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Journal of the Egyptian Mathematical Society (2013) xxx, xxx–xxx

Egyptian Mathematical Society

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Subclasses of bi-univalent functions defined by convolution R.M. El-Ashwah Department of Mathematics, Faculty of Science, Damietta University, New Damietta 34517, Egypt Received 10 April 2013; accepted 8 June 2013

KEYWORDS Analytic and univalent functions; Bi-univalent functions; Starlike and convex functions; Coefficients bounds

Abstract In this paper, we introduced two new subclasses of the function class R of bi-univalent functions analytic in the open unit disc defined by convolution. Furthermore, we find estimates on the coefficients Œa2 Œ and Œa3 Œ for functions in these new subclasses. 2000 MATHEMATICS SUBJECT CLASSIFICATION:

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For 0 6 a < 1 and k P 0, we let Qk(h, a) be the subclass of A consisting of functions f(z) of the form (1.1) and functions h(z) given by

1. Introduction and definitions Let A denote the class of functions of the form: fðzÞ ¼ z þ

30C45; 30C55; 30C80

1 X an zn ;

ð1:1Þ

hðzÞ ¼ z þ

1 X hn zn

ðhn > 0Þ

ð1:4Þ

n¼2

n¼2

which are analytic in the open unit disc U ¼ fz : jzj < 1g. Further, by S we shall denote the class of all functions in A which are univalent in U. For f(z) defined by (1.1) and U(z) defined by 1 X UðzÞ ¼ z þ /n zn

ð/n P 0Þ;

the Hadamard product (f \ U)(z) of the functions f(z) and U(z) defined by 1 X an /n zn ¼ ðU  fÞðzÞ:

ð1:3Þ

n¼2

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    ðf  hÞðzÞ Qk ðh; aÞ ¼ f 2 A : Re ð1  kÞ þ kðf  hÞ0 ðzÞ > a; 0 6 a < 1; k P 0 : z ð1:5Þ

ð1:2Þ

n¼2

ðf  UÞðzÞ ¼ z þ

and satisfying the analytic criterion:

It is easy to see that Qk1 ðh; aÞ  Qk2 ðh; aÞ for k1 > k2 P 0. Thus, for k P 1, 0 6 a < 1, Qk ðh; aÞ  Q1 ðh; aÞ ¼ ff; h 2 A : Reðf  hÞ0 ðzÞ > a; 0 6 a < 1g and hence Qk(h, a) is univalent class (see [1–3]).  z  We note that Qk 1z ; a ¼ Qk ðaÞ (see Ding et al. [4]). It is well known that every function f 2 S has an inverse f 1, defined by f1 ðfðzÞÞ ¼ z

ðz 2 UÞ

and fðf1 ðwÞÞ ¼ w

  1 jwj < r0 ðfÞ; r0 ðfÞ P ; 4

1110-256X ª 2013 Production and hosting by Elsevier B.V. on behalf of Egyptian Mathematical Society. http://dx.doi.org/10.1016/j.joems.2013.06.017

Please cite this article in press as: R.M. El-Ashwah, Subclasses of bi-univalent functions defined by convolution, Journal of the Egyptian Mathematical Society (2013), http://dx.doi.org/10.1016/j.joems.2013.06.017

2

R.M. El-Ashwah  !   0 ðf  hÞ1 ðwÞ   þ kððf  hÞ1 Þ ðwÞ  arg ð1  kÞ   w

where     f1 ðwÞ ¼ w  a2 w2 þ 2a22  a3 w3  5a32  5a2 a3 þ a4 w4 þ :


0 and have the form p(z) = 1 + p1z + p2z2 + p3z3 +    for z 2 U. Then ŒpnŒ 6 2, for each n.

qðwÞ ¼ 1 þ q1 w þ q2 w2 þ q3 w3 þ    : Now, equating the coefficients in (2.6) and (2.7), we get ðk þ 1Þa2 h2 ¼ ap1 ;

2. Coefficient bounds for the function class Bðh; a; kÞ

f2R ap < 2 and

     ðf  hÞðzÞ and arg ð1  kÞ þ kðf  hÞ0 ðzÞ  z ð0 < a 6 1; k P 1; z 2 UÞ

ð2:10Þ

aða  1Þ 2 ð2k þ 1Þa3 h3 ¼ ap2 þ p1 ; 2  ðk þ 1Þa2 h2 ¼ aq1

Definition 1. A function f(z) given by (1.1) is said to be in the class BR ðh; a; kÞ if the following conditions are satisfied:

