901)

(3) rArr (1) Suppose that 119909 isin 119878(119871Φ119901) is not a complex

strongly extreme point of the unit ball 119861(119871Φ119901) then there

exists 1205760gt 0 such thatΔ

119888(119909 1205760) = 0That is there exist120582

119899isin C

with |120582119899| rarr 1 and119910

119899isin 119871Φ119901

satisfying 119910119899Φ119901

ge 1205760such that

10038171003817100381710038171205821198991199090 plusmn 1199101198991003817100381710038171003817Φ119901 le 1

10038171003817100381710038171205821198991199090 plusmn 1198941199101198991003817100381710038171003817Φ119901 le 1 (19)

which gives100381710038171003817100381710038171003817100381710038171199090plusmn119910119899

120582119899

10038171003817100381710038171003817100381710038171003817Φ119901

le110038161003816100381610038161205821198991003816100381610038161003816

100381710038171003817100381710038171003817100381710038171199090plusmn 119894119910119899

120582119899

10038171003817100381710038171003817100381710038171003817Φ119901

le110038161003816100381610038161205821198991003816100381610038161003816

(20)

Setting 119911119899= 119910119899120582119899 we have10038171003817100381710038171199111198991003817100381710038171003817Φ119901 ge

10038171003817100381710038171199101198991003817100381710038171003817Φ119901 ge 1205760

10038171003817100381710038171199090 plusmn 1199111198991003817100381710038171003817Φ119901 le

110038161003816100381610038161205821198991003816100381610038161003816

10038171003817100381710038171199090 plusmn 1198941199111198991003817100381710038171003817Φ119901 le

110038161003816100381610038161205821198991003816100381610038161003816

(21)

For the above 1205760gt 0 by Lemma 1 there exists 120575

0isin

(0 12) such that if 119906 V isin C and

|V| ge1205760

8max119890|119906 + 119890V| (22)

then

|119906| le1 minus 2120575

0

4Σ119890 |119906 + 119890V| (23)

For each 119899 isin N let

119860119899= 119905 isin 119879

1003816100381610038161003816119911119899 (119905)1003816100381610038161003816 ge

1205760

8max119890

10038161003816100381610038161199090 (119905) + 119890119911119899 (119905)1003816100381610038161003816

119911(1)

119899 119911(1)

119899(119905) = 119911

119899(119905) (119905 notin 119860

119899) 119911

(1)

119899(119905) = 0 (119905 isin 119860

119899)

119911(2)

119899 119911(2)

119899(119905) = 0 (119905 notin 119860

119899) 119911

(2)

119899(119905) = 119911

119899(119905) (119905 isin 119860

119899)

(24)

It is easy to see that 119911119899= 119911(1)

119899+119911(2)

119899forall119899 isin N Since |120582

119899| rarr 1

when 119899 rarr infin the following inequalities

10038171003817100381710038171003817119911(1)

119899

10038171003817100381710038171003817Φ119901

= inf119896gt0

1

1198961 + [int

119905notin119860119899

Φ(119905 1198961003816100381610038161003816119911119899 (119905)

1003816100381610038161003816) 119889119905]

119901

1119901

le inf119896gt0

1

1198961+

[int119905notin119860119899

Φ(119905 1198961205760

8max119890

10038161003816100381610038161199090 (119905) + 119890119911119899 (119905)1003816100381610038161003816) 119889119905]

119901

1119901

le1205760

8

100381710038171003817100381710038171003817max119890

10038161003816100381610038161199090 + 1198901199111198991003816100381610038161003816

100381710038171003817100381710038171003817Φ119901

le1205760

8

1003817100381710038171003817Σ11989010038161003816100381610038161199090 + 119890119911119899

10038161003816100381610038161003817100381710038171003817Φ119901

le1205760

210038161003816100381610038161205821198991003816100381610038161003816

lt31205760

4

(25)

hold for 119899 large enoughTherefore 119911(2)

119899Φ119901

gt 12057604 which shows that 120583(119860

119899) gt 0

for 119899 large enough Furthermore for any 119905 isin 119860119899 we have

10038161003816100381610038161199090 (119905)1003816100381610038161003816 le

1 minus 21205750

4Σ119890

10038161003816100381610038161199090 (119905) + 119890119911119899 (119905)1003816100381610038161003816

(26)

To complete the proof we consider the following twocases

(I) One has 119896119899rarr infin(119899 rarr infin) where 119896

119899isin 119870119901((14)

Σ119890|1199090+ 119890119911119899|)

4 Abstract and Applied Analysis

For each 119899 isin N we get

1 =10038171003817100381710038171199090

1003817100381710038171003817Φ119901

le1

119896119899

1 + 119868119901

Φ(1198961198991199090)1119901

=1

119896119899

1 + [int119905isin119860119899

Φ(119905 119896119899

10038161003816100381610038161199090 (119905)1003816100381610038161003816) 119889119905

+int119905notin119860119899

Φ(119905 119896119899

10038161003816100381610038161199090 (119905)1003816100381610038161003816) 119889119905]

119901

1119901

le1

119896119899

1 + [int119905isin119860119899

Φ(119905 119896119899(1 minus 2120575

0

4

timesΣ119890

10038161003816100381610038161199090 (119905) + 119890119911119899 (119905)1003816100381610038161003816 )) 119889119905

+ int119905notin119860119899

Φ(119905 119896119899

times (1

4Σ119890

10038161003816100381610038161199090 (119905) + 119890119911119899 (119905)1003816100381610038161003816)) 119889119905]

119901

1119901

le1

119896119899

1

+ [ (1 minus 21205750)

times int119905isin119860119899

Φ(119905 119896119899(1

4Σ119890

10038161003816100381610038161199090 (119905) + 119890119911119899 (119905)1003816100381610038161003816)) 119889119905

+ int119905notin119860119899

Φ(119905 119896119899

times(1

4Σ119890

10038161003816100381610038161199090 (119905) + 119890119911119899 (119905)1003816100381610038161003816)) 119889119905]

119901

1119901

=1

119896119899

1

+ [119868Φ(119896119899(1

4Σ119890

10038161003816100381610038161199090 + 1198901199111198991003816100381610038161003816)) minus 21205750

times int119905isin119860119899

Φ(119905 119896119899

times(1

4Σ119890

10038161003816100381610038161199090 (119905) + 119890119911119899 (119905)1003816100381610038161003816)) 119889119905]

119901

1119901

(27)

Furthermore we notice that

int119905isin119860119899

Φ(119905 119896119899(1

4Σ119890

10038161003816100381610038161199090 (119905) + 119890119911119899 (119905)1003816100381610038161003816)) 119889119905

ge int119905isin119860119899

Φ(119905 119896119899

1003816100381610038161003816119911119899 (119905)1003816100381610038161003816) 119889119905

ge [119896119901

119899

10038171003817100381710038171003817119911(2)

119899

10038171003817100381710038171003817

119901

Φ119901minus 1]1119901

ge [(1205760

4119896119899)

119901

minus 1]

1119901

(28)

Since 119896119899rarr infinwhen 119899 rarr infin we can see that the inequality

((12057604)119896119899)119901minus 1 gt 0 holds for 119899 large enough Moreover we

find that

119868Φ(119896119899(1

4Σ119890

10038161003816100381610038161199090 + 1198901199111198991003816100381610038161003816))

ge 119868Φ(1198961198991199090)

ge (119896119901

119899minus 1)1119901

ge 21205750((1205760

4119896119899)

119901

minus 1)

1119901

(29)

Then we deduce

1 =10038171003817100381710038171199090

1003817100381710038171003817Φ119901

le1

119896119899

1 + [119868Φ(119896119899(1

4Σ119890

10038161003816100381610038161199090 + 1198901199111198991003816100381610038161003816)) minus 2120575

0

times int119905isin119860119899

Φ(119905 119896119899

times(1

4Σ119890

10038161003816100381610038161199090 (119905) + 119890119911119899 (119905)1003816100381610038161003816)) 119889119905]

119901

1119901

le1

119896119899

1 + [119868Φ(119896119899(1

4Σ119890

10038161003816100381610038161199090 + 1198901199111198991003816100381610038161003816))

minus21205750((1205760

4119896119899)

119901

minus 1)

1119901

]

119901

1119901

lt1

119896119899

1 + 119868119901

Φ(119896119899(1

4Σ119890

10038161003816100381610038161199090 + 1198901199111198991003816100381610038161003816))

1119901

= 1

(30)

a contradiction(II) One has 119896

119899rarr 119896

0(119899 rarr infin) where 119896

119899isin

119870119901((14)Σ

119890|1199090+ 119890119911119899|) 1198960isin R

Then for each 119899 isin N

1

120582119899

ge1003817100381710038171003817100381710038171003817

1

4Σ119890

10038161003816100381610038161199090 + 1198901199111198991003816100381610038161003816

1003817100381710038171003817100381710038171003817Φ119901

=1

119896119899

1 + 119868119901

Φ(119896119899

4Σ119890

10038161003816100381610038161199090 + 1198901199111198991003816100381610038161003816)

1119901

ge1

119896119899

1 + 119868119901

Φ(1198961198991199090)1119901

ge10038171003817100381710038171199090

1003817100381710038171003817Φ119901 = 1

(31)

Let 119899 rarr infin we deduce that 1 = (11198960)1 + 119868

119901

Φ(11989601199090)1119901

=

1199090Φ119901

Abstract and Applied Analysis 5

From (I) we obtain that

1 = 1199090Φ119901

le1

119896119899

1+ [119868Φ(119896119899(1

4Σ119890

10038161003816100381610038161199090 + 1198901199111198991003816100381610038161003816)) minus 21205750

times int119905isin119860119899

Φ(119905 119896119899

times (1

4Σ119890

10038161003816100381610038161199090 (119905) + 119890119911119899 (119905)1003816100381610038161003816)) 119889119905]

119901

1119901

(32)

Now we consider the following two subcases(a) Consider int

119905isin119860119899

Φ(119905 119896119899((14)Σ

119890|1199090(119905) + 119890119911

119899(119905)|))119889119905 gt 0

for some 119899 isin NThen we obtain

1 = 1199090Φ119901

le1

119896119899

1 + [119868Φ(119896119899(1

4Σ119890

10038161003816100381610038161199090 + 1198901199111198991003816100381610038161003816)) minus 2120575

0

times int119905isin119860119899

Φ(119905 119896119899

times(1

4Σ119890

10038161003816100381610038161199090 (119905) + 119890119911119899 (119905)1003816100381610038161003816)) 119889119905]

119901

1119901

lt1

119896119899

1 + 119868119901

Φ(119896119899(1

4Σ119890

10038161003816100381610038161199090 + 1198901199111198991003816100381610038161003816))

