research article growth of logarithmic derivatives and...

Research ArticleGrowth of Logarithmic Derivatives and Their Applications inComplex Differential Equations

Zinelacircabidine Latreuch and Benharrat Beladi

Department of Mathematics Laboratory of Pure and Applied Mathematics University of Mostaganem (UMAB)B P 227 Mostaganem Algeria

Correspondence should be addressed to Benharrat Belaıdi belaidiuniv-mostadz

Received 29 November 2013 Accepted 21 January 2014 Published 4 March 2014

Academic Editor Chuanxi Qian

Copyright copy 2014 Z Latreuch and B Belaıdi This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

We continue the study of the behavior of the growth of logarithmic derivatives In fact we prove some relations between the valuedistribution of solutions of linear differential equations and growth of their logarithmic derivatives We also give an estimate of thegrowth of the quotient of two differential polynomials generated by solutions of the equation 119891

10158401015840

+119860(119911)1198911015840 +119861(119911)119891 = 0where119860(119911)and 119861(119911) are entire functions

1 Introduction and Main Results

Throughout this paper we assume that the reader is familiarwith the fundamental results and the standard notations ofNevanlinnarsquos value distribution theory (see [1ndash4]) In addi-tion we will denote by 120582(119891) (resp 120582(119891)) and 120582(1119891) (resp120582(1119891)) the exponent of convergence of zeros (resp distinctzeros) and poles (resp distinct poles) of a meromorphicfunction 119891 120588(119891) to denote the order of growth of 119891 Ameromorphic function 120593(119911) is called a small function withrespect to 119891(119911) if 119879(119903 120593) = 119900(119879(119903 119891)) as 119903 rarr +infin exceptpossibly a set of 119903 of finite linear measure where 119879(119903 119891) isthe Nevanlinna characteristic function of 119891

Definition 1 (see [4]) Let119891 be ameromorphic functionThenthe hyperorder of 119891(119911) is defined as

1205882(119891) = lim sup

119903rarr+infin

log log119879 (119903 119891)

log 119903 (1)

Definition 2 (see [1 3]) The type of a meromorphic function119891 of order 120588(0 lt 120588 lt infin) is defined as

120591 (119891) = lim sup119903rarr+infin

119879 (119903 119891)

119903120588 (2)

If 119891 is an entire function then the type of 119891 of order 120588(0 lt120588 lt infin) is defined as

120591119872

(119891) = lim sup119903rarr+infin

log119872(119903 119891)

119903120588 (3)

where 119872(119903 119891) = max|119911|=119903

|119891(119911)| is the maximum modulusfunction

Remark 3 For entire function we can have 120591119872(119891) = 120591(119891) For

example if 119891(119911) = 119890119911 then we have 120591119872(119891) = 1 and 120591(119891) =

1120587

Growth of logarithmic derivative of meromorphic func-tions has been generously considered during the last decadesamong others by Golrsquodberg and Grinsteın [5] Benbourenaneand Korhonen [6] Hinkkanen [7] and Heittokangas et al[8] All these considerations have been devoted to gettingmore detailed estimates to the proximity function119898(119903 1198911015840119891)than what is the original Nevanlinna estimate that essentiallycan be written as 119898(119903 1198911015840119891) = 119878(119903 119891) The same appliesfor the case of higher logarithmic derivatives 119898(119903 119891(119896)119891)as well The estimation of logarithmic derivatives plays thekey role in theory of differential equations In his paperGundersen [9] proved some interesting inequalities on themodule of logarithmic derivatives ofmeromorphic functions

Hindawi Publishing CorporationInternational Journal of AnalysisVolume 2014 Article ID 342974 6 pageshttpdxdoiorg1011552014342974

2 International Journal of Analysis

Recently [10] the authors have studied some properties of thebehavior of growth of logarithmic derivatives of entire andmeromorphic functions and have obtained some relationsbetween the zeros of entire functions and the growth oftheir logarithmic derivatives In fact they have proved thefollowing

Theorem A (see [10]) Let 119891 be an entire function with finitenumber of zeros Then for any integer 119896 ge 1

120588(119891(119896)

119891) = 120588

2(119891) (4)

Theorem B (see [10]) Suppose that 119896 ge 2 is an integer and let119891 be a meromorphic function Then

120588(1198911015840

119891) = max120588(

119891(119896)

119891) 119896 ge 2

= max120588(119891(119896)

119891) 120588(

119891(119896+1)

119891) 119896 ge 2

(5)

In [11 pages 457ndash458] Bank and Langley have obtainedthe following result

Theorem C (see [11]) All nontrivial solutions of

11989110158401015840 + 119890119875(119911)119891 = 0 (6)

where 119875(119911) is a nonconstant polynomial satisfy 120582(119891) = infin

Theorem D (see [4] (E Picard)) Any transcendental entirefunction119891must take every finite complex value infinitely manytimes with at most one exception

Example 4 The entire function 119890119911 omits the value zero andevery other finite value is assumed infinitely often and zero isthe exceptional value of Picard

Question 1 Under what conditions can we ensure that 119891 hasno Picardrsquos exceptional value

Question 2 Under what conditions can we obtain the sameresult as Theorem C for higher order linear differentialequations with entire coefficients

The subject of this paper is to give answers to the abovequestions In fact we prove some relations between the valuedistribution of solutions of linear differential equations andgrowth of their logarithmic derivatives Firstly we study therelationship between the growth of logarithmic derivatives ofsolutions of complex differential equation

119891(119896) + 119860119896minus1

119891(119896minus1) + sdot sdot sdot + 1198600119891 = 0 (7)

and their exponents of convergence We will also prove thenonexistence of any Picard exceptional value of solutions of(7) and we obtain the following

Theorem 5 Let 1198600 1198601 119860

119896minus1be entire functions satisfy-

ing 120588(1198600) = 120588 (0 lt 120588 lt infin) 120591

119872(1198600) = 120591 (0 lt 120591 lt infin) and

let 120588(119860119895) le 120588 120591

119872(119860119895) lt 120591 if 120588(119860

119895) = 120588 (119895 = 1 119896 minus 1) If

119891 is a nontrivial solution of (7) such that 120588(1198911015840119891) gt 120588(1198600)

then 119891 assume every finite complex value infinitely oftenFurthermore if 120588(1198911015840119891) = infin then

120582 (119891) = 120582 (119891) = infin (8)

Corollary 6 Under the hypotheses of Theorem 5 let 120593(119911) bean entire function with finite order If 119891 is a nontrivial solutionof (7) such that 120588(119891(119895+1)119891(119895)) = infin for some 119895 isin N then

120582 (119891(119895) minus 120593) = 120582 (119891(119895) minus 120593) = infin (9)

Corollary 7 Under the hypotheses of Theorem 5 every non-trivial solution of (7) does not have any Picard exceptionalvalues

Example 8 It is clear that the function 119891(119911) = sin 119890119911 satisfiesthe differential equation

11989110158401015840 minus 1198911015840 + 1198902119911119891 = 0 (10)

and 120588(1198911015840119891) = 120588(119891) = infin Obviously the conditions ofTheorem 5 are satisfied Then 120582(119891) = 120582(119891) = infin

Secondly we consider the differential equation

11989110158401015840 + 119860 (119911) 1198911015840 + 119861 (119911) 119891 = 0 (11)

where 119860(119911) and 119861(119911) are entire functions and let 119875119899[119891] and

119876119898[119891] be two differential polynomials defined by

119875119899[119891] = 119889

119899119891(119899) + 119889

119899minus1119891(119899minus1) + sdot sdot sdot + 119889

0119891 (119899 ge 1)

