coefficient estimate for a subclass of close to convex functions

8/13/2019 Coefficient Estimate for a Subclass of Close to Convex Functions

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Copyright 2012 by Modern Scientific Press Company, Florida, USA

International Journal of Modern Mathematical Sciences, 2012, 4(2): 71-83I nternati onal Jour nal of M oder n M athematical Sciences

Journal homepage:www.ModernScientificPress.com/Journals/ijmms.aspx

ISSN: 2166-286X Florida, USA

Article

Coefficient Estimate for a Subclass of Close-to-convex Functions

B. S. Mehrok 1, Gagandeep Singh 2, Deepak Gupta

1# 643 E, B.R.S. Nagar, Ludhiana (Punjab), India 2 Department of Mathematics, DIPS College (Co-Educational), Dhilwan(Kapurthala), Punjab, India3 Department of Mathematics, M.M.University, Mullana-Ambala (Haryana), India

* Author to whom correspondence should be addressed; Email: [email protected]

Article history : Received 1October 2012, Received in revised form 27 October 2012, Accepted 29October 2012, Published 30 October 2012.

Abstract: In this paper, we introduce a subclass of the Close-to-convex functions. We

derive inclusion relation, establish integral representation formula and obtain some

coefficient estimates for such functions.

Keywords : Subordination, Starlike functions, Convex functions, Close-to-convexfunctions, -Close-to-convex functions.

Mathematics Subject Classification: 30C45

1. Introduction

Let U be the class of functions

1

k k

k

w z c z (1.1)

which are regular in the unit disc : 1 E z z and satisfying the conditions

(0) 0w and 1w z , z E .

Let A denote the class of functions

2k

k k z a z z f (1.2)

which are analytic in E .

Let S be the class of functions A z f which are regular and univalent in . E


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Int. J. Modern Math. Sci. 2012 , 4(2): 71-83


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Let S be the class of starlike functions with the condition

.,0Re E z z f z f z

Let K be the class of convex functions with the condition

.,01Re E z z f z f z

If f and g are analytic functions in E , then we say that f is subordinate to g , written as f g or

f z g z , if there exists a function ( )w z U such that ( ) ( ( )) f z g w z . If g is univalent then

f g if and only if 0 0 f g and f E g E .

A function

A z f is said to be in the class

B A J , if there exists a convex function

z h such that

Bz

Az z h z f z

1

1, E z A B ,11 .

This class was introduced and studied by Mehrok and Singh[7]. Also J J 1,1 , the class ofclose-to-convex functions introduced by Gawad and Thomas[5].

Let B A J , be the class of functions A z f with the condition

Bz

Az z h z f z

1

1, E z A B K z h ,11, .

By

C , we denote the class of functions A z f for which 0.

z z f z f

and

001Re

z g

z f z

z g

z f z , where .S z g

This class was introduced by Chichra[2] and functions of this class are called -Close-to-

convex functions.

Since K is a subclass of S , we define the following subclass.

Let B A J ,; be the class of functions A z f for which 0.

z z f z f

and


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E z A B Bz Az

z h z f z

z h z f z

,0,11,1

11 , (1.3)

where 2 .k k

k K z b z z h (1.4)

In particular .1,1; J J

The following observations are obvious:

(i) J J 1,1;0 .

(ii) .,,;0 B A J B A J

(iii) .,,;1 B A J B A J

In this paper we obtain sharp bounds for some coefficients of the functions belonging to

the class .,; B A J We also find an integral representation formula and inclusion relation for the

class .,; B A J

2. Preliminary Lemmas

Lemma 2.1[6] If

z Bw z Aw

z p1

1

1

,,1k

k k U z w z p

(2.1)

then B A pn , 1n .

The bounds are sharp, being attained for the functions

.1,1,1

1 n

z B

z A z P

n

n

n

Lemma 2.2[8] Let P be the class of analytic functions of the form

z w z w

z p1

1 1

,,1k

k k U z w z p

which have positive real part in E, then

2n p , 1n .