1119901

= 1

(33)

a contradiction(b) Consider int

119905isin119860119899

Φ(119905 119896119899((14)Σ

119890|1199090(119905) + 119890119911

119899(119905)|))119889119905 = 0

for any 119899 isin NFor 119899 large enough we observe that

119896119899gt1 minus 2120575

0

1 minus 1205750

1198960 (34)

Hence we get

0 = int119905isin119860119899

Φ(119905 119896119899(1

4Σ119890

10038161003816100381610038161199090 (119905) + 119890119911119899 (119905)1003816100381610038161003816)) 119889119905

ge int119905isin119860119899

Φ(119905

10038161003816100381610038161003816100381610038161003816

119896119899

1 minus 21205750

1199090(119905)

10038161003816100381610038161003816100381610038161003816) 119889119905

ge int119905isin119860119899

Φ(119905

10038161003816100381610038161003816100381610038161003816

1198960

1 minus 1205750

1199090(119905)

10038161003816100381610038161003816100381610038161003816) 119889119905 ge 0

(35)

Thus we see the equalityint119905isin119860119899

Φ(119905 |(1198960(1minus120575

0))1199090(119905)|) 119889119905 = 0

holds for 119899 large enough It follows that10038161003816100381610038161003816100381610038161003816

1198960

1 minus 1205750

1199090(119905)

10038161003816100381610038161003816100381610038161003816le 119890 (119905) 120583-ae 119905 isin 119860

119899 (36)

since 120583(119860119899) gt 0 for 119899 large enough

On the other hand by (3) we deduce that

100381610038161003816100381611989601199090 (119905)1003816100381610038161003816 ge 119890 (119905) 120583-ae 119905 isin 119860

119899 (37)

Hence we get a contradiction

10038161003816100381610038161003816100381610038161003816

1198960

1 minus 1205750

1199090(119905)

10038161003816100381610038161003816100381610038161003816ge

119890 (119905)

1 minus 1205750

gt 119890 (119905) 120583-ae 119905 isin 119860119899 (38)

Theorem 4 Assume 1 le 119901 lt infin then the following assertionsare equivalent

(1) 119871Φ119901

is complex midpoint locally uniformly convex

(2) 119871Φ119901

is complex strictly convex

(3) 119890(119905) = 0 for 120583-ae 119905 isin 119879

Proof The implication (1) rArr (2) is trivial Now assume that119871Φ119901

is complex strictly convex If 120583119905 isin 119879 119890(119905) gt 0 gt 0 let1198790= 119905 isin 119879 119890(119905) gt 0 and it is not difficult to find an element

119909 isin 119878(119871Φ119901) satisfying supp 119909 = 119879 119879

0 Take 119896 isin 119870

119901(119909) and

define

119910 (119905) =

119890 (119905)

2119896for 119905 isin 119879

0

119909 (119905) for 119905 isin 119879 1198790

(39)

Obviously 119910Φ119901

ge 119909Φ119901

= 1 On the other hand

10038171003817100381710038171199101003817100381710038171003817Φ119901 le

1

1198961 + [int

119905isin1198790

Φ(119905119890 (119905)

2) 119889119905

+int1198791198790

Φ (119905 119896119909 (119905)) 119889119905]

119901

1119901

=1

119896(1 + 119868

119901

Φ(119896119909))1119901

= 1

(40)

Thus 119910Φ119901

= (1119896)(1 + 119868119901

Φ(119896119910))1119901

= 1 However for 119905 isin1198790 we find 119896|119910(119905)| = 119890(119905)2 lt 119890(119905) which implies 119910 isin 119878(119871

Φ119901)

is not a complex extreme point of 119861(119871Φ119901) fromTheorem 3

(3) rArr (1) Suppose that 119909 isin 119878(119871Φ119901) is not a complex

strongly extreme point of 119861(119871Φ119901) It follows fromTheorem 3

that 120583119905 isin 119879 1198960|119909(119905)| lt 119890(119905) gt 0 for some 119896

0isin 119870119901(119909)

consequently 120583119905 isin 119879 119890(119905) gt 0 gt 0which is a contradiction

Remark 5 If 119901 = infin then 119901-Amemiya norm equals Luxem-burg norm the problem of complex convexity of Musielak-Orlicz function spaces equipped with the Luxemburg normhas been investigated in [8]

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

6 Abstract and Applied Analysis

Acknowledgments

This work is supported by Grants from HeilongjiangProvincial Natural Science Foundation for Youths (noQC2013C001) Natural Science Foundation of HeilongjiangEducational Committee (no 12531099) Youth Science Fundof Harbin University of Science and Technology (no2011YF002) and Tianyuan Funds of the National NaturalScience Foundation of China (no 11226127)

References

[1] O Blasco and M Pavlovic ldquoComplex convexity and vector-valued Littlewood-Paley inequalitiesrdquo Bulletin of the LondonMathematical Society vol 35 no 6 pp 749ndash758 2003

[2] C Choi A Kaminska and H J Lee ldquoComplex convexity ofOrlicz-Lorentz spaces and its applicationsrdquo Bulletin of the PolishAcademy of SciencesMathematics vol 52 no 1 pp 19ndash38 2004

[3] H Hudzik and A Narloch ldquoRelationships betweenmonotonic-ity and complex rotundity properties with some consequencesrdquoMathematica Scandinavica vol 96 no 2 pp 289ndash306 2005

[4] H J Lee ldquoMonotonicity and complex convexity in Banachlatticesrdquo Journal of Mathematical Analysis and Applications vol307 no 1 pp 86ndash101 2005

[5] H J Lee ldquoComplex convexity and monotonicity in Quasi-Banach latticesrdquo Israel Journal of Mathematics vol 159 no 1 pp57ndash91 2007

[6] M M Czerwinska and A Kaminska ldquoComplex rotunditiesand midpoint local uniform rotundity in symmetric spaces ofmeasurable operatorsrdquo Studia Mathematica vol 201 no 3 pp253ndash285 2010

[7] E Thorp and R Whitley ldquoThe strong maximum modulustheorem for analytic functions into a Banach spacerdquo Proceedingsof the AmericanMathematical Society vol 18 pp 640ndash646 1967

[8] L Chen Y Cui and H Hudzik ldquoCriteria for complex stronglyextreme points of Musielak-Orlicz function spacesrdquo NonlinearAnalysis Theory Methods amp Applications vol 70 no 6 pp2270ndash2276 2009

[9] S Chen Geometry of Orlicz Spaces Dissertationes Mathemati-cae 1996

[10] Y Cui L Duan H Hudzik and M Wisła ldquoBasic theory of p-Amemiya norm in Orlicz spaces (1 le 119901 le infin) extreme pointsand rotundity in Orlicz spaces endowed with these normsrdquoNonlinear Analysis Theory Methods amp Applications vol 69 no5-6 pp 1796ndash1816 2008

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 Abstract and Applied Analysis

We define supp 119909 = 119905 isin 119879 |119909(119905)| = 0 and the Musielak-Orlicz space 119871

Φgenerated by the formula

119871Φ= 119909 isin 119871

119888(120583) 119868

Φ(120582119909) lt infin for some 120582 gt 0 (3)

Set

119890 (119905) = sup 119906 ge 0 Φ (119905 119906) = 0

119864 (119905) = sup 119906 ge 0 Φ (119905 119906) lt infin (4)

then 119890(119905) and 119864(119905) are 120583-measurable (see [9])The notion of 119901-Amemiya norm has been introduced in

[10] for any 1 le 119901 le infin and 119906 gt 0 define

119904119901(119906) =

(1 + 119906119901)1119901

for 1 le 119901 lt infin

max 1 119906 for 119901 = infin(5)

Furthermore define 119904Φ119901(119909) = 119904

119901∘119868Φ(119909) for all 1 le 119901 le infin

and 119909 isin 119871119888(120583) Notice that the function 119904119901is increasing on 119877

+

for 1 le 119901 lt infin however the function 119904infinis increasing on the

interval [1infin) onlyFor 119909 isin 119871

119888(120583) define the 119901-Amemiya norm by the

formula

119909Φ119901 = inf119896gt0

1

119896119904Φ119901(119896119909) 1 le 119901 le infin (6)

For convenience from now on we write 119871Φ119901

=

(119871Φ sdot Φ119901) it is easy to see that 119871

Φ119901is a Banach space For

1 le 119901 lt infin 119909 isin 119871Φ119901

set

119870119901(119909) = 119896 gt 0 119909Φ119901 =

1

1198961 + 119868

119901

Φ(119896119909)1119901

(7)

2 Main Results

We begin this section from the following useful lemmas

Lemma 1 (see [9]) For any 120576 gt 0 there exists 120575 isin (0 12)such that if 119906 V isin C and

|V| ge120576

8max119890|119906 + 119890V| (8)

then

|119906| le1 minus 2120575

4Σ119890 |119906 + 119890V| (9)

where

max119890|119906 + 119890V| = max |119906 + V| |119906 minus V| |119906 + 119894V| |119906 minus 119894V|

Σ119890 |119906 + 119890V| = |119906 + V| + |119906 minus V| + |119906 + 119894V| + |119906 minus 119894V|

(10)

Lemma 2 If lim119906rarrinfin

(Φ(119905 119906)119906) = infin for 120583-ae 119905 isin 119879 then119870119901(119909) = 0 for any 119909 isin 119871

Φ119901 0 where 1 le 119901 lt infin

Proof For any 119909 isin 119871Φ119901

0 there exists 120572 gt 0 such that120583119905 isin 119879 |119909(119905)| ge 120572 gt 0 Let 119879

1= 119905 isin 119879 |119909(119905)| ge 120572 and

define the subsets

119865119899= 119905 isin 119879

1Φ (119905 119906)

119906ge4119909Φ119901

120572 sdot 1205831198791

119906 ge 119899 (11)

for each 119899 isin N It is easy to see that 1198651sube F2sube sdot sdot sdot sube 119865

119899sube sdot sdot sdot

and lim119899rarrinfin

120583119865119899= 1205831198791 Then there exists 119899

0isin N such that

1205831198651198990gt (12)120583119879

1

It follows from the definition of 119901-Amemiya norm thatthere is a sequence 119896

119899 satisfying

119909Φ119901 = lim119899rarrinfin

1

119896119899

1 + 119868119901

Φ(119896119899119909)1119901

(12)

For any 119896 gt 0 we have

1

1198961 + 119868

119901

Φ(119896119909)1119901

=1

1198961 + [int

119879

Φ (119905 119896 |119909 (119905)|) 119889119905]

119901

1119901

ge

int1198651198990

Φ (119905 119896 |119909 (119905)|) 119889119905

119896

ge int1198651198990

Φ (119905 119896120572)

119896119889119905

= 120572int1198651198990

Φ (119905 119896120572)