119876119898[119891] = 119887

119898119891(119898) + 119887


0119891 (119898 ge 1)

(12)

where 119889119894(119894 = 0 119899) and 119887

119895(119895 = 0 119898) are entire

functions not all vanishing identically In recent years thereis a great interest to investigate the growth and oscillation ofdifferential polynomials generated by solutions of differentialequations in the complex plane (see [12ndash14]) In this partwe give under some conditions an estimate on the growthof the quotient of two differential polynomials generated bysolutions of (11) and we obtain the following results

Theorem 9 Let 119891 be a nontrivial solution of (11) and let119889119894(119894 = 0 119899) and 119887

119895(119895 = 0 119898) be entire functions

not all vanishing identically such that 119892119891

= 119875119899[119891]119876

119898[119891] is

irreducible function in 1198911015840119891 If

max 120588 (119860) 120588 (119861) 120588 (119889119894) (119894 = 0 119899)

120588 (119887119895) (119895 = 0 119898) lt 120588(

1198911015840

119891)

(13)

then

120588 (119892119891) = 120588(

1198911015840

119891) (14)

International Journal of Analysis 3

Remark 10 In Theorem 9 the condition (13) is necessaryFor example 119891(119911) = exp(119890119911) is a solution of the differentialequation

11989110158401015840 minus (119890119911 + 1198902119911) 119891 = 0 (15)

with

120588(1198911015840

119891) = 1 (16)

and let

119892119891=

119889211989110158401015840 + 119889

11198911015840 + 119889

0119891

119887211989110158401015840 + 119887

11198911015840 + 119887

0119891

=1198892(119890119911 + 1198902119911) + 119889

1119890119911 + 119889

0

1198872(119890119911 + 1198902119911) + 119887

1119890119911 + 1198870

(17)

If we take 1198892= 119890119911

2

1198891= 1198890= 1 and 119887

2= 1198871= 1198870= 1 then

120588 (119892119891) = 2 gt 120588(

1198911015840

119891) = 1 (18)

Remark 11 InTheorem 9 the condition that 119892119891is irreducible

function in 1198911015840119891 is necessary If 119889119894= 119887119895(119899 = 119898) then 119892

119891equiv 1

and so

120588 (119892119891) = 0 = 120588(

1198911015840

119891) (19)

By setting 119899 = 119898 = 2 in Theorem 9 we can obtain thefollowing special case

Corollary 12 Let 119860(119911) and 119861(119911) be entire functions of finiteorder such that 120588(119860) lt 120588(119861) and 0 lt 120591(119860) lt 120591(119861) lt infin if120588(119861) = 120588(119860) gt 0 Let 119889

119894(119911)(119894 = 0 1 2) 119887

119895(119911)(119895 = 0 1 2) be

entire functions that are not all vanishing identically such that

max 120588 (119889119894) 120588 (119887

119895) 0 le 119894 119895 le 2 lt 120588 (119861) (20)

and 1198892(119911)b1(119911)minus119889

1(119911)1198872(119911) equiv 0 If119891 equiv 0 is a solution of (11)

such that 120588(119861) lt 120588(1198911015840119891) then

120588(119889211989110158401015840 + 119889

11198911015840 + 119889

0119891

119887211989110158401015840 + 119887

11198911015840 + 119887

0119891

) = 120588(1198911015840

119891) (21)

Corollary 13 Under the hypotheses of Corollary 12 if 119891 is anontrivial solution of (11) such that 120588(1198911015840119891) = infin then

120582 (119891) = 120582 (1198911015840) = infin (22)

2 Auxiliary Lemmas

Lemma 14 (see [1]) Let 119891 be a meromorphic function and let119896 isin N Then

119898(119903119891(119896)

119891) = 119878 (119903 119891) (23)

where 119878(119903 119891) = 119874(log119879(119903 119891) + log 119903) possibly outside a set119864 sub (0infin) with a finite linear measure If 119891 is a finite order ofgrowth then

119898(119903119891(119896)

119891) = 119874 (log 119903) (24)

Lemma 15 (see [15]) Let 1198600(119911) 119860

119896minus1(119911) be entire func-

tions of finite order such that

max 120588 (119860119895) 119895 = 1 119896 minus 1 lt 120588 (119860

0) (25)

Then every solution 119891 equiv 0 of (7) satisfies 120588(119891) = +infin and1205882(119891) = 120588(119860

0)

Lemma 16 (see [16]) Let1198600 1198601 119860

119896minus1be entire functions

satisfying 120588(1198600) = 120588 (0 lt 120588 lt infin) 120591

119872(1198600) = 120591 (0 lt 120591 lt infin)

and let 120588(119860119895) le 120588 120591

119872(119860119895) lt 120591 if 120588(119860

119895) = 120588 (119895 = 1 119896 minus 1)


0)

Here we give a special case of the result given by Xu et alin [17]

Lemma 17 Let 1198600 1198601 119860

119896minus1be entire functions with

finite order and let them satisfy one of the following conditions

(i) max120588(119860119895) 119895 = 1 119896 minus 1 lt 120588(119860

0) lt infin

(ii) 0 lt 120588(119860119896minus1

) = sdot sdot sdot = 120588(1198600) lt infin and max120591

119872(119860119895)

119895 = 1 119896 minus 1 lt 120591119872(1198600) lt infin

Then for every solution 119891 equiv 0 of (7) and for any entirefunction 120593(119911) equiv 0 satisfying 120588(120593) lt infin we have

120582 (119891(119895) minus 120593) = 120582 (119891(119895) minus 120593) = 120588 (119891) = infin (119895 isin N)

1205822(119891(119895) minus 120593) = 120582

2(119891(119895) minus 120593) = 120588 (119860

0) (119895 isin N)

(26)

Lemma 18 (see [10]) Let 119891 be a meromorphic function Ifthere exists an integer 119896 ge 1 such that 120588(119891(119896)119891) = 120588(119891) and120588(119891) gt 120588

2(119891) then

max120582 (119891) 120582 (1

119891) = max120582 (119891) 120582 (

1

119891) = 120588 (119891)

(27)

Furthermore if 119891 is entire function then

120582 (119891) = 120582 (119891) = 120588 (119891) (28)

By the principle of mathematical induction we can obtaineasily the following lemma

Lemma 19 Suppose that 11989110158401015840 = minus119860(119911)1198911015840 minus 119861(119911)119891 where 119860(119911)and 119861(119911) are entire functions Then for an integer 119895 ge 2 thereexist entire functions 120572

119895and 120573

119895such that

119891(119895) = 1205721198951198911015840 + 120573

119895119891 (29)


Lemma 20 (see [18]) Let 119891 be a solution of (7) where thecoefficients119860

119895(119911) (119895 = 0 119896minus1) are analytic functions in the

disc Δ119877= 119911 isin C |119911| lt 119877 0 lt 119877 le infin Let 119899

119888isin 1 119896

be the number of nonzero coefficients 119860119895(119911)(119895 = 0 119896 minus 1)

and let 120579 isin [0 2120587[ and 120576 gt 0 If 119911120579

= ]119890119894120579 isin Δ119877such that

119860119895(119911120579) = 0 for some 119895 = 0 119896 minus 1 then for all ] lt 119903 lt 119877

10038161003816100381610038161003816119891 (119903119890119894120579)10038161003816100381610038161003816 le 119862 exp(119899

119888int119903

]max119895=0119896minus1

10038161003816100381610038161003816119860119895 (119905119890119894120579)