Equality is attained for the Mobius function

.11

0 z z z L

Lemma 2.3[4] Let , P z p then


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74

.2

22

2

12

1

2

p p p

(2.2)

Note that the inequality (2.2) can be written as

.1,21

221 2

12

12

p p p (2.3)

Lemma 2.4 For every convex function z h and for real numbers ,,, , such that ,23

56 , ,2415

1

3640

1 and

181620

1 are all nonnegative, we have the sharp inequalities:

1324 bbb ; ,65

(2.4)

132324

bbbb ; ,0665

(2.5)

132

25 bbb ; ,

15

8 (2.6)

12

3425 bbbb ; .0252012

(2.7)

Proof of Lemma. Since z h is convex, there exists

1

1

k

k k z p z p (2.8)

such that

.1 z p z h z h z

(2.9)

Comparing coefficients of both sides of (2.9) using (1.4) and (2.8), we see that

122

1 pb

66

2

12

3

p pb

24812

21123

4

p p p pb

12040201520

4

1

2

2

2

12134

5

p p p p p p pb

..

so that

324 bbb 223212

2423

12

2

1213 p p

p p

(2.10)


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3

2324 bbbb 223

3212

2423

12

21

213 p p

p p

(2.11)

3225

bbb

256512

120

56

401520

2

12

2

1

2

2134 p p

p p p p p

(2.12)

2

3425 bbbb

.2

181620

1

3648120

12

181620

1

3640

1

2415

1

20

2

1

2

2

1

2

213

4

p

p p p p p p

(2.13)

Recall that the real numbers ,23 56 , ,

2415

1

3640

1 and

181620

1 are nonnegative.

We eliminate 2 p in each of the terms in the curly brackets in (2.10) - (2.13) using the equality (2.3).

For instance, we have from (2.10),

.2

2223

65

223

2122

12

1

2

12

p p p p

(2.14)

Since ,02

2

2

1 p the absolute value of (2.14) attains its maximum for 21 p provided .6

5

Thus (2.14) yields

,23

652

223

212 212

p

p

(2.15)

so that, by triangle inequality and lemma 2.2, (2.10) yields the first inequality of the theorem.

Similar arguments and computation from (2.11) to (2.13), lead to the remaining inequalities

respectively.

For each of the real numbers ,,, and , equality is attained in each case by the function given

by

.1

0

z z

z K

Lemma 2.5[3] Let 0 and z D be starlike function in E . Let z N be analytic in E

and 000 D N , 100 D N , then

0Re z D z N

for z in E , whenever

.,01Re E z z D z N

z D z N


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Lemma 2.6. Under the same conditions of lemma 2.5,

Bz

Az

z D

z N

1

1

whenever

Bz Az z D z N z D z N 111

.,11, E z A B

Proof. By definition of subordination,

.,1

11 U z w

z Bw

z Aw

z D

z N

z D

z N

Taking real part,

z Bw

z Aw

z D

z N

z D

z N

1

1Re1Re

r z Br

Ar ,

1

1

B

A

1

1

(say)

.10

The above result can be written as

01Re1

1

z D z N

z D

z N (2.16)

Setting

1

z D z N z M

, (2.17)

(2.16) takes the form

.01Re z D

z M

z D

z M

So using lemma 2.5,we get

.0Re z D

z M

From (2.17) , it yields

.1

1Re

B A

z D z N

So

.1

1

Bz Az

z D z N

Lemma 2.7[1] Let z g z f , and z h are analytic in E and z h is convex univalent such that


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z g z f and z h z g ,

then z h z g z f 1 for .10

Except the above lemmas, we shall depend on the following well known coefficient functionals for

convex functions:

;2,1 nb n (2.18)

12

23 bb provided .

3

2 (2.19)

3. Integral Representation Formula

Theorem 3.1. A function z f is in B A J ,; if and only if there exists a convex function z h and

a function z p which is regular and has a positive real part in E such that

z t cc dt duu puhuht ht

c z f

0 0

.1

(3.1)

where .0,11

c

If0

, then

.

0 dt t pt

t h

z f

z

(3.2)

( powers in (3.1) are meant as principal values)

Proof. Let ,0,; B A J z f then

z p z h z f z

z h z f z

1 ,

where z p is a function with positive real part in E.

Dividing by and putting ,11

c we get

.1 z pc z h z f z

z h z f z

c (3.3)

Multiplying (3.3) by z h z h c , we obtain

z p z h z hc z f z z h z h z h z f cz ccc 11

which implies

.1 z p z h z hc z h z f z cc

(3.4)

Therefore on integrating (3.4) with respect to z , we obtain (3.1).


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Conversely, if z f satisfies (3.1), then it is easy to see that .,; B A J z f

On choosing z z

z h1

and Bz

Az z P

1

1 in (3.1), we obtain the following function of

:,; B A J

z t

c

ccc dt du

Buu

Auut t c z f

0 0

2

1

11

111 ,

11

c . (3.5)

4. Coefficient Bounds

Theorem 4.1. If B A J z f , , then

,2

)1(1 B Ann

an 2n .