119896120572119889119905

(13)

If 119896 gt 1198990120572 notice that

1

1198961 + 119868

119901

Φ(119896119909)1119901

ge 120572 sdot4119909Φ119901

120572 sdot 1205831198791

sdot 1205831198651198990gt 2119909Φ119901 (14)

which means the sequence 119896119899 is bounded Hence without

loss of generality we assume that 119896119899rarr 1198960as 119899 rarr infin

We can also choose the monotonic increasing or decreasingsubsequence of 119896

119899 that converges to the number 119896

0 Apply-

ing Levi Theorem and Lebesgue Dominated ConvergenceTheorem we obtain

119909Φ119901 = lim119899rarrinfin

1

119896119899

1 + 119868119901

Φ(119896119899119909)1119901

=1

1198960

1 + 119868119901

Φ(1198960119909)1119901

(15)

which implies 1198960isin 119870119901(119909)

In order to exclude the case when such sets 119870119901(119909) are

empty for 119909 isin 119871Φ1199010 in the sequel we will assume without

any special mention that lim119906rarrinfin

(Φ(119905 119906)119906) = infin for 120583-ae119905 isin 119879

Theorem 3 Assume 1 le 119901 lt infin 119909 isin 119878(119871Φ119901) Then the

following assertions are equivalent

(1) 119909 is a complex strongly extreme point of the unit ball119861(119871Φ119901)

(2) 119909 is a complex extreme point of the unit ball 119861(119871Φ119901)

(3) for any 119896 isin 119870119901(119909) 120583119905 isin 119879 119896|119909(119905)| lt 119890(119905) = 0

Proof The implication (1) rArr (2) is trivial Let 119909 be a complexextreme point of the unit ball 119861(119871

Φ119901) and there exists 119896

0isin

119870119901(119909) such that 120583119905 isin 119879 119896

0|119909(119905)| lt 119890(119905) gt 0 Then we can

Abstract and Applied Analysis 3

find 119889 gt 0 such that 120583119905 isin 119879 1198960|119909(119905)| + 119889 lt 119890(119905) gt 0

Indeed if 120583119905 isin 119879 1198960|119909(119905)| + 119889 lt 119890(119905) = 0 for any 119889 gt 0

then we set 119889119899= 1119899 and define the subsets

119879119899= 119905 isin 119879 119896

0 |119909 (119905)| + 119889119899 lt 119890 (119905) (16)

for each 119899 isin N Notice that 1198791sube 1198792sube sdot sdot sdot sube 119879

119899sube sdot sdot sdot and

119905 isin 119879 1198960|119909(119905)| le 119890(119905) = ⋃

infin

119899=1119879119899 we can find that

120583 119905 isin 119879 1198960 |119909 (119905)| le 119890 (119905) = 0 (17)

which is a contradictionLet 1198790= 119905 isin 119879 119896

0|119909(119905)| + 119889 lt 119890(119905) and define 119910 =

(1198891198960)1205941198790 we have 119910 = 0 and for any 120582 isin C with |120582| le 1

1003817100381710038171003817119909 + 1205821199101003817100381710038171003817Φ119901 le

1

1198960

1 + 119868119901

Φ(1198960(119909 + 120582119910))

1119901

=1

1198960

1 + [119868Φ(11989601199091205941198791198790

)

+119868Φ(11989601199091205941198790+ 120582119889120594

1198790)]119901

1119901

le1

1198960

1 + [119868Φ(11989601199091205941198791198790

)

+119868Φ((1198960 |119909| + 119889) 1205941198790

)]119901

1119901

=1

1198960

1 + [119868Φ(11989601199091205941198791198790

)]119901

1119901

=1

1198960

(1 + 119868119901

Φ(1198960119909))1119901

= 1

(18)

which shows that 119909 is not a complex extreme point of the unitball 119861(119871

Φ119901)

(3) rArr (1) Suppose that 119909 isin 119878(119871Φ119901) is not a complex

strongly extreme point of the unit ball 119861(119871Φ119901) then there

exists 1205760gt 0 such thatΔ

119888(119909 1205760) = 0That is there exist120582

119899isin C

with |120582119899| rarr 1 and119910

119899isin 119871Φ119901

satisfying 119910119899Φ119901

ge 1205760such that

10038171003817100381710038171205821198991199090 plusmn 1199101198991003817100381710038171003817Φ119901 le 1

10038171003817100381710038171205821198991199090 plusmn 1198941199101198991003817100381710038171003817Φ119901 le 1 (19)

which gives100381710038171003817100381710038171003817100381710038171199090plusmn119910119899

120582119899

10038171003817100381710038171003817100381710038171003817Φ119901

le110038161003816100381610038161205821198991003816100381610038161003816

100381710038171003817100381710038171003817100381710038171199090plusmn 119894119910119899

120582119899

10038171003817100381710038171003817100381710038171003817Φ119901

le110038161003816100381610038161205821198991003816100381610038161003816

(20)

Setting 119911119899= 119910119899120582119899 we have10038171003817100381710038171199111198991003817100381710038171003817Φ119901 ge

10038171003817100381710038171199101198991003817100381710038171003817Φ119901 ge 1205760

10038171003817100381710038171199090 plusmn 1199111198991003817100381710038171003817Φ119901 le

110038161003816100381610038161205821198991003816100381610038161003816

10038171003817100381710038171199090 plusmn 1198941199111198991003817100381710038171003817Φ119901 le

110038161003816100381610038161205821198991003816100381610038161003816

(21)

For the above 1205760gt 0 by Lemma 1 there exists 120575

0isin

(0 12) such that if 119906 V isin C and

|V| ge1205760

8max119890|119906 + 119890V| (22)

then

|119906| le1 minus 2120575

0

4Σ119890 |119906 + 119890V| (23)

For each 119899 isin N let

119860119899= 119905 isin 119879

1003816100381610038161003816119911119899 (119905)1003816100381610038161003816 ge

1205760

8max119890

10038161003816100381610038161199090 (119905) + 119890119911119899 (119905)1003816100381610038161003816

119911(1)

119899 119911(1)

119899(119905) = 119911

119899(119905) (119905 notin 119860

119899) 119911

(1)

119899(119905) = 0 (119905 isin 119860

119899)

119911(2)

119899 119911(2)

119899(119905) = 0 (119905 notin 119860

119899) 119911

(2)

119899(119905) = 119911

119899(119905) (119905 isin 119860

119899)

(24)

It is easy to see that 119911119899= 119911(1)

119899+119911(2)

119899forall119899 isin N Since |120582

119899| rarr 1

when 119899 rarr infin the following inequalities

10038171003817100381710038171003817119911(1)

119899

10038171003817100381710038171003817Φ119901

= inf119896gt0

1

1198961 + [int

119905notin119860119899

Φ(119905 1198961003816100381610038161003816119911119899 (119905)

1003816100381610038161003816) 119889119905]

119901

1119901

le inf119896gt0

1

1198961+

[int119905notin119860119899

Φ(119905 1198961205760

8max119890

10038161003816100381610038161199090 (119905) + 119890119911119899 (119905)1003816100381610038161003816) 119889119905]

119901

1119901

le1205760

8

100381710038171003817100381710038171003817max119890

10038161003816100381610038161199090 + 1198901199111198991003816100381610038161003816

100381710038171003817100381710038171003817Φ119901

le1205760

8

1003817100381710038171003817Σ11989010038161003816100381610038161199090 + 119890119911119899

10038161003816100381610038161003817100381710038171003817Φ119901

le1205760

210038161003816100381610038161205821198991003816100381610038161003816

lt31205760

4

(25)

hold for 119899 large enoughTherefore 119911(2)

119899Φ119901

gt 12057604 which shows that 120583(119860

119899) gt 0

for 119899 large enough Furthermore for any 119905 isin 119860119899 we have

10038161003816100381610038161199090 (119905)1003816100381610038161003816 le

1 minus 21205750

4Σ119890

10038161003816100381610038161199090 (119905) + 119890119911119899 (119905)1003816100381610038161003816

(26)

To complete the proof we consider the following twocases

(I) One has 119896119899rarr infin(119899 rarr infin) where 119896

119899isin 119870119901((14)

Σ119890|1199090+ 119890119911119899|)

4 Abstract and Applied Analysis

For each 119899 isin N we get

1 =10038171003817100381710038171199090

1003817100381710038171003817Φ119901

le1

119896119899

1 + 119868119901

Φ(1198961198991199090)1119901

=1

119896119899

1 + [int119905isin119860119899

Φ(119905 119896119899

10038161003816100381610038161199090 (119905)1003816100381610038161003816) 119889119905

+int119905notin119860119899

Φ(119905 119896119899

10038161003816100381610038161199090 (119905)1003816100381610038161003816) 119889119905]

119901

1119901

le1

119896119899

1 + [int119905isin119860119899

Φ(119905 119896119899(1 minus 2120575

0

4

timesΣ119890

10038161003816100381610038161199090 (119905) + 119890119911119899 (119905)1003816100381610038161003816 )) 119889119905

+ int119905notin119860119899

Φ(119905 119896119899

times (1

4Σ119890

10038161003816100381610038161199090 (119905) + 119890119911119899 (119905)1003816100381610038161003816)) 119889119905]

119901

1119901

le1

119896119899

1

+ [ (1 minus 21205750)

times int119905isin119860119899

Φ(119905 119896119899(1

4Σ119890

10038161003816100381610038161199090 (119905) + 119890119911119899 (119905)1003816100381610038161003816)) 119889119905

+ int119905notin119860119899

Φ(119905 119896119899

times(1

4Σ119890

10038161003816100381610038161199090 (119905) + 119890119911119899 (119905)1003816100381610038161003816)) 119889119905]

119901

1119901

=1

119896119899

1

+ [119868Φ(119896119899(1

4Σ119890

10038161003816100381610038161199090 + 1198901199111198991003816100381610038161003816)) minus 21205750

times int119905isin119860119899

Φ(119905 119896119899

times(1

4Σ119890

10038161003816100381610038161199090 (119905) + 119890119911119899 (119905)1003816100381610038161003816)) 119889119905]

119901

1119901

(27)

Furthermore we notice that

int119905isin119860119899

Φ(119905 119896119899(1

4Σ119890

10038161003816100381610038161199090 (119905) + 119890119911119899 (119905)1003816100381610038161003816)) 119889119905

ge int119905isin119860119899

Φ(119905 119896119899

1003816100381610038161003816119911119899 (119905)1003816100381610038161003816) 119889119905

ge [119896119901

119899

10038171003817100381710038171003817119911(2)

119899

10038171003817100381710038171003817

119901

Φ119901minus 1]1119901

ge [(1205760

4119896119899)

119901

minus 1]

1119901

(28)