100381610038161003816100381610038161(119896minus119895)

119889119905)

(30)

where 119862 gt 0 is a constant satisfying

119862 le (1 + 120576) max119895=0119896minus1

(

10038161003816100381610038161003816119891(119895) (119911120579)10038161003816100381610038161003816

(119899119888)119895max119899=0119896minus1

1003816100381610038161003816119860119899 (119911120579)1003816100381610038161003816119895(119896minus119899)

)

(31)

Here we give a special case of the result given by Cao etal in [19]

Lemma 21 Let 119891 be a meromorphic function with finite order0 lt 120588(119891) lt infin and type 0 lt 120591(119891) lt infin Then for any given120573 lt 120591(119891) there exists a subset 119868 of [1 +infin) that has infinitelogarithmic measure such that 119879(119903 119891) gt 120573119903120588(119891) holds for all119903 isin 119868

Lemma 22 Let 119860(119911) and 119861(119911) be entire functions such that120588(119861) = 120588 (0 lt 120588 lt infin) 120591(119861) = 120591 (0 lt 120591 lt infin) and let 120588(119860) lt120588(119861) and 120591(119860) lt 120591(119861) if 120588(119860) = 120588(119861) If 119891 equiv 0 is a solution of

11989110158401015840 + 119860 (119911) 1198911015840 + 119861 (119911) 119891 = 0 (32)

then 120588(119891) = infin and 1205882(119891) = 120588(119861)

Proof If 120588(119860) lt 120588(119861) then by Lemma 15 we obtain theresult We prove only the case when 120588(119860) = 120588(119861) = 120588 and120591(119860) lt 120591(119861) Since 119891 equiv 0 then by (32) we have

119861 = minus(11989110158401015840

119891+ 119860

1198911015840

119891) (33)

Suppose that 119891 is of finite order then by (33) and Lemma 14we obtain

119879 (119903 119861) le 119879 (119903 119860) + 119874 (log 119903) (34)

which implies the contradiction

120591 (119861) le 120591 (119860) (35)

Hence 120588(119891) = infin By using inequality (30) for 119877 = infin wehave

1205882(119891) le max 120588 (119860) 120588 (119861) = 120588 (119861) (36)

On the other hand since 120588(119891) = infin then by (33) andLemma 14

119879 (119903 119861) le 119879 (119903 119860) + 119878 (119903 119891) (37)

where 119878(119903 119891) = 119874(log119879(119903 119891) + log 119903) possibly outside a set119864 sub (0 +infin) with a finite linear measure By 120591(119860) lt 120591(119861) wechoose 120572

0 1205721satisfying 120591(119860) lt 120572

1lt 1205720lt 120591(119861) such that for

119903 rarr +infin we have

119879 (119903 119860) le 1205721119903120588 (38)

By Lemma 21 there exists a subset 1198641sub [1 +infin) of infinite

logarithmic measure such that

119879 (119903 119861) gt 1205720119903120588 (39)

By (37)ndash(39) we obtain for all 119903 isin 1198641 119864

(1205720minus 1205721) 119903120588 le 119874 (log119879 (119903 119891)) + 119874 (log 119903) (40)

which implies

120588 (119861) = 120588 le 1205882(119891) (41)

By using (36) and (41) we obtain 1205882(119891) = 120588(119861)

Remark 23 Lemma 22 was obtained by Tu and Yi in [16] forhigher order linear differential equations by using the type120591119872(Lemma 16 in the present paper)

Lemma 24 (see [20]) Let 119891 and 119892 be meromorphic functionssuch that 0 lt 120588(119891) 120588(119892) lt infin and 0 lt 120591(119891) 120591(119892) lt infin Thenwe have the following

(i) If 120588(119891) gt 120588(119892) then we obtain

120591 (119891 + 119892) = 120591 (119891119892) = 120591 (119891) (42)

(ii) If 120588(119891) = 120588(119892) and 120591(119891) = 120591(119892) then we get

120588 (119891 + 119892) = 120588 (119891119892) = 120588 (119891) = 120588 (119892) (43)

3 Proof of Theorems and Corollaries

Proof of Theorem 5 By the conditions of Theorem 5 andLemma 16 every nontrivial solution 119891 of (7) is of infiniteorder and 120588

2(119891) = 120588(119860

0) We need to prove that every

solution 119891 attains every complex value without exceptionthat is for all 119886 isin C the equation 119891(119911) = 119886 has infinitelymany solutions We have two cases

(i) If 119886 = 0 then by Lemma 17 we have 120582(119891 minus 119886) =+infin and the equation 119891(119911) = 119886 has infinitely manysolutions

(ii) If 119886 = 0 then we suppose that 119891 has a finite numberof zeros ByTheorem A

120588 (1198600) lt 120588(

1198911015840

119891) = 120588

2(119891) = 120588 (119860

0) (44)

which is a contradiction and hence 119891 assume every finitecomplex value infinitely often Now if 120588(1198911015840119891) = 120588(119891) = infinand since 120588

2(119891) = 120588(119860

0) lt 120588(119891) = infin then by using

Lemma 18 we have

120582 (119891) = 120582 (119891) = 120588 (119891) = infin (45)


Proof of Corollary 6 By using Lemma 17 we have for anyfinite order entire function 120593(119911) equiv 0

120582 (119891(119895) minus 120593) = 120582 (119891(119895) minus 120593) = 120588 (119891) = infin (119895 isin N) (46)

Now if 120593(119911) equiv 0 then we put 119892(119911) = 119891(119895)(119911) (119895 isin N) Itfollows that

120588(119891(119895+1)

119891(119895)) = 120588(

1198921015840

119892) = 120588 (119892) = infin (47)

and since 1205882(119892) = 120588

2(119891) = 120588(119860

0) lt 120588(119892) = infin then by

applying Lemma 18 we obtain

120582 (119892) = 120582 (119892) = infin (48)

that is 120582(119891(119895)) = 120582(119891(119895)) = infin (119895 isin N) and the proof ofCorollary 6 is completed

Proof of Theorem 9 By using Lemma 19 we have

119892119891=

119875119899[119891]

119876119898[119891]

=sum119899

119894=0119889119894119891(119894)

sum119898

119895=0119887119895119891(119895)

=1198890119891 + 11988911198911015840 + sum

119899

119894=2119889119894(1205721198941198911015840 + 120573

119894119891)

1198870119891 + 11988711198911015840 + sum

119898

119895=2119887119895(1205721198951198911015840 + 120573

119895119891)

(49)

which we can write as

119892119891=

119875 (119911) 1198911015840119891 + 119876 (119911)

119877 (119911) 1198911015840119891 + 119878 (119911) (50)

where

119875 = 1198891+119899

sum119894=2

119889119894120572119894 119876 = 119889

0+119899

sum119894=2

119889119894120573119894

119877 = 1198871+119898

sum119895=2

119887119895120572119895 119878 = 119887

0+119898

sum119895=2

119887119895120573119895

(51)

It is clear that

max 120588 (119875) 120588 (119876) 120588 (119877) 120588 (119878) lt 120588(1198911015840

119891) (52)

and since 119892119891is irreducible function in1198911015840119891 then 119875119878minus119876119877 equiv

0 So

120588 (119892119891) le max120588 (119875) 120588 (119876) 120588 (119877) 120588 (119878) 120588 (

1198911015840

119891)

= 120588(1198911015840

119891)

(53)

On the other hand we have

1198911015840

119891=

minus119878 (119911) 119892119891+ 119876 (119911)

119877 (119911) 119892119891minus 119875 (119911)