ð2:9Þ

ð2:11Þ ð2:12Þ

and ð2k þ 1Þð2a22 h22  a3 h3 Þ ¼ aq2 þ

aða  1Þ 2 q1 : 2

ð2:13Þ

From (2.10) and (2.12), we get p1 ¼ q1 ð2:1Þ

ð2:14Þ

and 2ðk þ 1Þ2 a22 h22 ¼ a2 ðp21 þ q21 Þ:

ð2:15Þ

Please cite this article in press as: R.M. El-Ashwah, Subclasses of bi-univalent functions defined by convolution, Journal of the Egyptian Mathematical Society (2013), http://dx.doi.org/10.1016/j.joems.2013.06.017

Subclasses of bi-univalent functions defined by convolution Now from (2.11), (2.13) and (2.15), we obtain

1

ja3 j 6 h

aða  1Þ 2 ðp1 þ q21 Þ 2 aða  1Þ 2ðk þ 1Þ2 a22 h22 : ¼ aðp2 þ q2 Þ þ a2 2

1þ‘þ2c 1þ‘

2ð2k þ 1Þa22 h22 ¼ aðp2 þ q2 Þ þ

im

4a2 ðk þ 1Þ

2

þ

2a : ð2k þ 1Þ

(ii) For hðzÞ ¼ z þ

Therefore, we have a22 ¼

3 !

1 X Cn1 ða1 Þzn ;

ð2:18Þ

n¼2

a2 ðp2 þ q2 Þ 2að2k þ 1Þh22  ða  1Þðk þ 1Þ2 h22

where

:

Applying Lemma 1 for the coefficients p2 and q2, we immediately have

ða1 Þn1    ðaq Þn1 ðb1 Þn1    ðbs Þn1 ð1Þn1

Cn1 ða1 Þ ¼

ðn P 2Þ;

ð2:19Þ

2a ja2 j 6 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi : 2 h2 ðk þ 1Þ þ að1 þ 2k  k2 Þ

q 6 s þ 1; ai 2 Cði ¼ 1; 2; . . . ; qÞ and b1 2 C n Z ðj ¼ 1; 2; . . . ; sÞ, this operator contains in turn 0 many interesting operators (see Dziok and Srivastav [12]), Theorem 1 becomes

This gives the bound on Œa2Œ as asserted in (2.4). Next, in order to find the bound on Œa3Œ, by subtracting (2.13) from (2.4), we get

ja2 j 6

2ð2k þ 1Þa3 h3  2ð2k þ 1Þa22 h22   aða  1Þ 2 aða  1Þ 2 ¼ ap2 þ p1  aq2 þ q1 : 2 2

2a qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi jC1 ða1 Þj ðk þ 1Þ2 þ að1 þ 2k  k2 Þ

and ð2:16Þ

! 1 4a2 2a : ja3 j 6 þ jC2 ða1 Þj ðk þ 1Þ2 ð2k þ 1Þ

It follows from (2.14), (2.15) and (2.16) that 2ð2k þ 1Þa3 h3 ¼

a2 ð2k þ 1Þðp21 þ q21 Þ ðk þ 1Þ2

3. Coefficient bounds for the function class Bðh; b; kÞ

þ aðp2  q2 Þ

or, equivalently,

Definition 2. A function f(z) given by (1.1) is said to be in the class BR ðh; b; kÞ if the following conditions are satisfied:

a2 ðp21 þ q21 Þ

aðp2  q2 Þ a3 ¼ þ 2ðk þ 1Þ2 h3 2ð2k þ 1Þh3 applying Lemma 1 once again for the coefficients p1, p2, q1 and q2, we readily get ! 1 4a2 2a : ja3 j 6 þ h3 ðk þ 1Þ2 ð2k þ 1Þ This completes the proof of Theorem 1.

h

Example 1

1 þ ‘ þ cðn  1Þ 1þ‘

m

zn ð‘;c P 0; m 2 N0 ¼ N [ f0gÞ;

ð3:1Þ

and ! >b

ð0 ð3:2Þ

where the functions h(z) and (f \ h)1(w) are defined by (1.4) and (2.3) respectively. z We note that for k = 1 and hðzÞ ¼ 1z , the class BR ðh; b; kÞ reduces to the class HR ðbÞ introduced and studied by Srivastz ava et al. [5]. Also for hðzÞ ¼ 1z , the class BR ðh; b; kÞ reduces to the class BR ðb; kÞ introduced and studied by Frasin and Aouf [10].