Since 119896119899rarr infinwhen 119899 rarr infin we can see that the inequality

((12057604)119896119899)119901minus 1 gt 0 holds for 119899 large enough Moreover we

find that

119868Φ(119896119899(1

4Σ119890

10038161003816100381610038161199090 + 1198901199111198991003816100381610038161003816))

ge 119868Φ(1198961198991199090)

ge (119896119901

119899minus 1)1119901

ge 21205750((1205760

4119896119899)

119901

minus 1)

1119901

(29)

Then we deduce

1 =10038171003817100381710038171199090

1003817100381710038171003817Φ119901

le1

119896119899

1 + [119868Φ(119896119899(1

4Σ119890

10038161003816100381610038161199090 + 1198901199111198991003816100381610038161003816)) minus 2120575

0

times int119905isin119860119899

Φ(119905 119896119899

times(1

4Σ119890

10038161003816100381610038161199090 (119905) + 119890119911119899 (119905)1003816100381610038161003816)) 119889119905]

119901

1119901

le1

119896119899

1 + [119868Φ(119896119899(1

4Σ119890

10038161003816100381610038161199090 + 1198901199111198991003816100381610038161003816))

minus21205750((1205760

4119896119899)

119901

minus 1)

1119901

]

119901

1119901

lt1

119896119899

1 + 119868119901

Φ(119896119899(1

4Σ119890

10038161003816100381610038161199090 + 1198901199111198991003816100381610038161003816))

1119901

= 1

(30)

a contradiction(II) One has 119896

119899rarr 119896

0(119899 rarr infin) where 119896

119899isin

119870119901((14)Σ

119890|1199090+ 119890119911119899|) 1198960isin R

Then for each 119899 isin N

1

120582119899

ge1003817100381710038171003817100381710038171003817

1

4Σ119890

10038161003816100381610038161199090 + 1198901199111198991003816100381610038161003816

1003817100381710038171003817100381710038171003817Φ119901

=1

119896119899

1 + 119868119901

Φ(119896119899

4Σ119890

10038161003816100381610038161199090 + 1198901199111198991003816100381610038161003816)

1119901

ge1

119896119899

1 + 119868119901

Φ(1198961198991199090)1119901

ge10038171003817100381710038171199090

1003817100381710038171003817Φ119901 = 1

(31)

Let 119899 rarr infin we deduce that 1 = (11198960)1 + 119868

119901

Φ(11989601199090)1119901

=

1199090Φ119901

Abstract and Applied Analysis 5

From (I) we obtain that

1 = 1199090Φ119901

le1

119896119899

1+ [119868Φ(119896119899(1

4Σ119890

10038161003816100381610038161199090 + 1198901199111198991003816100381610038161003816)) minus 21205750

times int119905isin119860119899

Φ(119905 119896119899

times (1

4Σ119890

10038161003816100381610038161199090 (119905) + 119890119911119899 (119905)1003816100381610038161003816)) 119889119905]

119901

1119901

(32)

Now we consider the following two subcases(a) Consider int

119905isin119860119899

Φ(119905 119896119899((14)Σ

119890|1199090(119905) + 119890119911

119899(119905)|))119889119905 gt 0

for some 119899 isin NThen we obtain

1 = 1199090Φ119901

le1

119896119899

1 + [119868Φ(119896119899(1

4Σ119890

10038161003816100381610038161199090 + 1198901199111198991003816100381610038161003816)) minus 2120575

0

times int119905isin119860119899

Φ(119905 119896119899

times(1

4Σ119890

10038161003816100381610038161199090 (119905) + 119890119911119899 (119905)1003816100381610038161003816)) 119889119905]

119901

1119901

lt1

119896119899

1 + 119868119901

Φ(119896119899(1

4Σ119890

10038161003816100381610038161199090 + 1198901199111198991003816100381610038161003816))

1119901

= 1

(33)

a contradiction(b) Consider int

119905isin119860119899

Φ(119905 119896119899((14)Σ

119890|1199090(119905) + 119890119911

119899(119905)|))119889119905 = 0

for any 119899 isin NFor 119899 large enough we observe that

119896119899gt1 minus 2120575

0

1 minus 1205750

1198960 (34)

Hence we get

0 = int119905isin119860119899

Φ(119905 119896119899(1

4Σ119890

10038161003816100381610038161199090 (119905) + 119890119911119899 (119905)1003816100381610038161003816)) 119889119905

ge int119905isin119860119899

Φ(119905

10038161003816100381610038161003816100381610038161003816

119896119899

1 minus 21205750

1199090(119905)

10038161003816100381610038161003816100381610038161003816) 119889119905

ge int119905isin119860119899

Φ(119905

10038161003816100381610038161003816100381610038161003816

1198960

1 minus 1205750

1199090(119905)

10038161003816100381610038161003816100381610038161003816) 119889119905 ge 0

(35)

Thus we see the equalityint119905isin119860119899

Φ(119905 |(1198960(1minus120575

0))1199090(119905)|) 119889119905 = 0

holds for 119899 large enough It follows that10038161003816100381610038161003816100381610038161003816

1198960

1 minus 1205750

1199090(119905)

10038161003816100381610038161003816100381610038161003816le 119890 (119905) 120583-ae 119905 isin 119860

119899 (36)

since 120583(119860119899) gt 0 for 119899 large enough

On the other hand by (3) we deduce that

100381610038161003816100381611989601199090 (119905)1003816100381610038161003816 ge 119890 (119905) 120583-ae 119905 isin 119860

119899 (37)

Hence we get a contradiction

10038161003816100381610038161003816100381610038161003816

1198960

1 minus 1205750

1199090(119905)

10038161003816100381610038161003816100381610038161003816ge

119890 (119905)

1 minus 1205750

gt 119890 (119905) 120583-ae 119905 isin 119860119899 (38)

Theorem 4 Assume 1 le 119901 lt infin then the following assertionsare equivalent

(1) 119871Φ119901

is complex midpoint locally uniformly convex

(2) 119871Φ119901

is complex strictly convex

(3) 119890(119905) = 0 for 120583-ae 119905 isin 119879

Proof The implication (1) rArr (2) is trivial Now assume that119871Φ119901

is complex strictly convex If 120583119905 isin 119879 119890(119905) gt 0 gt 0 let1198790= 119905 isin 119879 119890(119905) gt 0 and it is not difficult to find an element

119909 isin 119878(119871Φ119901) satisfying supp 119909 = 119879 119879

0 Take 119896 isin 119870

119901(119909) and

define

119910 (119905) =

119890 (119905)

2119896for 119905 isin 119879

0

119909 (119905) for 119905 isin 119879 1198790

(39)

Obviously 119910Φ119901

ge 119909Φ119901

= 1 On the other hand

10038171003817100381710038171199101003817100381710038171003817Φ119901 le

1

1198961 + [int

119905isin1198790

Φ(119905119890 (119905)

2) 119889119905

+int1198791198790

Φ (119905 119896119909 (119905)) 119889119905]

119901

1119901

=1

119896(1 + 119868

119901

Φ(119896119909))1119901

= 1

(40)

Thus 119910Φ119901

= (1119896)(1 + 119868119901

Φ(119896119910))1119901

= 1 However for 119905 isin1198790 we find 119896|119910(119905)| = 119890(119905)2 lt 119890(119905) which implies 119910 isin 119878(119871

Φ119901)

is not a complex extreme point of 119861(119871Φ119901) fromTheorem 3

(3) rArr (1) Suppose that 119909 isin 119878(119871Φ119901) is not a complex

strongly extreme point of 119861(119871Φ119901) It follows fromTheorem 3

that 120583119905 isin 119879 1198960|119909(119905)| lt 119890(119905) gt 0 for some 119896

0isin 119870119901(119909)

consequently 120583119905 isin 119879 119890(119905) gt 0 gt 0which is a contradiction

Remark 5 If 119901 = infin then 119901-Amemiya norm equals Luxem-burg norm the problem of complex convexity of Musielak-Orlicz function spaces equipped with the Luxemburg normhas been investigated in [8]

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

6 Abstract and Applied Analysis

Acknowledgments

This work is supported by Grants from HeilongjiangProvincial Natural Science Foundation for Youths (noQC2013C001) Natural Science Foundation of HeilongjiangEducational Committee (no 12531099) Youth Science Fundof Harbin University of Science and Technology (no2011YF002) and Tianyuan Funds of the National NaturalScience Foundation of China (no 11226127)

References

[1] O Blasco and M Pavlovic ldquoComplex convexity and vector-valued Littlewood-Paley inequalitiesrdquo Bulletin of the LondonMathematical Society vol 35 no 6 pp 749ndash758 2003

[2] C Choi A Kaminska and H J Lee ldquoComplex convexity ofOrlicz-Lorentz spaces and its applicationsrdquo Bulletin of the PolishAcademy of SciencesMathematics vol 52 no 1 pp 19ndash38 2004

[3] H Hudzik and A Narloch ldquoRelationships betweenmonotonic-ity and complex rotundity properties with some consequencesrdquoMathematica Scandinavica vol 96 no 2 pp 289ndash306 2005

[4] H J Lee ldquoMonotonicity and complex convexity in Banachlatticesrdquo Journal of Mathematical Analysis and Applications vol307 no 1 pp 86ndash101 2005

[5] H J Lee ldquoComplex convexity and monotonicity in Quasi-Banach latticesrdquo Israel Journal of Mathematics vol 159 no 1 pp57ndash91 2007

[6] M M Czerwinska and A Kaminska ldquoComplex rotunditiesand midpoint local uniform rotundity in symmetric spaces ofmeasurable operatorsrdquo Studia Mathematica vol 201 no 3 pp253ndash285 2010

[7] E Thorp and R Whitley ldquoThe strong maximum modulustheorem for analytic functions into a Banach spacerdquo Proceedingsof the AmericanMathematical Society vol 18 pp 640ndash646 1967

[8] L Chen Y Cui and H Hudzik ldquoCriteria for complex stronglyextreme points of Musielak-Orlicz function spacesrdquo NonlinearAnalysis Theory Methods amp Applications vol 70 no 6 pp2270ndash2276 2009

[9] S Chen Geometry of Orlicz Spaces Dissertationes Mathemati-cae 1996

[10] Y Cui L Duan H Hudzik and M Wisła ldquoBasic theory of p-Amemiya norm in Orlicz spaces (1 le 119901 le infin) extreme pointsand rotundity in Orlicz spaces endowed with these normsrdquoNonlinear Analysis Theory Methods amp Applications vol 69 no5-6 pp 1796ndash1816 2008

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Abstract and Applied Analysis 3

find 119889 gt 0 such that 120583119905 isin 119879 1198960|119909(119905)| + 119889 lt 119890(119905) gt 0