(54)

which implies

120588(1198911015840

119891) le max 120588 (119875) 120588 (119876) 120588 (119877) 120588 (119878) 120588 (119892

119891)

= 120588 (119892119891)

(55)

By (53) and (55) we obtain 120588(119892119891) = 120588(1198911015840119891)

Proof of Corollary 12 Under the conditions of Corollary 12and Lemma 22 we have 120588(119891) = infin and 120588

2(119891) = 120588(119861) Set

119892119891=

119889211989110158401015840 + 119889

11198911015840 + 119889

0119891

119887211989110158401015840 + 119887

11198911015840 + 119887

0119891

(56)

Substituting 11989110158401015840 = minus119860(119911)1198911015840 minus 119861(119911)119891 into 119892119891 we obtain

119892119891=

119889211989110158401015840 + 119889

11198911015840 + 119889

0119891

119887211989110158401015840 + 119887

11198911015840 + 119887

0119891

=(1198891minus 1198892119860)1198911015840 + (119889

0minus 1198892119861)119891

(1198871minus 1198872119860)1198911015840 + (119887

0minus 1198872119861)119891

=(1198891minus 1198892119860)119908 + (119889

0minus 1198892119861)

(1198871minus 1198872119860)119908 + (119887

0minus 1198872119861)

(57)

where 119908 = 1198911015840119891 In order to prove that 119892119891is irreducible

function in 119908 we need only to prove that ℎ equiv 0 where

ℎ = (1198891minus 1198892119860) (1198870minus 1198872119861) minus (119889

0minus 1198892119861) (1198871minus 1198872119860)

= 11988701198891minus 11988711198890+ 119860 (119887

21198890minus 11988701198892) + 119861 (119887

11198892minus 11988721198891)

(58)

Since 120588(119860) lt 120588(119861) and 0 lt 120591(119860) lt 120591(119861) lt infin if 120588(119861) =120588(119860) gt 0 119889

2(119911)1198871(119911) minus 119889

1(119911)1198872(119911) equiv 0 then by Lemma 24

we have 120588(ℎ) = 120588(119861) gt 0 so ℎ equiv 0 By applying Theorem 9we obtain 120588(1198911015840119891) = 120588(119892

119891)

Proof of Corollary 13 Suppose that 119891 is a nontrivial solutionof (11) such that 120588(1198911015840119891) = infin Then by Lemma 22 we have

120588 (119891) = +infin 1205882(119891) = 120588 (119861) (59)

By setting 1198892= 1198871equiv 1 and 119889

1= 1198890= 1198872= 1198870equiv 0 and using

Corollary 12 we have

120588(1198911015840

119891) = 120588(

11989110158401015840

1198911015840) = infin (60)

which implies by using Lemma 18 that

120582 (119891) = 120582 (1198911015840) = infin (61)

Conflict of Interests

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


References

[1] W K Hayman Meromorphic Functions Oxford MathematicalMonographs Clarendon Press Oxford UK 1964

[2] I Laine Nevanlinna theory and complex differential equationsvol 15 of de Gruyter Studies in Mathematics Walter de GruyterBerlin Germany 1993

[3] B Ya LevinLectures on Entire Functions vol 150 ofTranslationsof Mathematical Monographs American Mathematical SocietyProvidence RI USA 1996

[4] C-C Yang and H-X Yi Uniqueness Theory of MeromorphicFunctions vol 557 ofMathematics and Its Applications KluwerAcademic PublishersGroupDordrechtTheNetherlands 2003

[5] A A Golrsquodberg andV A Grinsteın ldquoThe logarithmic derivativeof a meromorphic functionrdquo Matematicheskie Zametki vol 19no 4 pp 525ndash530 1976 (Russian)

[6] D Benbourenane and R Korhonen ldquoOn the growth of thelogarithmic derivativerdquo Computational Methods and FunctionTheory vol 1 no 2 pp 301ndash310 2001

[7] A Hinkkanen ldquoA sharp form of Nevanlinnarsquos second funda-mental theoremrdquo Inventiones Mathematicae vol 108 no 3 pp549ndash574 1992

[8] J Heittokangas R Korhonen and J Rattya ldquoGeneralizedlogarithmic derivative estimates of Golrsquodberg-Grinshtein typerdquoThe Bulletin of the London Mathematical Society vol 36 no 1pp 105ndash114 2004

[9] G G Gundersen ldquoEstimates for the logarithmic derivative ofa meromorphic function plus similar estimatesrdquo Journal of theLondon Mathematical Society vol 37 no 1 pp 88ndash104 1988

[10] Z Latreuch and B Belaıdi ldquoGrowth of logarithmic derivativeofmeromorphic functionsrdquoMathematica Scandinavica vol 113no 2 pp 248ndash261 2013

[11] S B Bank and J K Langley ldquoOn the oscillation of solutionsof certain linear differential equations in the complex domainrdquoProceedings of the EdinburghMathematical Society II vol 30 no3 pp 455ndash469 1987

[12] B Belaıdi ldquoThe fixed points and iterated order of some differen-tial polynomialsrdquo Commentationes Mathematicae UniversitatisCarolinae vol 50 no 2 pp 209ndash219 2009

[13] I Laine and J Rieppo ldquoDifferential polynomials generated bylinear differential equationsrdquo Complex Variables vol 49 no 12pp 897ndash911 2004

[14] Z Latreuch and B Belaıdi ldquoGrowth and oscillation of differen-tial polynomials generated by complex differential equationsrdquoElectronic Journal of Differential Equations vol 2013 no 16 pp1ndash14 2013

[15] Z-X Chen and C-C Yang ldquoQuantitative estimations on thezeros and growths of entire solutions of linear differentialequationsrdquo Complex Variables vol 42 no 2 pp 119ndash133 2000

[16] J Tu andC-F Yi ldquoOn the growth of solutions of a class of higherorder linear differential equations with coefficients having thesame orderrdquo Journal of Mathematical Analysis and Applicationsvol 340 no 1 pp 487ndash497 2008

[17] H-Y Xu J Tu and X-M Zheng ldquoOn the hyper exponent ofconvergence of zeros of119891(119895)minus120593 of higher order linear differentialequationsrdquo Advances in Difference Equations vol 2012 article114 2012

[18] J Heittokangas R Korhonen and J Rattya ldquoGrowth estimatesfor solutions of linear complex differential equationsrdquo AnnalesAcademiaelig Scientiarum Fennicaelig Mathematica vol 29 no 1 pp233ndash246 2004

[19] T-B Cao J-F Xu and Z-X Chen ldquoOn the meromorphicsolutions of linear differential equations on the complex planerdquoJournal of Mathematical Analysis and Applications vol 364 no1 pp 130ndash142 2010

[20] Z Latreuch and B Belaıdi ldquoEstimations about the order ofgrowth and the type of meromorphic functions in the complexplanerdquo Analele Universitatii Din Oradea Fascicola Matematicavol 20 no 1 pp 179ndash186 2013

Submit your manuscripts athttpwwwhindawicom

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Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


Recently [10] the authors have studied some properties of thebehavior of growth of logarithmic derivatives of entire andmeromorphic functions and have obtained some relationsbetween the zeros of entire functions and the growth oftheir logarithmic derivatives In fact they have proved thefollowing

Theorem A (see [10]) Let 119891 be an entire function with finitenumber of zeros Then for any integer 119896 ge 1

120588(119891(119896)

119891) = 120588

2(119891) (4)

Theorem B (see [10]) Suppose that 119896 ge 2 is an integer and let119891 be a meromorphic function Then