This result was proved by Mehrok and Singh in [7].

Theorem 4.2. If B A J z f , , then

,

2)1(

11 B Ann

an 2n . (4.1)

Proof. As B A J z f , , then by definition of subordination

z h

z f z

z Bw

z Aw

1

1 1

.,1k

k k U z w z p

(4.2)

Using (1.2) and (1.4) in (4.2), we get

......321......941 123212232 nnnn z nb z b z b z an z a z a . ......1 11

2

21

n

n z p z p z p . (4.3)

On equating the coefficients of 1n z in (4.3), we have

nnnnn nb pbn pb pan 112212 1...2 .

So

nnnnn bn pbn pb pan 112212 1...2 . (4.4)

Using (2.2) and lemma 2.1 in (4.4), (4.1) can be easily obtained.

For 1,1 B A , theorem 4.2 gives the following result :

Corollary 4.3. If J z f , then

.2,1 na n

Theorem 4.4. If B A J z f ,; , then


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,122

12

B Aa

(4.5)

,13

2

3

13

B Aa

(4.6)

,14

3

4

14

B Aa

(4.7)

,15

4

5

15

B Aa

(4.8)

.16

5

6

16

B Aa

(4.9)

The bounds are sharp.

Proof. Since B A J z f ,; , so by definition of subordination and using (2.1),

z h z h z p z f z z h z h z f z 1

(4.10)

Using (1.2),(1.4) and (2.1) in (4.10) , it yields

.. .654321... .65432

15

6

4

5

3

4

2

32

6

6

5

5

4

4

3

3

2

2

z b z b z b z b z b

z a z a z a z a z a z

.. .... .362516941

6

6

5

5

4

4

3

3

2

2

5

6

4

5

3

4

2

32

z b z b z b z b z b z

z a z a z a z a z a

...1 66

5

5

4

4

3

3

2

21 z p z p z p z p z p z p

. ...654321... 5645342326655443322 z b z b z b z b z b z b z b z b z b z b z (4.11)On equating the coefficients of z 2 , z 3 , z 4 , z 5 and z 6 respectively in (4.11) , we get

,112 122 pba

(4.12)

,222123223 23214213 pb pbbbaa

(4.13)

324432234 55436236314 bbbbbabaa ,324 3221223 p pb pbb (4.14)

1325422354233245 56634524918415 pbbbbbbbbababaa

,3225 4322322214 p pb pb pb pb (4.15)

6524334256 563524334325516 bbabababaa

1514212362534 663777 pb pbb pbbbbbb

.34255 5423332224232 p pb pb pb pb pbb (4.16)

Using (4.12) in (4.13) , we get


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.1312112113 21233 p pbba

(4.17)

Again using (4.12) and (4.13) in (4.14) , it yields

1344 512131211312114 pbba

.1211511 122322 pb p pb (4.18)

Then using (4.12), (4.13) and (4.14) in (4.15), it gives

,5 13272

2261325443232321415050 pb A pb A pbb A p A pb A pb A pb Ab Aa A (4.19)

where ,41312110 A ,7131211 A

,813112 A

,712113 A

,312114 A

,52125 A

,12 26 A ,67 A A

From (4.12) , (4.13) , (4.14) , (4.15) and (4.16), we have

13

2

281

2

3712465542433324215160606 pbb B pb B pbb B p B pb B pb B pb B pb Bb Ba B

,322122

3211223101

429 pb B pb B pbb B pb B

(4.20)

where ,5141312110 B ,914131211 B

,111413112 B

,111412113 B

,91312114 B

,41312115 B

,3116 26 B

,4121147 B

,3445111 28 B

,3212 29 B

,7214 210 B

,911 B B

.2113 212 B

From (4.12) and (4.13) , using lemma 2.1 and inequality (2.16), we can easily obtain (4.5) and (4.6).

Now from (4.14) , using lemma 2.1 and inequality (2.16), we get

211312114312114 4 B Aa

.

5121

151215112 2

23 bb B A B A

(4.21)

It is easily verified that

32

5121

1

for all .0

So using inequality (2.17) in (4.14) , we obtain (4.7).