Indeed if 120583119905 isin 119879 1198960|119909(119905)| + 119889 lt 119890(119905) = 0 for any 119889 gt 0

then we set 119889119899= 1119899 and define the subsets

119879119899= 119905 isin 119879 119896

0 |119909 (119905)| + 119889119899 lt 119890 (119905) (16)

for each 119899 isin N Notice that 1198791sube 1198792sube sdot sdot sdot sube 119879

119899sube sdot sdot sdot and

119905 isin 119879 1198960|119909(119905)| le 119890(119905) = ⋃

infin

119899=1119879119899 we can find that

120583 119905 isin 119879 1198960 |119909 (119905)| le 119890 (119905) = 0 (17)

which is a contradictionLet 1198790= 119905 isin 119879 119896

0|119909(119905)| + 119889 lt 119890(119905) and define 119910 =

(1198891198960)1205941198790 we have 119910 = 0 and for any 120582 isin C with |120582| le 1

1003817100381710038171003817119909 + 1205821199101003817100381710038171003817Φ119901 le

1

1198960

1 + 119868119901

Φ(1198960(119909 + 120582119910))

1119901

=1

1198960

1 + [119868Φ(11989601199091205941198791198790

)

+119868Φ(11989601199091205941198790+ 120582119889120594

1198790)]119901

1119901

le1

1198960

1 + [119868Φ(11989601199091205941198791198790

)

+119868Φ((1198960 |119909| + 119889) 1205941198790

)]119901

1119901

=1

1198960

1 + [119868Φ(11989601199091205941198791198790

)]119901

1119901

=1

1198960

(1 + 119868119901

Φ(1198960119909))1119901

= 1

(18)

which shows that 119909 is not a complex extreme point of the unitball 119861(119871

Φ119901)

(3) rArr (1) Suppose that 119909 isin 119878(119871Φ119901) is not a complex

strongly extreme point of the unit ball 119861(119871Φ119901) then there

exists 1205760gt 0 such thatΔ

119888(119909 1205760) = 0That is there exist120582

119899isin C

with |120582119899| rarr 1 and119910

119899isin 119871Φ119901

satisfying 119910119899Φ119901

ge 1205760such that

10038171003817100381710038171205821198991199090 plusmn 1199101198991003817100381710038171003817Φ119901 le 1

10038171003817100381710038171205821198991199090 plusmn 1198941199101198991003817100381710038171003817Φ119901 le 1 (19)

which gives100381710038171003817100381710038171003817100381710038171199090plusmn119910119899

120582119899

10038171003817100381710038171003817100381710038171003817Φ119901

le110038161003816100381610038161205821198991003816100381610038161003816

100381710038171003817100381710038171003817100381710038171199090plusmn 119894119910119899

120582119899

10038171003817100381710038171003817100381710038171003817Φ119901

le110038161003816100381610038161205821198991003816100381610038161003816

(20)

Setting 119911119899= 119910119899120582119899 we have10038171003817100381710038171199111198991003817100381710038171003817Φ119901 ge

10038171003817100381710038171199101198991003817100381710038171003817Φ119901 ge 1205760

10038171003817100381710038171199090 plusmn 1199111198991003817100381710038171003817Φ119901 le

110038161003816100381610038161205821198991003816100381610038161003816

10038171003817100381710038171199090 plusmn 1198941199111198991003817100381710038171003817Φ119901 le

110038161003816100381610038161205821198991003816100381610038161003816

(21)

For the above 1205760gt 0 by Lemma 1 there exists 120575

0isin

(0 12) such that if 119906 V isin C and

|V| ge1205760

8max119890|119906 + 119890V| (22)

then

|119906| le1 minus 2120575

0

4Σ119890 |119906 + 119890V| (23)

For each 119899 isin N let

119860119899= 119905 isin 119879

1003816100381610038161003816119911119899 (119905)1003816100381610038161003816 ge

1205760

8max119890

10038161003816100381610038161199090 (119905) + 119890119911119899 (119905)1003816100381610038161003816

119911(1)

119899 119911(1)

119899(119905) = 119911

119899(119905) (119905 notin 119860

119899) 119911

(1)

119899(119905) = 0 (119905 isin 119860

119899)

119911(2)

119899 119911(2)

119899(119905) = 0 (119905 notin 119860

119899) 119911

(2)

119899(119905) = 119911

119899(119905) (119905 isin 119860

119899)

(24)

It is easy to see that 119911119899= 119911(1)

119899+119911(2)

119899forall119899 isin N Since |120582

119899| rarr 1

when 119899 rarr infin the following inequalities

10038171003817100381710038171003817119911(1)

119899

10038171003817100381710038171003817Φ119901

= inf119896gt0

1

1198961 + [int

119905notin119860119899

Φ(119905 1198961003816100381610038161003816119911119899 (119905)

1003816100381610038161003816) 119889119905]

119901

1119901

le inf119896gt0

1

1198961+

[int119905notin119860119899

Φ(119905 1198961205760

8max119890

10038161003816100381610038161199090 (119905) + 119890119911119899 (119905)1003816100381610038161003816) 119889119905]

119901

1119901

le1205760

8

100381710038171003817100381710038171003817max119890

10038161003816100381610038161199090 + 1198901199111198991003816100381610038161003816

100381710038171003817100381710038171003817Φ119901

le1205760

8

1003817100381710038171003817Σ11989010038161003816100381610038161199090 + 119890119911119899

10038161003816100381610038161003817100381710038171003817Φ119901

le1205760

210038161003816100381610038161205821198991003816100381610038161003816

lt31205760

4

(25)

hold for 119899 large enoughTherefore 119911(2)

119899Φ119901

gt 12057604 which shows that 120583(119860

119899) gt 0

for 119899 large enough Furthermore for any 119905 isin 119860119899 we have

10038161003816100381610038161199090 (119905)1003816100381610038161003816 le

1 minus 21205750

4Σ119890

10038161003816100381610038161199090 (119905) + 119890119911119899 (119905)1003816100381610038161003816

(26)

To complete the proof we consider the following twocases

(I) One has 119896119899rarr infin(119899 rarr infin) where 119896

119899isin 119870119901((14)

Σ119890|1199090+ 119890119911119899|)

4 Abstract and Applied Analysis

For each 119899 isin N we get

1 =10038171003817100381710038171199090

1003817100381710038171003817Φ119901

le1

119896119899

1 + 119868119901

Φ(1198961198991199090)1119901

=1

119896119899

1 + [int119905isin119860119899

Φ(119905 119896119899

10038161003816100381610038161199090 (119905)1003816100381610038161003816) 119889119905

+int119905notin119860119899

Φ(119905 119896119899

10038161003816100381610038161199090 (119905)1003816100381610038161003816) 119889119905]

119901

1119901

le1

119896119899

1 + [int119905isin119860119899

Φ(119905 119896119899(1 minus 2120575

0

4

timesΣ119890

10038161003816100381610038161199090 (119905) + 119890119911119899 (119905)1003816100381610038161003816 )) 119889119905

+ int119905notin119860119899

Φ(119905 119896119899

times (1

4Σ119890

10038161003816100381610038161199090 (119905) + 119890119911119899 (119905)1003816100381610038161003816)) 119889119905]

119901

1119901

le1

119896119899

1

+ [ (1 minus 21205750)

times int119905isin119860119899

Φ(119905 119896119899(1

4Σ119890

10038161003816100381610038161199090 (119905) + 119890119911119899 (119905)1003816100381610038161003816)) 119889119905

+ int119905notin119860119899

Φ(119905 119896119899

times(1

4Σ119890

10038161003816100381610038161199090 (119905) + 119890119911119899 (119905)1003816100381610038161003816)) 119889119905]

119901

1119901

=1

119896119899

1

+ [119868Φ(119896119899(1

4Σ119890

10038161003816100381610038161199090 + 1198901199111198991003816100381610038161003816)) minus 21205750

times int119905isin119860119899

Φ(119905 119896119899

times(1

4Σ119890

10038161003816100381610038161199090 (119905) + 119890119911119899 (119905)1003816100381610038161003816)) 119889119905]

119901

1119901

(27)

Furthermore we notice that

int119905isin119860119899

Φ(119905 119896119899(1

4Σ119890

10038161003816100381610038161199090 (119905) + 119890119911119899 (119905)1003816100381610038161003816)) 119889119905

ge int119905isin119860119899

Φ(119905 119896119899

1003816100381610038161003816119911119899 (119905)1003816100381610038161003816) 119889119905

ge [119896119901

119899

10038171003817100381710038171003817119911(2)

119899

10038171003817100381710038171003817

119901

Φ119901minus 1]1119901

ge [(1205760

4119896119899)

119901

minus 1]

1119901

(28)

Since 119896119899rarr infinwhen 119899 rarr infin we can see that the inequality

((12057604)119896119899)119901minus 1 gt 0 holds for 119899 large enough Moreover we

find that

119868Φ(119896119899(1

4Σ119890

10038161003816100381610038161199090 + 1198901199111198991003816100381610038161003816))

ge 119868Φ(1198961198991199090)

ge (119896119901

119899minus 1)1119901

ge 21205750((1205760

4119896119899)

119901

minus 1)

1119901

(29)

Then we deduce

1 =10038171003817100381710038171199090

1003817100381710038171003817Φ119901

le1

119896119899

1 + [119868Φ(119896119899(1

4Σ119890

10038161003816100381610038161199090 + 1198901199111198991003816100381610038161003816)) minus 2120575

0

times int119905isin119860119899

Φ(119905 119896119899

times(1

4Σ119890

10038161003816100381610038161199090 (119905) + 119890119911119899 (119905)1003816100381610038161003816)) 119889119905]

119901

1119901

le1

119896119899

1 + [119868Φ(119896119899(1

4Σ119890

10038161003816100381610038161199090 + 1198901199111198991003816100381610038161003816))

minus21205750((1205760

4119896119899)

119901

minus 1)

1119901

]

119901

1119901

lt1

119896119899

1 + 119868119901

Φ(119896119899(1

4Σ119890

10038161003816100381610038161199090 + 1198901199111198991003816100381610038161003816))

1119901

= 1

(30)

a contradiction(II) One has 119896

119899rarr 119896

0(119899 rarr infin) where 119896

119899isin

119870119901((14)Σ

119890|1199090+ 119890119911119899|) 1198960isin R

Then for each 119899 isin N

1

120582119899

ge1003817100381710038171003817100381710038171003817

1

4Σ119890

10038161003816100381610038161199090 + 1198901199111198991003816100381610038161003816

1003817100381710038171003817100381710038171003817Φ119901

=1

119896119899

1 + 119868119901

Φ(119896119899

4Σ119890

10038161003816100381610038161199090 + 1198901199111198991003816100381610038161003816)