120588(1198911015840

119891) = max120588(

119891(119896)

119891) 119896 ge 2

= max120588(119891(119896)

119891) 120588(

119891(119896+1)

119891) 119896 ge 2

(5)

In [11 pages 457ndash458] Bank and Langley have obtainedthe following result

Theorem C (see [11]) All nontrivial solutions of

11989110158401015840 + 119890119875(119911)119891 = 0 (6)

where 119875(119911) is a nonconstant polynomial satisfy 120582(119891) = infin

Theorem D (see [4] (E Picard)) Any transcendental entirefunction119891must take every finite complex value infinitely manytimes with at most one exception

Example 4 The entire function 119890119911 omits the value zero andevery other finite value is assumed infinitely often and zero isthe exceptional value of Picard

Question 1 Under what conditions can we ensure that 119891 hasno Picardrsquos exceptional value

Question 2 Under what conditions can we obtain the sameresult as Theorem C for higher order linear differentialequations with entire coefficients

The subject of this paper is to give answers to the abovequestions In fact we prove some relations between the valuedistribution of solutions of linear differential equations andgrowth of their logarithmic derivatives Firstly we study therelationship between the growth of logarithmic derivatives ofsolutions of complex differential equation

119891(119896) + 119860119896minus1

119891(119896minus1) + sdot sdot sdot + 1198600119891 = 0 (7)

and their exponents of convergence We will also prove thenonexistence of any Picard exceptional value of solutions of(7) and we obtain the following

Theorem 5 Let 1198600 1198601 119860

119896minus1be entire functions satisfy-

ing 120588(1198600) = 120588 (0 lt 120588 lt infin) 120591

119872(1198600) = 120591 (0 lt 120591 lt infin) and

let 120588(119860119895) le 120588 120591

119872(119860119895) lt 120591 if 120588(119860

119895) = 120588 (119895 = 1 119896 minus 1) If

119891 is a nontrivial solution of (7) such that 120588(1198911015840119891) gt 120588(1198600)

then 119891 assume every finite complex value infinitely oftenFurthermore if 120588(1198911015840119891) = infin then

120582 (119891) = 120582 (119891) = infin (8)

Corollary 6 Under the hypotheses of Theorem 5 let 120593(119911) bean entire function with finite order If 119891 is a nontrivial solutionof (7) such that 120588(119891(119895+1)119891(119895)) = infin for some 119895 isin N then

120582 (119891(119895) minus 120593) = 120582 (119891(119895) minus 120593) = infin (9)

Corollary 7 Under the hypotheses of Theorem 5 every non-trivial solution of (7) does not have any Picard exceptionalvalues

Example 8 It is clear that the function 119891(119911) = sin 119890119911 satisfiesthe differential equation

11989110158401015840 minus 1198911015840 + 1198902119911119891 = 0 (10)

and 120588(1198911015840119891) = 120588(119891) = infin Obviously the conditions ofTheorem 5 are satisfied Then 120582(119891) = 120582(119891) = infin

Secondly we consider the differential equation

11989110158401015840 + 119860 (119911) 1198911015840 + 119861 (119911) 119891 = 0 (11)

where 119860(119911) and 119861(119911) are entire functions and let 119875119899[119891] and

119876119898[119891] be two differential polynomials defined by

119875119899[119891] = 119889

119899119891(119899) + 119889


0119891 (119899 ge 1)

119876119898[119891] = 119887

119898119891(119898) + 119887


0119891 (119898 ge 1)

(12)

where 119889119894(119894 = 0 119899) and 119887

119895(119895 = 0 119898) are entire

functions not all vanishing identically In recent years thereis a great interest to investigate the growth and oscillation ofdifferential polynomials generated by solutions of differentialequations in the complex plane (see [12ndash14]) In this partwe give under some conditions an estimate on the growthof the quotient of two differential polynomials generated bysolutions of (11) and we obtain the following results

Theorem 9 Let 119891 be a nontrivial solution of (11) and let119889119894(119894 = 0 119899) and 119887

119895(119895 = 0 119898) be entire functions

not all vanishing identically such that 119892119891

= 119875119899[119891]119876

119898[119891] is

irreducible function in 1198911015840119891 If

max 120588 (119860) 120588 (119861) 120588 (119889119894) (119894 = 0 119899)

120588 (119887119895) (119895 = 0 119898) lt 120588(

1198911015840

119891)

(13)

then

120588 (119892119891) = 120588(

1198911015840

119891) (14)



11989110158401015840 minus (119890119911 + 1198902119911) 119891 = 0 (15)

with

120588(1198911015840

119891) = 1 (16)

and let

119892119891=

119889211989110158401015840 + 119889

11198911015840 + 119889

0119891

119887211989110158401015840 + 119887

11198911015840 + 119887

0119891

=1198892(119890119911 + 1198902119911) + 119889

1119890119911 + 119889

0

1198872(119890119911 + 1198902119911) + 119887

1119890119911 + 1198870

(17)

If we take 1198892= 119890119911

2

1198891= 1198890= 1 and 119887

2= 1198871= 1198870= 1 then

120588 (119892119891) = 2 gt 120588(

1198911015840

119891) = 1 (18)



119891equiv 1

and so

120588 (119892119891) = 0 = 120588(

1198911015840

119891) (19)



119894(119911)(119894 = 0 1 2) 119887

119895(119911)(119895 = 0 1 2) be


max 120588 (119889119894) 120588 (119887

119895) 0 le 119894 119895 le 2 lt 120588 (119861) (20)

and 1198892(119911)b1(119911)minus119889


such that 120588(119861) lt 120588(1198911015840119891) then

120588(119889211989110158401015840 + 119889

11198911015840 + 119889

0119891

119887211989110158401015840 + 119887

11198911015840 + 119887

0119891

) = 120588(1198911015840

119891) (21)


120582 (119891) = 120582 (1198911015840) = infin (22)

2 Auxiliary Lemmas


119898(119903119891(119896)

119891) = 119878 (119903 119891) (23)


119898(119903119891(119896)

119891) = 119874 (log 119903) (24)

Lemma 15 (see [15]) Let 1198600(119911) 119860



max 120588 (119860119895) 119895 = 1 119896 minus 1 lt 120588 (119860

0) (25)


0)

Lemma 16 (see [16]) Let1198600 1198601 119860



119872(1198600) = 120591 (0 lt 120591 lt infin)

and let 120588(119860119895) le 120588 120591

119872(119860119895) lt 120591 if 120588(119860

119895) = 120588 (119895 = 1 119896 minus 1)


0)


Lemma 17 Let 1198600 1198601 119860



(i) max120588(119860119895) 119895 = 1 119896 minus 1 lt 120588(119860

0) lt infin

(ii) 0 lt 120588(119860119896minus1


119872(119860119895)

119895 = 1 119896 minus 1 lt 120591119872(1198600) lt infin



1205822(119891(119895) minus 120593) = 120582

2(119891(119895) minus 120593) = 120588 (119860

0) (119895 isin N)

(26)


2(119891) then

max120582 (119891) 120582 (1

119891) = max120582 (119891) 120582 (

1

119891) = 120588 (119891)

(27)


120582 (119891) = 120582 (119891) = 120588 (119891) (28)



119895and 120573

119895such that

119891(119895) = 1205721198951198911015840 + 120573

119895119891 (29)





119888isin 1 119896



= ]119890119894120579 isin Δ119877such that


10038161003816100381610038161003816119891 (119903119890119894120579)10038161003816100381610038161003816 le 119862 exp(119899