Now for n=5 , if ,1 we define

2

61

A A

and .5

72 A

A


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Therefore (4.19) can be written as

,5 2223125443232213221415050 bb pb A p A pb Abb p A pb Ab Aa A (4.22)

and for ,10 we define

1

5

A

A and rewrite (4.19) as

.5 132744323221322324115050 pb A p A pb Abb p Abbb p Ab Aa A (4.23)

We can easily verify that3

2, 21 for all .0

So also6

5 , for .10

Using lemma 2.1, lemma 2.4, inequalities (2.16) and (2.17) in (4.22) and (4.23) , we obtain (4.8). Now we compute the bound for .6a

We define

1

6

1

8

2

11

2

10

8

92

3

121 ,,,,, B

B

B

B

B

B

B

B

B

B

B

B and .

1

7

B

B

For ,1 we write (4.20) as

55424333323242232251160606 p B pb B pb Bbbbb p Bbbb p Bb Ba B ,` 3

22121

4291

2371246 pb B pb B pb B pbb B

(4.24)

and for ,10 we rewrite (4.20) as

42422133332422234251160606 pb Bbb p Bbbb p Bbbbb p Bb Ba B .`

2

3

211

2

2231

2

2855 pb Bbb pb B p B

(4.25)

Again, it is easy to verify that for all ,0 the real numbers ,,,,, 21 and defined in (4.24)

and (4.25) all satisfy (as appropriate) the conditions of lemma 2.4 and inequality (2.17).

Thus using lemma 2.1 , lemma 2.4 , inequalities (2.16) and (2.17) in (4.24) and (4.25) , we obtain

the bound for .6a

The extremal function is obtained by choosing z z

z g 1

and Bz Az

z p1

1in the integral

representation formula proved in Theorem 3.1.

Conjecture: If B A J z f ,; , then

.2,1

11 n

n

B An

na

n


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On putting 1,1 B A in Theorem 4.4, we obtain the following result:

Corollary 4.5. If J z f , then

,

12

32

a

,

13

53

a

,

14

74

a

,

15

95

a

.1611

6

a

5. Inclusion Relations

Theorem 5.1. If B A J z f ,; , then ., B A J z f

Proof. It is well known that

Bz

Az z f

1

1 if and only if .

1

1Re

B

A z f

As B A J z f ,, , so

E z Bz Az

z h z f z

z h z f z

,1

11 , (5.1)

which implies that

.1

11Re

B A

z h z f z

z h z f z

Also for 2k

k k z a z z f is analytic in E , ,1000 f f z h z D is convex(as every

convex function is starlike) in E ,

,1000 D D from lemma 2.5 and (5.1) , we have

,,1

1Re E z

B A

z h z f z

which implies that ., B A J z f Theorem 5.2. Let ,10 , then


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Copyright 2012 by Modern Scientific Press Company Florida USA

83

.,;,; B A J B A J Proof. Let ,,; B A J f then

.1

11 Bz

Az z h z f z

z h z f z

Also by Theorem 5.1,

.

1

1

Bz Az

z h z f z

As Bz

Az

1

1

is convex univalent , so by lemma 2.7, it yields

Bz

Az z h z f z

z h z f z

z h z f z

1

111

for .10

So

,1

11

Bz Az

z h z f z

z h z f z

which implies that .,; B A J f Hence .,;,; B A J B A J

References

[1] S. Bernardi, Special classes of subordinate functions, Duke Math. J ., 33(1)( 1993 ): 13-23.

[2] P. N. Chichra, New subclasses of the class of close-to-convex functions, Proc. Amer. Math. Soc .,

62(1)( 1977 ): 37-43.

[3] Gao Chunyi and Shigeyoshi Owa, Certain class of analytic functions in the unit disc,

Kyungpook Math. Journal , 33(1)( 1993 ): 13-23.

[4] P. L. Duren, Coefficients of univalent functions, Bull . Amer. Math. Soc ., 83( 1977 ): 891-911.

[5] H. R. Abdel-Gawad and D. K. Thomas, A subclass of close-to-convex functions, Publications DeLInstitut Mathmatique , Nouvelle srie tome , 49(63)( 1991 ): 61-66.

[6] R. M. Goel and B. S. Mehrok, A subclass of univalent functions, Houston J.Math .,8(3)( 1982 ):

343-357.

[7] B. S. Mehrok and Gagandeep Singh, A Subclass of close-to-convex functions, Int. Journal of

Math. Analysis, 4 (2010 ): 1319-1327.

[8] Ch. Pommerenke, Univalent functions . Vandenhoeck and Ruprecht, Gttingen, 1975 .

coefficient estimate for a subclass of close to convex functions

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