1119901

ge1

119896119899

1 + 119868119901

Φ(1198961198991199090)1119901

ge10038171003817100381710038171199090

1003817100381710038171003817Φ119901 = 1

(31)

Let 119899 rarr infin we deduce that 1 = (11198960)1 + 119868

119901

Φ(11989601199090)1119901

=

1199090Φ119901

Abstract and Applied Analysis 5

From (I) we obtain that

1 = 1199090Φ119901

le1

119896119899

1+ [119868Φ(119896119899(1

4Σ119890

10038161003816100381610038161199090 + 1198901199111198991003816100381610038161003816)) minus 21205750

times int119905isin119860119899

Φ(119905 119896119899

times (1

4Σ119890

10038161003816100381610038161199090 (119905) + 119890119911119899 (119905)1003816100381610038161003816)) 119889119905]

119901

1119901

(32)

Now we consider the following two subcases(a) Consider int

119905isin119860119899

Φ(119905 119896119899((14)Σ

119890|1199090(119905) + 119890119911

119899(119905)|))119889119905 gt 0

for some 119899 isin NThen we obtain

1 = 1199090Φ119901

le1

119896119899

1 + [119868Φ(119896119899(1

4Σ119890

10038161003816100381610038161199090 + 1198901199111198991003816100381610038161003816)) minus 2120575

0

times int119905isin119860119899

Φ(119905 119896119899

times(1

4Σ119890

10038161003816100381610038161199090 (119905) + 119890119911119899 (119905)1003816100381610038161003816)) 119889119905]

119901

1119901

lt1

119896119899

1 + 119868119901

Φ(119896119899(1

4Σ119890

10038161003816100381610038161199090 + 1198901199111198991003816100381610038161003816))

1119901

= 1

(33)

a contradiction(b) Consider int

119905isin119860119899

Φ(119905 119896119899((14)Σ

119890|1199090(119905) + 119890119911

119899(119905)|))119889119905 = 0

for any 119899 isin NFor 119899 large enough we observe that

119896119899gt1 minus 2120575

0

1 minus 1205750

1198960 (34)

Hence we get

0 = int119905isin119860119899

Φ(119905 119896119899(1

4Σ119890

10038161003816100381610038161199090 (119905) + 119890119911119899 (119905)1003816100381610038161003816)) 119889119905

ge int119905isin119860119899

Φ(119905

10038161003816100381610038161003816100381610038161003816

119896119899

1 minus 21205750

1199090(119905)

10038161003816100381610038161003816100381610038161003816) 119889119905

ge int119905isin119860119899

Φ(119905

10038161003816100381610038161003816100381610038161003816

1198960

1 minus 1205750

1199090(119905)

10038161003816100381610038161003816100381610038161003816) 119889119905 ge 0

(35)

Thus we see the equalityint119905isin119860119899

Φ(119905 |(1198960(1minus120575

0))1199090(119905)|) 119889119905 = 0

holds for 119899 large enough It follows that10038161003816100381610038161003816100381610038161003816

1198960

1 minus 1205750

1199090(119905)

10038161003816100381610038161003816100381610038161003816le 119890 (119905) 120583-ae 119905 isin 119860

119899 (36)

since 120583(119860119899) gt 0 for 119899 large enough

On the other hand by (3) we deduce that

100381610038161003816100381611989601199090 (119905)1003816100381610038161003816 ge 119890 (119905) 120583-ae 119905 isin 119860

119899 (37)

Hence we get a contradiction

10038161003816100381610038161003816100381610038161003816

1198960

1 minus 1205750

1199090(119905)

10038161003816100381610038161003816100381610038161003816ge

119890 (119905)

1 minus 1205750

gt 119890 (119905) 120583-ae 119905 isin 119860119899 (38)

Theorem 4 Assume 1 le 119901 lt infin then the following assertionsare equivalent

(1) 119871Φ119901

is complex midpoint locally uniformly convex

(2) 119871Φ119901

is complex strictly convex

(3) 119890(119905) = 0 for 120583-ae 119905 isin 119879

Proof The implication (1) rArr (2) is trivial Now assume that119871Φ119901

is complex strictly convex If 120583119905 isin 119879 119890(119905) gt 0 gt 0 let1198790= 119905 isin 119879 119890(119905) gt 0 and it is not difficult to find an element

119909 isin 119878(119871Φ119901) satisfying supp 119909 = 119879 119879

0 Take 119896 isin 119870

119901(119909) and

define

119910 (119905) =

119890 (119905)

2119896for 119905 isin 119879

0

119909 (119905) for 119905 isin 119879 1198790

(39)

Obviously 119910Φ119901

ge 119909Φ119901

= 1 On the other hand

10038171003817100381710038171199101003817100381710038171003817Φ119901 le

1

1198961 + [int

119905isin1198790

Φ(119905119890 (119905)

2) 119889119905

+int1198791198790

Φ (119905 119896119909 (119905)) 119889119905]

119901

1119901

=1

119896(1 + 119868

119901

Φ(119896119909))1119901

= 1

(40)

Thus 119910Φ119901

= (1119896)(1 + 119868119901

Φ(119896119910))1119901

= 1 However for 119905 isin1198790 we find 119896|119910(119905)| = 119890(119905)2 lt 119890(119905) which implies 119910 isin 119878(119871

Φ119901)

is not a complex extreme point of 119861(119871Φ119901) fromTheorem 3

(3) rArr (1) Suppose that 119909 isin 119878(119871Φ119901) is not a complex

strongly extreme point of 119861(119871Φ119901) It follows fromTheorem 3

that 120583119905 isin 119879 1198960|119909(119905)| lt 119890(119905) gt 0 for some 119896

0isin 119870119901(119909)

consequently 120583119905 isin 119879 119890(119905) gt 0 gt 0which is a contradiction

Remark 5 If 119901 = infin then 119901-Amemiya norm equals Luxem-burg norm the problem of complex convexity of Musielak-Orlicz function spaces equipped with the Luxemburg normhas been investigated in [8]

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

6 Abstract and Applied Analysis

Acknowledgments

This work is supported by Grants from HeilongjiangProvincial Natural Science Foundation for Youths (noQC2013C001) Natural Science Foundation of HeilongjiangEducational Committee (no 12531099) Youth Science Fundof Harbin University of Science and Technology (no2011YF002) and Tianyuan Funds of the National NaturalScience Foundation of China (no 11226127)

References

[1] O Blasco and M Pavlovic ldquoComplex convexity and vector-valued Littlewood-Paley inequalitiesrdquo Bulletin of the LondonMathematical Society vol 35 no 6 pp 749ndash758 2003

[2] C Choi A Kaminska and H J Lee ldquoComplex convexity ofOrlicz-Lorentz spaces and its applicationsrdquo Bulletin of the PolishAcademy of SciencesMathematics vol 52 no 1 pp 19ndash38 2004

[3] H Hudzik and A Narloch ldquoRelationships betweenmonotonic-ity and complex rotundity properties with some consequencesrdquoMathematica Scandinavica vol 96 no 2 pp 289ndash306 2005

[4] H J Lee ldquoMonotonicity and complex convexity in Banachlatticesrdquo Journal of Mathematical Analysis and Applications vol307 no 1 pp 86ndash101 2005

[5] H J Lee ldquoComplex convexity and monotonicity in Quasi-Banach latticesrdquo Israel Journal of Mathematics vol 159 no 1 pp57ndash91 2007

[6] M M Czerwinska and A Kaminska ldquoComplex rotunditiesand midpoint local uniform rotundity in symmetric spaces ofmeasurable operatorsrdquo Studia Mathematica vol 201 no 3 pp253ndash285 2010

[7] E Thorp and R Whitley ldquoThe strong maximum modulustheorem for analytic functions into a Banach spacerdquo Proceedingsof the AmericanMathematical Society vol 18 pp 640ndash646 1967

[8] L Chen Y Cui and H Hudzik ldquoCriteria for complex stronglyextreme points of Musielak-Orlicz function spacesrdquo NonlinearAnalysis Theory Methods amp Applications vol 70 no 6 pp2270ndash2276 2009

[9] S Chen Geometry of Orlicz Spaces Dissertationes Mathemati-cae 1996

[10] Y Cui L Duan H Hudzik and M Wisła ldquoBasic theory of p-Amemiya norm in Orlicz spaces (1 le 119901 le infin) extreme pointsand rotundity in Orlicz spaces endowed with these normsrdquoNonlinear Analysis Theory Methods amp Applications vol 69 no5-6 pp 1796ndash1816 2008

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 Abstract and Applied Analysis

For each 119899 isin N we get

1 =10038171003817100381710038171199090

1003817100381710038171003817Φ119901

le1

119896119899

1 + 119868119901

Φ(1198961198991199090)1119901

=1

119896119899

1 + [int119905isin119860119899

Φ(119905 119896119899

10038161003816100381610038161199090 (119905)1003816100381610038161003816) 119889119905

+int119905notin119860119899

Φ(119905 119896119899

10038161003816100381610038161199090 (119905)1003816100381610038161003816) 119889119905]

119901

1119901

le1

119896119899

1 + [int119905isin119860119899

Φ(119905 119896119899(1 minus 2120575

0

4

timesΣ119890

10038161003816100381610038161199090 (119905) + 119890119911119899 (119905)1003816100381610038161003816 )) 119889119905

+ int119905notin119860119899

Φ(119905 119896119899

times (1

4Σ119890

10038161003816100381610038161199090 (119905) + 119890119911119899 (119905)1003816100381610038161003816)) 119889119905]

119901

1119901

le1

119896119899

1

+ [ (1 minus 21205750)

times int119905isin119860119899

Φ(119905 119896119899(1

4Σ119890

10038161003816100381610038161199090 (119905) + 119890119911119899 (119905)1003816100381610038161003816)) 119889119905

+ int119905notin119860119899

Φ(119905 119896119899

times(1

4Σ119890

10038161003816100381610038161199090 (119905) + 119890119911119899 (119905)1003816100381610038161003816)) 119889119905]

119901

1119901

=1

119896119899

1

+ [119868Φ(119896119899(1

4Σ119890

10038161003816100381610038161199090 + 1198901199111198991003816100381610038161003816)) minus 21205750

times int119905isin119860119899

Φ(119905 119896119899

times(1

4Σ119890

10038161003816100381610038161199090 (119905) + 119890119911119899 (119905)1003816100381610038161003816)) 119889119905]

119901

1119901

(27)

Furthermore we notice that

int119905isin119860119899

Φ(119905 119896119899(1

4Σ119890

10038161003816100381610038161199090 (119905) + 119890119911119899 (119905)1003816100381610038161003816)) 119889119905

ge int119905isin119860119899

Φ(119905 119896119899

1003816100381610038161003816119911119899 (119905)1003816100381610038161003816) 119889119905

ge [119896119901

119899

10038171003817100381710038171003817119911(2)