119888int119903

]max119895=0119896minus1

10038161003816100381610038161003816119860119895 (119905119890119894120579)

100381610038161003816100381610038161(119896minus119895)

119889119905)

(30)


119862 le (1 + 120576) max119895=0119896minus1

(

10038161003816100381610038161003816119891(119895) (119911120579)10038161003816100381610038161003816

(119899119888)119895max119899=0119896minus1

1003816100381610038161003816119860119899 (119911120579)1003816100381610038161003816119895(119896minus119899)

)

(31)




11989110158401015840 + 119860 (119911) 1198911015840 + 119861 (119911) 119891 = 0 (32)

then 120588(119891) = infin and 1205882(119891) = 120588(119861)


119861 = minus(11989110158401015840

119891+ 119860

1198911015840

119891) (33)


119879 (119903 119861) le 119879 (119903 119860) + 119874 (log 119903) (34)


120591 (119861) le 120591 (119860) (35)


1205882(119891) le max 120588 (119860) 120588 (119861) = 120588 (119861) (36)


119879 (119903 119861) le 119879 (119903 119860) + 119878 (119903 119891) (37)


0 1205721satisfying 120591(119860) lt 120572



119879 (119903 119860) le 1205721119903120588 (38)



119879 (119903 119861) gt 1205720119903120588 (39)


(1205720minus 1205721) 119903120588 le 119874 (log119879 (119903 119891)) + 119874 (log 119903) (40)

which implies

120588 (119861) = 120588 le 1205882(119891) (41)





120591 (119891 + 119892) = 120591 (119891119892) = 120591 (119891) (42)


120588 (119891 + 119892) = 120588 (119891119892) = 120588 (119891) = 120588 (119892) (43)



2(119891) = 120588(119860





120588 (1198600) lt 120588(

1198911015840

119891) = 120588

2(119891) = 120588 (119860

0) (44)


2(119891) = 120588(119860


Lemma 18 we have

120582 (119891) = 120582 (119891) = 120588 (119891) = infin (45)





120588(119891(119895+1)

119891(119895)) = 120588(

1198921015840

119892) = 120588 (119892) = infin (47)

and since 1205882(119892) = 120588

2(119891) = 120588(119860

0) lt 120588(119892) = infin then by


120582 (119892) = 120582 (119892) = infin (48)



119892119891=

119875119899[119891]

119876119898[119891]

=sum119899

119894=0119889119894119891(119894)

sum119898

119895=0119887119895119891(119895)

=1198890119891 + 11988911198911015840 + sum

119899

119894=2119889119894(1205721198941198911015840 + 120573

119894119891)

1198870119891 + 11988711198911015840 + sum

119898

119895=2119887119895(1205721198951198911015840 + 120573

119895119891)

(49)


119892119891=

119875 (119911) 1198911015840119891 + 119876 (119911)

119877 (119911) 1198911015840119891 + 119878 (119911) (50)

where

119875 = 1198891+119899

sum119894=2

119889119894120572119894 119876 = 119889

0+119899

sum119894=2

119889119894120573119894

119877 = 1198871+119898

sum119895=2

119887119895120572119895 119878 = 119887

0+119898

sum119895=2

119887119895120573119895

(51)

It is clear that

max 120588 (119875) 120588 (119876) 120588 (119877) 120588 (119878) lt 120588(1198911015840

119891) (52)


0 So

120588 (119892119891) le max120588 (119875) 120588 (119876) 120588 (119877) 120588 (119878) 120588 (

1198911015840

119891)

= 120588(1198911015840

119891)

(53)


1198911015840

119891=

minus119878 (119911) 119892119891+ 119876 (119911)

119877 (119911) 119892119891minus 119875 (119911)

(54)

which implies

120588(1198911015840

119891) le max 120588 (119875) 120588 (119876) 120588 (119877) 120588 (119878) 120588 (119892

119891)

= 120588 (119892119891)

(55)

By (53) and (55) we obtain 120588(119892119891) = 120588(1198911015840119891)


2(119891) = 120588(119861) Set

119892119891=

119889211989110158401015840 + 119889

11198911015840 + 119889

0119891

119887211989110158401015840 + 119887

11198911015840 + 119887

0119891

(56)


119892119891=

119889211989110158401015840 + 119889

11198911015840 + 119889

0119891

119887211989110158401015840 + 119887

11198911015840 + 119887

0119891

=(1198891minus 1198892119860)1198911015840 + (119889

0minus 1198892119861)119891

(1198871minus 1198872119860)1198911015840 + (119887

0minus 1198872119861)119891

=(1198891minus 1198892119860)119908 + (119889

0minus 1198892119861)

(1198871minus 1198872119860)119908 + (119887

0minus 1198872119861)

(57)




0minus 1198892119861) (1198871minus 1198872119860)

= 11988701198891minus 11988711198890+ 119860 (119887

21198890minus 11988701198892) + 119861 (119887

11198892minus 11988721198891)

(58)


2(119911)1198871(119911) minus 119889



119891)


120588 (119891) = +infin 1205882(119891) = 120588 (119861) (59)


1= 1198890= 1198872= 1198870equiv 0 and using


120588(1198911015840

119891) = 120588(

11989110158401015840

1198911015840) = infin (60)


120582 (119891) = 120582 (1198911015840) = infin (61)




References




























Volume 2014




Journal of











Journal of


Function Spaces






Algebra











11989110158401015840 minus (119890119911 + 1198902119911) 119891 = 0 (15)

with

120588(1198911015840

119891) = 1 (16)

and let

119892119891=

119889211989110158401015840 + 119889

11198911015840 + 119889

0119891

119887211989110158401015840 + 119887

11198911015840 + 119887

0119891

=1198892(119890119911 + 1198902119911) + 119889

1119890119911 + 119889

0

1198872(119890119911 + 1198902119911) + 119887

1119890119911 + 1198870

(17)

If we take 1198892= 119890119911

2

1198891= 1198890= 1 and 119887

2= 1198871= 1198870= 1 then

120588 (119892119891) = 2 gt 120588(

1198911015840

119891) = 1 (18)



119891equiv 1

and so

120588 (119892119891) = 0 = 120588(

1198911015840

119891) (19)



119894(119911)(119894 = 0 1 2) 119887

119895(119911)(119895 = 0 1 2) be


max 120588 (119889119894) 120588 (119887

119895) 0 le 119894 119895 le 2 lt 120588 (119861) (20)

and 1198892(119911)b1(119911)minus119889


such that 120588(119861) lt 120588(1198911015840119891) then

120588(119889211989110158401015840 + 119889

11198911015840 + 119889

0119891

119887211989110158401015840 + 119887

11198911015840 + 119887

0119891

) = 120588(1198911015840

119891) (21)


120582 (119891) = 120582 (1198911015840) = infin (22)

2 Auxiliary Lemmas


119898(119903119891(119896)

119891) = 119878 (119903 119891) (23)


119898(119903119891(119896)

119891) = 119874 (log 119903) (24)

Lemma 15 (see [15]) Let 1198600(119911) 119860



max 120588 (119860119895) 119895 = 1 119896 minus 1 lt 120588 (119860

0) (25)


0)

Lemma 16 (see [16]) Let1198600 1198601 119860



119872(1198600) = 120591 (0 lt 120591 lt infin)

and let 120588(119860119895) le 120588 120591

119872(119860119895) lt 120591 if 120588(119860

119895) = 120588 (119895 = 1 119896 minus 1)


0)