119899

10038171003817100381710038171003817

119901

Φ119901minus 1]1119901

ge [(1205760

4119896119899)

119901

minus 1]

1119901

(28)

Since 119896119899rarr infinwhen 119899 rarr infin we can see that the inequality

((12057604)119896119899)119901minus 1 gt 0 holds for 119899 large enough Moreover we

find that

119868Φ(119896119899(1

4Σ119890

10038161003816100381610038161199090 + 1198901199111198991003816100381610038161003816))

ge 119868Φ(1198961198991199090)

ge (119896119901

119899minus 1)1119901

ge 21205750((1205760

4119896119899)

119901

minus 1)

1119901

(29)

Then we deduce

1 =10038171003817100381710038171199090

1003817100381710038171003817Φ119901

le1

119896119899

1 + [119868Φ(119896119899(1

4Σ119890

10038161003816100381610038161199090 + 1198901199111198991003816100381610038161003816)) minus 2120575

0

times int119905isin119860119899

Φ(119905 119896119899

times(1

4Σ119890

10038161003816100381610038161199090 (119905) + 119890119911119899 (119905)1003816100381610038161003816)) 119889119905]

119901

1119901

le1

119896119899

1 + [119868Φ(119896119899(1

4Σ119890

10038161003816100381610038161199090 + 1198901199111198991003816100381610038161003816))

minus21205750((1205760

4119896119899)

119901

minus 1)

1119901

]

119901

1119901

lt1

119896119899

1 + 119868119901

Φ(119896119899(1

4Σ119890

10038161003816100381610038161199090 + 1198901199111198991003816100381610038161003816))

1119901

= 1

(30)

a contradiction(II) One has 119896

119899rarr 119896

0(119899 rarr infin) where 119896

119899isin

119870119901((14)Σ

119890|1199090+ 119890119911119899|) 1198960isin R

Then for each 119899 isin N

1

120582119899

ge1003817100381710038171003817100381710038171003817

1

4Σ119890

10038161003816100381610038161199090 + 1198901199111198991003816100381610038161003816

1003817100381710038171003817100381710038171003817Φ119901

=1

119896119899

1 + 119868119901

Φ(119896119899

4Σ119890

10038161003816100381610038161199090 + 1198901199111198991003816100381610038161003816)

1119901

ge1

119896119899

1 + 119868119901

Φ(1198961198991199090)1119901

ge10038171003817100381710038171199090

1003817100381710038171003817Φ119901 = 1

(31)

Let 119899 rarr infin we deduce that 1 = (11198960)1 + 119868

119901

Φ(11989601199090)1119901

=

1199090Φ119901

Abstract and Applied Analysis 5

From (I) we obtain that

1 = 1199090Φ119901

le1

119896119899

1+ [119868Φ(119896119899(1

4Σ119890

10038161003816100381610038161199090 + 1198901199111198991003816100381610038161003816)) minus 21205750

times int119905isin119860119899

Φ(119905 119896119899

times (1

4Σ119890

10038161003816100381610038161199090 (119905) + 119890119911119899 (119905)1003816100381610038161003816)) 119889119905]

119901

1119901

(32)

Now we consider the following two subcases(a) Consider int

119905isin119860119899

Φ(119905 119896119899((14)Σ

119890|1199090(119905) + 119890119911

119899(119905)|))119889119905 gt 0

for some 119899 isin NThen we obtain

1 = 1199090Φ119901

le1

119896119899

1 + [119868Φ(119896119899(1

4Σ119890

10038161003816100381610038161199090 + 1198901199111198991003816100381610038161003816)) minus 2120575

0

times int119905isin119860119899

Φ(119905 119896119899

times(1

4Σ119890

10038161003816100381610038161199090 (119905) + 119890119911119899 (119905)1003816100381610038161003816)) 119889119905]

119901

1119901

lt1

119896119899

1 + 119868119901

Φ(119896119899(1

4Σ119890

10038161003816100381610038161199090 + 1198901199111198991003816100381610038161003816))

1119901

= 1

(33)

a contradiction(b) Consider int

119905isin119860119899

Φ(119905 119896119899((14)Σ

119890|1199090(119905) + 119890119911

119899(119905)|))119889119905 = 0

for any 119899 isin NFor 119899 large enough we observe that

119896119899gt1 minus 2120575

0

1 minus 1205750

1198960 (34)

Hence we get

0 = int119905isin119860119899

Φ(119905 119896119899(1

4Σ119890

10038161003816100381610038161199090 (119905) + 119890119911119899 (119905)1003816100381610038161003816)) 119889119905

ge int119905isin119860119899

Φ(119905

10038161003816100381610038161003816100381610038161003816

119896119899

1 minus 21205750

1199090(119905)

10038161003816100381610038161003816100381610038161003816) 119889119905

ge int119905isin119860119899

Φ(119905

10038161003816100381610038161003816100381610038161003816

1198960

1 minus 1205750

1199090(119905)

10038161003816100381610038161003816100381610038161003816) 119889119905 ge 0

(35)

Thus we see the equalityint119905isin119860119899

Φ(119905 |(1198960(1minus120575

0))1199090(119905)|) 119889119905 = 0

holds for 119899 large enough It follows that10038161003816100381610038161003816100381610038161003816

1198960

1 minus 1205750

1199090(119905)

10038161003816100381610038161003816100381610038161003816le 119890 (119905) 120583-ae 119905 isin 119860

119899 (36)

since 120583(119860119899) gt 0 for 119899 large enough

On the other hand by (3) we deduce that

100381610038161003816100381611989601199090 (119905)1003816100381610038161003816 ge 119890 (119905) 120583-ae 119905 isin 119860

119899 (37)

Hence we get a contradiction

10038161003816100381610038161003816100381610038161003816

1198960

1 minus 1205750

1199090(119905)

10038161003816100381610038161003816100381610038161003816ge

119890 (119905)

1 minus 1205750

gt 119890 (119905) 120583-ae 119905 isin 119860119899 (38)

Theorem 4 Assume 1 le 119901 lt infin then the following assertionsare equivalent

(1) 119871Φ119901

is complex midpoint locally uniformly convex

(2) 119871Φ119901

is complex strictly convex

(3) 119890(119905) = 0 for 120583-ae 119905 isin 119879

Proof The implication (1) rArr (2) is trivial Now assume that119871Φ119901

is complex strictly convex If 120583119905 isin 119879 119890(119905) gt 0 gt 0 let1198790= 119905 isin 119879 119890(119905) gt 0 and it is not difficult to find an element

119909 isin 119878(119871Φ119901) satisfying supp 119909 = 119879 119879

0 Take 119896 isin 119870

119901(119909) and

define

119910 (119905) =

119890 (119905)

2119896for 119905 isin 119879

0

119909 (119905) for 119905 isin 119879 1198790

(39)

Obviously 119910Φ119901

ge 119909Φ119901

= 1 On the other hand

10038171003817100381710038171199101003817100381710038171003817Φ119901 le

1

1198961 + [int

119905isin1198790

Φ(119905119890 (119905)

2) 119889119905

+int1198791198790

Φ (119905 119896119909 (119905)) 119889119905]

119901

1119901

=1

119896(1 + 119868

119901

Φ(119896119909))1119901

= 1

(40)

Thus 119910Φ119901

= (1119896)(1 + 119868119901

Φ(119896119910))1119901

= 1 However for 119905 isin1198790 we find 119896|119910(119905)| = 119890(119905)2 lt 119890(119905) which implies 119910 isin 119878(119871

Φ119901)

is not a complex extreme point of 119861(119871Φ119901) fromTheorem 3

(3) rArr (1) Suppose that 119909 isin 119878(119871Φ119901) is not a complex

strongly extreme point of 119861(119871Φ119901) It follows fromTheorem 3

that 120583119905 isin 119879 1198960|119909(119905)| lt 119890(119905) gt 0 for some 119896

0isin 119870119901(119909)

consequently 120583119905 isin 119879 119890(119905) gt 0 gt 0which is a contradiction

Remark 5 If 119901 = infin then 119901-Amemiya norm equals Luxem-burg norm the problem of complex convexity of Musielak-Orlicz function spaces equipped with the Luxemburg normhas been investigated in [8]

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

6 Abstract and Applied Analysis

Acknowledgments

This work is supported by Grants from HeilongjiangProvincial Natural Science Foundation for Youths (noQC2013C001) Natural Science Foundation of HeilongjiangEducational Committee (no 12531099) Youth Science Fundof Harbin University of Science and Technology (no2011YF002) and Tianyuan Funds of the National NaturalScience Foundation of China (no 11226127)

References

[1] O Blasco and M Pavlovic ldquoComplex convexity and vector-valued Littlewood-Paley inequalitiesrdquo Bulletin of the LondonMathematical Society vol 35 no 6 pp 749ndash758 2003

[2] C Choi A Kaminska and H J Lee ldquoComplex convexity ofOrlicz-Lorentz spaces and its applicationsrdquo Bulletin of the PolishAcademy of SciencesMathematics vol 52 no 1 pp 19ndash38 2004

[3] H Hudzik and A Narloch ldquoRelationships betweenmonotonic-ity and complex rotundity properties with some consequencesrdquoMathematica Scandinavica vol 96 no 2 pp 289ndash306 2005

[4] H J Lee ldquoMonotonicity and complex convexity in Banachlatticesrdquo Journal of Mathematical Analysis and Applications vol307 no 1 pp 86ndash101 2005

[5] H J Lee ldquoComplex convexity and monotonicity in Quasi-Banach latticesrdquo Israel Journal of Mathematics vol 159 no 1 pp57ndash91 2007

[6] M M Czerwinska and A Kaminska ldquoComplex rotunditiesand midpoint local uniform rotundity in symmetric spaces ofmeasurable operatorsrdquo Studia Mathematica vol 201 no 3 pp253ndash285 2010

[7] E Thorp and R Whitley ldquoThe strong maximum modulustheorem for analytic functions into a Banach spacerdquo Proceedingsof the AmericanMathematical Society vol 18 pp 640ndash646 1967

[8] L Chen Y Cui and H Hudzik ldquoCriteria for complex stronglyextreme points of Musielak-Orlicz function spacesrdquo NonlinearAnalysis Theory Methods amp Applications vol 70 no 6 pp2270ndash2276 2009

[9] S Chen Geometry of Orlicz Spaces Dissertationes Mathemati-cae 1996

[10] Y Cui L Duan H Hudzik and M Wisła ldquoBasic theory of p-Amemiya norm in Orlicz spaces (1 le 119901 le infin) extreme pointsand rotundity in Orlicz spaces endowed with these normsrdquoNonlinear Analysis Theory Methods amp Applications vol 69 no5-6 pp 1796ndash1816 2008

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Abstract and Applied Analysis 5

From (I) we obtain that

1 = 1199090Φ119901

le1

119896119899

1+ [119868Φ(119896119899(1

4Σ119890

10038161003816100381610038161199090 + 1198901199111198991003816100381610038161003816)) minus 21205750

times int119905isin119860119899

Φ(119905 119896119899

times (1

4Σ119890

10038161003816100381610038161199090 (119905) + 119890119911119899 (119905)1003816100381610038161003816)) 119889119905]

119901

1119901

(32)

Now we consider the following two subcases(a) Consider int

119905isin119860119899

Φ(119905 119896119899((14)Σ

119890|1199090(119905) + 119890119911

119899(119905)|))119889119905 gt 0

for some 119899 isin NThen we obtain

1 = 1199090Φ119901

le1

119896119899

1 + [119868Φ(119896119899(1

4Σ119890

10038161003816100381610038161199090 + 1198901199111198991003816100381610038161003816)) minus 2120575

0

times int119905isin119860119899

Φ(119905 119896119899

times(1

4Σ119890

10038161003816100381610038161199090 (119905) + 119890119911119899 (119905)1003816100381610038161003816)) 119889119905]

119901

1119901

lt1

119896119899

1 + 119868119901

Φ(119896119899(1

4Σ119890

10038161003816100381610038161199090 + 1198901199111198991003816100381610038161003816))

1119901

= 1

(33)

a contradiction(b) Consider int

119905isin119860119899

Φ(119905 119896119899((14)Σ

119890|1199090(119905) + 119890119911

119899(119905)|))119889119905 = 0

for any 119899 isin NFor 119899 large enough we observe that

119896119899gt1 minus 2120575

0

1 minus 1205750

1198960 (34)

Hence we get

0 = int119905isin119860119899

Φ(119905 119896119899(1

4Σ119890

10038161003816100381610038161199090 (119905) + 119890119911119899 (119905)1003816100381610038161003816)) 119889119905

ge int119905isin119860119899

Φ(119905

10038161003816100381610038161003816100381610038161003816

119896119899

1 minus 21205750

1199090(119905)

10038161003816100381610038161003816100381610038161003816) 119889119905

ge int119905isin119860119899

Φ(119905

10038161003816100381610038161003816100381610038161003816

1198960

1 minus 1205750

1199090(119905)

10038161003816100381610038161003816100381610038161003816) 119889119905 ge 0

(35)

Thus we see the equalityint119905isin119860119899

Φ(119905 |(1198960(1minus120575

0))1199090(119905)|) 119889119905 = 0

holds for 119899 large enough It follows that10038161003816100381610038161003816100381610038161003816

1198960

1 minus 1205750

1199090(119905)

10038161003816100381610038161003816100381610038161003816le 119890 (119905) 120583-ae 119905 isin 119860

119899 (36)

since 120583(119860119899) gt 0 for 119899 large enough

On the other hand by (3) we deduce that

100381610038161003816100381611989601199090 (119905)1003816100381610038161003816 ge 119890 (119905) 120583-ae 119905 isin 119860

119899 (37)

Hence we get a contradiction

10038161003816100381610038161003816100381610038161003816

1198960

1 minus 1205750

1199090(119905)

10038161003816100381610038161003816100381610038161003816ge

119890 (119905)

1 minus 1205750

gt 119890 (119905) 120583-ae 119905 isin 119860119899 (38)

Theorem 4 Assume 1 le 119901 lt infin then the following assertionsare equivalent

(1) 119871Φ119901

is complex midpoint locally uniformly convex

(2) 119871Φ119901

is complex strictly convex

(3) 119890(119905) = 0 for 120583-ae 119905 isin 119879

Proof The implication (1) rArr (2) is trivial Now assume that119871Φ119901

is complex strictly convex If 120583119905 isin 119879 119890(119905) gt 0 gt 0 let1198790= 119905 isin 119879 119890(119905) gt 0 and it is not difficult to find an element

119909 isin 119878(119871Φ119901) satisfying supp 119909 = 119879 119879

0 Take 119896 isin 119870

119901(119909) and

define

119910 (119905) =

119890 (119905)

2119896for 119905 isin 119879

0

119909 (119905) for 119905 isin 119879 1198790

(39)

Obviously 119910Φ119901

ge 119909Φ119901

= 1 On the other hand

10038171003817100381710038171199101003817100381710038171003817Φ119901 le

1

1198961 + [int

119905isin1198790

Φ(119905119890 (119905)

2) 119889119905

+int1198791198790

Φ (119905 119896119909 (119905)) 119889119905]

119901

1119901

=1

119896(1 + 119868

119901

Φ(119896119909))1119901

= 1

(40)

Thus 119910Φ119901

= (1119896)(1 + 119868119901

Φ(119896119910))1119901

= 1 However for 119905 isin1198790 we find 119896|119910(119905)| = 119890(119905)2 lt 119890(119905) which implies 119910 isin 119878(119871

Φ119901)

is not a complex extreme point of 119861(119871Φ119901) fromTheorem 3

(3) rArr (1) Suppose that 119909 isin 119878(119871Φ119901) is not a complex

strongly extreme point of 119861(119871Φ119901) It follows fromTheorem 3

that 120583119905 isin 119879 1198960|119909(119905)| lt 119890(119905) gt 0 for some 119896

0isin 119870119901(119909)

consequently 120583119905 isin 119879 119890(119905) gt 0 gt 0which is a contradiction

Remark 5 If 119901 = infin then 119901-Amemiya norm equals Luxem-burg norm the problem of complex convexity of Musielak-Orlicz function spaces equipped with the Luxemburg normhas been investigated in [8]

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

6 Abstract and Applied Analysis

Acknowledgments

This work is supported by Grants from HeilongjiangProvincial Natural Science Foundation for Youths (noQC2013C001) Natural Science Foundation of HeilongjiangEducational Committee (no 12531099) Youth Science Fundof Harbin University of Science and Technology (no2011YF002) and Tianyuan Funds of the National NaturalScience Foundation of China (no 11226127)

References

[1] O Blasco and M Pavlovic ldquoComplex convexity and vector-valued Littlewood-Paley inequalitiesrdquo Bulletin of the LondonMathematical Society vol 35 no 6 pp 749ndash758 2003

[2] C Choi A Kaminska and H J Lee ldquoComplex convexity ofOrlicz-Lorentz spaces and its applicationsrdquo Bulletin of the PolishAcademy of SciencesMathematics vol 52 no 1 pp 19ndash38 2004

[3] H Hudzik and A Narloch ldquoRelationships betweenmonotonic-ity and complex rotundity properties with some consequencesrdquoMathematica Scandinavica vol 96 no 2 pp 289ndash306 2005

[4] H J Lee ldquoMonotonicity and complex convexity in Banachlatticesrdquo Journal of Mathematical Analysis and Applications vol307 no 1 pp 86ndash101 2005

[5] H J Lee ldquoComplex convexity and monotonicity in Quasi-Banach latticesrdquo Israel Journal of Mathematics vol 159 no 1 pp57ndash91 2007

[6] M M Czerwinska and A Kaminska ldquoComplex rotunditiesand midpoint local uniform rotundity in symmetric spaces ofmeasurable operatorsrdquo Studia Mathematica vol 201 no 3 pp253ndash285 2010

[7] E Thorp and R Whitley ldquoThe strong maximum modulustheorem for analytic functions into a Banach spacerdquo Proceedingsof the AmericanMathematical Society vol 18 pp 640ndash646 1967

[8] L Chen Y Cui and H Hudzik ldquoCriteria for complex stronglyextreme points of Musielak-Orlicz function spacesrdquo NonlinearAnalysis Theory Methods amp Applications vol 70 no 6 pp2270ndash2276 2009

[9] S Chen Geometry of Orlicz Spaces Dissertationes Mathemati-cae 1996

[10] Y Cui L Duan H Hudzik and M Wisła ldquoBasic theory of p-Amemiya norm in Orlicz spaces (1 le 119901 le infin) extreme pointsand rotundity in Orlicz spaces endowed with these normsrdquoNonlinear Analysis Theory Methods amp Applications vol 69 no5-6 pp 1796ndash1816 2008

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 Abstract and Applied Analysis

Acknowledgments

This work is supported by Grants from HeilongjiangProvincial Natural Science Foundation for Youths (noQC2013C001) Natural Science Foundation of HeilongjiangEducational Committee (no 12531099) Youth Science Fundof Harbin University of Science and Technology (no2011YF002) and Tianyuan Funds of the National NaturalScience Foundation of China (no 11226127)

References

[1] O Blasco and M Pavlovic ldquoComplex convexity and vector-valued Littlewood-Paley inequalitiesrdquo Bulletin of the LondonMathematical Society vol 35 no 6 pp 749ndash758 2003

[2] C Choi A Kaminska and H J Lee ldquoComplex convexity ofOrlicz-Lorentz spaces and its applicationsrdquo Bulletin of the PolishAcademy of SciencesMathematics vol 52 no 1 pp 19ndash38 2004

[3] H Hudzik and A Narloch ldquoRelationships betweenmonotonic-ity and complex rotundity properties with some consequencesrdquoMathematica Scandinavica vol 96 no 2 pp 289ndash306 2005

[4] H J Lee ldquoMonotonicity and complex convexity in Banachlatticesrdquo Journal of Mathematical Analysis and Applications vol307 no 1 pp 86ndash101 2005

[5] H J Lee ldquoComplex convexity and monotonicity in Quasi-Banach latticesrdquo Israel Journal of Mathematics vol 159 no 1 pp57ndash91 2007

[6] M M Czerwinska and A Kaminska ldquoComplex rotunditiesand midpoint local uniform rotundity in symmetric spaces ofmeasurable operatorsrdquo Studia Mathematica vol 201 no 3 pp253ndash285 2010

[7] E Thorp and R Whitley ldquoThe strong maximum modulustheorem for analytic functions into a Banach spacerdquo Proceedingsof the AmericanMathematical Society vol 18 pp 640ndash646 1967

[8] L Chen Y Cui and H Hudzik ldquoCriteria for complex stronglyextreme points of Musielak-Orlicz function spacesrdquo NonlinearAnalysis Theory Methods amp Applications vol 70 no 6 pp2270ndash2276 2009

[9] S Chen Geometry of Orlicz Spaces Dissertationes Mathemati-cae 1996

[10] Y Cui L Duan H Hudzik and M Wisła ldquoBasic theory of p-Amemiya norm in Orlicz spaces (1 le 119901 le infin) extreme pointsand rotundity in Orlicz spaces endowed with these normsrdquoNonlinear Analysis Theory Methods amp Applications vol 69 no5-6 pp 1796ndash1816 2008

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of