Lemma 17 Let 1198600 1198601 119860



(i) max120588(119860119895) 119895 = 1 119896 minus 1 lt 120588(119860

0) lt infin

(ii) 0 lt 120588(119860119896minus1


119872(119860119895)

119895 = 1 119896 minus 1 lt 120591119872(1198600) lt infin



1205822(119891(119895) minus 120593) = 120582

2(119891(119895) minus 120593) = 120588 (119860

0) (119895 isin N)

(26)


2(119891) then

max120582 (119891) 120582 (1

119891) = max120582 (119891) 120582 (

1

119891) = 120588 (119891)

(27)


120582 (119891) = 120582 (119891) = 120588 (119891) (28)



119895and 120573

119895such that

119891(119895) = 1205721198951198911015840 + 120573

119895119891 (29)





119888isin 1 119896



= ]119890119894120579 isin Δ119877such that


10038161003816100381610038161003816119891 (119903119890119894120579)10038161003816100381610038161003816 le 119862 exp(119899

119888int119903

]max119895=0119896minus1

10038161003816100381610038161003816119860119895 (119905119890119894120579)

100381610038161003816100381610038161(119896minus119895)

119889119905)

(30)


119862 le (1 + 120576) max119895=0119896minus1

(

10038161003816100381610038161003816119891(119895) (119911120579)10038161003816100381610038161003816

(119899119888)119895max119899=0119896minus1

1003816100381610038161003816119860119899 (119911120579)1003816100381610038161003816119895(119896minus119899)

)

(31)




11989110158401015840 + 119860 (119911) 1198911015840 + 119861 (119911) 119891 = 0 (32)

then 120588(119891) = infin and 1205882(119891) = 120588(119861)


119861 = minus(11989110158401015840

119891+ 119860

1198911015840

119891) (33)


119879 (119903 119861) le 119879 (119903 119860) + 119874 (log 119903) (34)


120591 (119861) le 120591 (119860) (35)


1205882(119891) le max 120588 (119860) 120588 (119861) = 120588 (119861) (36)


119879 (119903 119861) le 119879 (119903 119860) + 119878 (119903 119891) (37)


0 1205721satisfying 120591(119860) lt 120572



119879 (119903 119860) le 1205721119903120588 (38)



119879 (119903 119861) gt 1205720119903120588 (39)


(1205720minus 1205721) 119903120588 le 119874 (log119879 (119903 119891)) + 119874 (log 119903) (40)

which implies

120588 (119861) = 120588 le 1205882(119891) (41)





120591 (119891 + 119892) = 120591 (119891119892) = 120591 (119891) (42)


120588 (119891 + 119892) = 120588 (119891119892) = 120588 (119891) = 120588 (119892) (43)



2(119891) = 120588(119860





120588 (1198600) lt 120588(

1198911015840

119891) = 120588

2(119891) = 120588 (119860

0) (44)


2(119891) = 120588(119860


Lemma 18 we have

120582 (119891) = 120582 (119891) = 120588 (119891) = infin (45)





120588(119891(119895+1)

119891(119895)) = 120588(

1198921015840

119892) = 120588 (119892) = infin (47)

and since 1205882(119892) = 120588

2(119891) = 120588(119860

0) lt 120588(119892) = infin then by


120582 (119892) = 120582 (119892) = infin (48)



119892119891=

119875119899[119891]

119876119898[119891]

=sum119899

119894=0119889119894119891(119894)

sum119898

119895=0119887119895119891(119895)

=1198890119891 + 11988911198911015840 + sum

119899

119894=2119889119894(1205721198941198911015840 + 120573

119894119891)

1198870119891 + 11988711198911015840 + sum

119898

119895=2119887119895(1205721198951198911015840 + 120573

119895119891)

(49)


119892119891=

119875 (119911) 1198911015840119891 + 119876 (119911)

119877 (119911) 1198911015840119891 + 119878 (119911) (50)

where

119875 = 1198891+119899

sum119894=2

119889119894120572119894 119876 = 119889

0+119899

sum119894=2

119889119894120573119894

119877 = 1198871+119898

sum119895=2

119887119895120572119895 119878 = 119887

0+119898

sum119895=2

119887119895120573119895

(51)

It is clear that

max 120588 (119875) 120588 (119876) 120588 (119877) 120588 (119878) lt 120588(1198911015840

119891) (52)


0 So

120588 (119892119891) le max120588 (119875) 120588 (119876) 120588 (119877) 120588 (119878) 120588 (

1198911015840

119891)

= 120588(1198911015840

119891)

(53)


1198911015840

119891=

minus119878 (119911) 119892119891+ 119876 (119911)

119877 (119911) 119892119891minus 119875 (119911)

(54)

which implies

120588(1198911015840

119891) le max 120588 (119875) 120588 (119876) 120588 (119877) 120588 (119878) 120588 (119892

119891)

= 120588 (119892119891)

(55)

By (53) and (55) we obtain 120588(119892119891) = 120588(1198911015840119891)


2(119891) = 120588(119861) Set

119892119891=

119889211989110158401015840 + 119889

11198911015840 + 119889

0119891

119887211989110158401015840 + 119887

11198911015840 + 119887

0119891

(56)


119892119891=

119889211989110158401015840 + 119889

11198911015840 + 119889

0119891

119887211989110158401015840 + 119887

11198911015840 + 119887

0119891

=(1198891minus 1198892119860)1198911015840 + (119889

0minus 1198892119861)119891

(1198871minus 1198872119860)1198911015840 + (119887

0minus 1198872119861)119891

=(1198891minus 1198892119860)119908 + (119889

0minus 1198892119861)

(1198871minus 1198872119860)119908 + (119887

0minus 1198872119861)

(57)




0minus 1198892119861) (1198871minus 1198872119860)

= 11988701198891minus 11988711198890+ 119860 (119887

21198890minus 11988701198892) + 119861 (119887

11198892minus 11988721198891)

(58)


2(119911)1198871(119911) minus 119889



119891)


120588 (119891) = +infin 1205882(119891) = 120588 (119861) (59)


1= 1198890= 1198872= 1198870equiv 0 and using


120588(1198911015840

119891) = 120588(

11989110158401015840

1198911015840) = infin (60)


120582 (119891) = 120582 (1198911015840) = infin (61)




References




























Volume 2014




Journal of











Journal of


Function Spaces






Algebra













119888isin 1 119896



= ]119890119894120579 isin Δ119877such that


10038161003816100381610038161003816119891 (119903119890119894120579)10038161003816100381610038161003816 le 119862 exp(119899

119888int119903

]max119895=0119896minus1

10038161003816100381610038161003816119860119895 (119905119890119894120579)

100381610038161003816100381610038161(119896minus119895)

119889119905)

(30)


119862 le (1 + 120576) max119895=0119896minus1

(

10038161003816100381610038161003816119891(119895) (119911120579)10038161003816100381610038161003816

(119899119888)119895max119899=0119896minus1

1003816100381610038161003816119860119899 (119911120579)1003816100381610038161003816119895(119896minus119899)

)

(31)




11989110158401015840 + 119860 (119911) 1198911015840 + 119861 (119911) 119891 = 0 (32)

then 120588(119891) = infin and 1205882(119891) = 120588(119861)


119861 = minus(11989110158401015840

119891+ 119860

1198911015840

119891) (33)


119879 (119903 119861) le 119879 (119903 119860) + 119874 (log 119903) (34)


120591 (119861) le 120591 (119860) (35)


1205882(119891) le max 120588 (119860) 120588 (119861) = 120588 (119861) (36)


119879 (119903 119861) le 119879 (119903 119860) + 119878 (119903 119891) (37)


0 1205721satisfying 120591(119860) lt 120572



119879 (119903 119860) le 1205721119903120588 (38)



119879 (119903 119861) gt 1205720119903120588 (39)


(1205720minus 1205721) 119903120588 le 119874 (log119879 (119903 119891)) + 119874 (log 119903) (40)

which implies

120588 (119861) = 120588 le 1205882(119891) (41)





120591 (119891 + 119892) = 120591 (119891119892) = 120591 (119891) (42)


120588 (119891 + 119892) = 120588 (119891119892) = 120588 (119891) = 120588 (119892) (43)



2(119891) = 120588(119860





120588 (1198600) lt 120588(

1198911015840

119891) = 120588

2(119891) = 120588 (119860

0) (44)


2(119891) = 120588(119860


Lemma 18 we have

120582 (119891) = 120582 (119891) = 120588 (119891) = infin (45)





120588(119891(119895+1)

119891(119895)) = 120588(

1198921015840

119892) = 120588 (119892) = infin (47)

and since 1205882(119892) = 120588

2(119891) = 120588(119860

0) lt 120588(119892) = infin then by


120582 (119892) = 120582 (119892) = infin (48)



119892119891=

119875119899[119891]

119876119898[119891]

=sum119899

119894=0119889119894119891(119894)

sum119898

119895=0119887119895119891(119895)

=1198890119891 + 11988911198911015840 + sum

119899

119894=2119889119894(1205721198941198911015840 + 120573

119894119891)

1198870119891 + 11988711198911015840 + sum

119898

119895=2119887119895(1205721198951198911015840 + 120573

119895119891)

(49)


119892119891=

119875 (119911) 1198911015840119891 + 119876 (119911)

119877 (119911) 1198911015840119891 + 119878 (119911) (50)

where

119875 = 1198891+119899

sum119894=2

119889119894120572119894 119876 = 119889

0+119899

sum119894=2

119889119894120573119894

119877 = 1198871+119898

sum119895=2

119887119895120572119895 119878 = 119887

0+119898

sum119895=2

119887119895120573119895

(51)

It is clear that

max 120588 (119875) 120588 (119876) 120588 (119877) 120588 (119878) lt 120588(1198911015840

119891) (52)


0 So

120588 (119892119891) le max120588 (119875) 120588 (119876) 120588 (119877) 120588 (119878) 120588 (

1198911015840

119891)

= 120588(1198911015840

119891)

(53)


1198911015840

119891=

minus119878 (119911) 119892119891+ 119876 (119911)

119877 (119911) 119892119891minus 119875 (119911)

(54)

which implies

120588(1198911015840

119891) le max 120588 (119875) 120588 (119876) 120588 (119877) 120588 (119878) 120588 (119892

119891)

= 120588 (119892119891)

(55)

By (53) and (55) we obtain 120588(119892119891) = 120588(1198911015840119891)


2(119891) = 120588(119861) Set

119892119891=

119889211989110158401015840 + 119889

11198911015840 + 119889

0119891

119887211989110158401015840 + 119887

11198911015840 + 119887

0119891

(56)


119892119891=

119889211989110158401015840 + 119889

11198911015840 + 119889

0119891

119887211989110158401015840 + 119887

11198911015840 + 119887

0119891

=(1198891minus 1198892119860)1198911015840 + (119889

0minus 1198892119861)119891

(1198871minus 1198872119860)1198911015840 + (119887

0minus 1198872119861)119891

=(1198891minus 1198892119860)119908 + (119889

0minus 1198892119861)

(1198871minus 1198872119860)119908 + (119887

0minus 1198872119861)

(57)




0minus 1198892119861) (1198871minus 1198872119860)

= 11988701198891minus 11988711198890+ 119860 (119887

21198890minus 11988701198892) + 119861 (119887

11198892minus 11988721198891)

(58)


2(119911)1198871(119911) minus 119889



119891)


120588 (119891) = +infin 1205882(119891) = 120588 (119861) (59)


1= 1198890= 1198872= 1198870equiv 0 and using


120588(1198911015840

119891) = 120588(

11989110158401015840

1198911015840) = infin (60)


120582 (119891) = 120582 (1198911015840) = infin (61)




References




























Volume 2014




Journal of











Journal of


Function Spaces






Algebra













120588(119891(119895+1)

119891(119895)) = 120588(

1198921015840

119892) = 120588 (119892) = infin (47)

and since 1205882(119892) = 120588

2(119891) = 120588(119860

0) lt 120588(119892) = infin then by


120582 (119892) = 120582 (119892) = infin (48)



119892119891=

119875119899[119891]

119876119898[119891]

=sum119899

119894=0119889119894119891(119894)

sum119898

119895=0119887119895119891(119895)

=1198890119891 + 11988911198911015840 + sum

119899

119894=2119889119894(1205721198941198911015840 + 120573

119894119891)

1198870119891 + 11988711198911015840 + sum

119898

119895=2119887119895(1205721198951198911015840 + 120573

119895119891)

(49)


119892119891=

119875 (119911) 1198911015840119891 + 119876 (119911)

119877 (119911) 1198911015840119891 + 119878 (119911) (50)

where

119875 = 1198891+119899

sum119894=2

119889119894120572119894 119876 = 119889

0+119899

sum119894=2

119889119894120573119894

119877 = 1198871+119898

sum119895=2

119887119895120572119895 119878 = 119887

0+119898

sum119895=2

119887119895120573119895

(51)

It is clear that

max 120588 (119875) 120588 (119876) 120588 (119877) 120588 (119878) lt 120588(1198911015840

119891) (52)


0 So

120588 (119892119891) le max120588 (119875) 120588 (119876) 120588 (119877) 120588 (119878) 120588 (

1198911015840

119891)

= 120588(1198911015840

119891)

(53)


1198911015840

119891=

minus119878 (119911) 119892119891+ 119876 (119911)

119877 (119911) 119892119891minus 119875 (119911)

(54)

which implies

120588(1198911015840

119891) le max 120588 (119875) 120588 (119876) 120588 (119877) 120588 (119878) 120588 (119892

119891)

= 120588 (119892119891)

(55)

By (53) and (55) we obtain 120588(119892119891) = 120588(1198911015840119891)


2(119891) = 120588(119861) Set

119892119891=

119889211989110158401015840 + 119889

11198911015840 + 119889

0119891

119887211989110158401015840 + 119887

11198911015840 + 119887

0119891

(56)


119892119891=

119889211989110158401015840 + 119889

11198911015840 + 119889

0119891

119887211989110158401015840 + 119887

11198911015840 + 119887

0119891

=(1198891minus 1198892119860)1198911015840 + (119889

0minus 1198892119861)119891

(1198871minus 1198872119860)1198911015840 + (119887

0minus 1198872119861)119891

=(1198891minus 1198892119860)119908 + (119889

0minus 1198892119861)

(1198871minus 1198872119860)119908 + (119887

0minus 1198872119861)

(57)




0minus 1198892119861) (1198871minus 1198872119860)

= 11988701198891minus 11988711198890+ 119860 (119887

21198890minus 11988701198892) + 119861 (119887

11198892minus 11988721198891)

(58)


2(119911)1198871(119911) minus 119889



119891)


120588 (119891) = +infin 1205882(119891) = 120588 (119861) (59)


1= 1198890= 1198872= 1198870equiv 0 and using


120588(1198911015840

119891) = 120588(

11989110158401015840

1198911015840) = infin (60)


120582 (119891) = 120582 (1198911015840) = infin (61)




References




























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










References




























Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









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