ð2:17Þ

this operator contains in turn many interesting operators (see Catas et al. [11]), Theorem 1, becomes 2a im qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1þ‘þc ðk þ 1Þ2 þ að1 þ 2k  k2 Þ 1þ‘

ja2 j 6 h

ð0 6 b < 1; k P 1; z 2 UÞ

Theorem 2. Let f(z) given by (1.1) be in the class BR ðh; b; kÞ; 0 6 b < 1 and k P 1. Then

(i) For n¼2

  ðf  hÞðzÞ Re ð1  kÞ þ kðf  hÞ0 ðzÞ > b z

6 b < 1; k P 1; w 2 UÞ;

z (i) Putting k = 1 and hðzÞ ¼ 1z in Theorem 1, we obtain the results obtained by Srivastava et al. [5, Theorem 1]. z in Theorem 1, we obtain the results (ii) Putting hðzÞ ¼ 1z obtained by Frasin and Aouf [10, Theorem 2.2].

hðzÞ ¼ z þ

and

0 ðf  hÞ1 ðwÞ Re ð1  kÞ þ kððf  hÞ1 Þ ðwÞ w

Remark 1

1

X

f2R

1 ja2 j 6 h2

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2ð1  bÞ ð2k þ 1Þ

ð3:3Þ

and 1 ja3 j 6 h3

! 2ð1  bÞ þ : ð2k þ 1Þ ðk þ 1Þ2

4ð1  bÞ2

ð3:4Þ

and

Please cite this article in press as: R.M. El-Ashwah, Subclasses of bi-univalent functions defined by convolution, Journal of the Egyptian Mathematical Society (2013), http://dx.doi.org/10.1016/j.joems.2013.06.017

4

R.M. El-Ashwah

Proof. It follows from (3.1) and (3.2) that there exist p(z) and qðzÞ 2 P such that ð1  kÞ

ðf  hÞðzÞ þ kðf  hÞ0 ðzÞ ¼ b þ ð1  bÞpðzÞ z

z (ii) Putting hðzÞ ¼ 1z in Theorem 1, we obtain the results obtained by Frasin and Aouf [10, Theorem 3.2].

ð3:5Þ Example 2

and 0 ðf  hÞ1 ðwÞ þ kððf  hÞ1 Þ ðwÞ w ¼ b þ ð1  bÞqðwÞ;

ð1  kÞ

ð3:6Þ

(i) For h(z) given by (2.17), Theorem 2, becomes sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 2ð1  bÞ im ja2 j 6 h 1þ‘þc ð2k þ 1Þ 1þ‘

where p(z) and q(w) have the forms (2.8) and (2.9), respectively. Equating coefficients in (3.5) and (3.6) yields

and

ðk þ 1Þa2 h2 ¼ ð1  bÞp1 ; ð2k þ 1Þa3 h3 ¼ ð1  bÞp2 ;  ðk þ 1Þa2 h2 ¼ ð1  bÞq1

ja3 j 6 h

ð3:7Þ ð3:8Þ ð3:9Þ

and ð2k þ 1Þð2a22 h22  a3 h3 Þ ¼ ð1  bÞq2 :

ð3:10Þ

From (3.7) and (3.9), we get

4ð1  bÞ2 ðk þ 1Þ2

þ

! 2ð1  bÞ : ð2k þ 1Þ

(ii) For h(z) given by (2.18), Theorem 2, becomes sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 2ð1  bÞ ja2 j 6 jC1 ða1 Þj ð2k þ 1Þ

ð3:11Þ ja3 j 6

and   2ðk þ 1Þ2 a22 h22 ¼ ð1  bÞ2 p21 þ q21 :

! 1 4ð1  bÞ2 2ð1  bÞ þ : jC2 ða1 Þj ðk þ 1Þ2 ð2k þ 1Þ

ð3:12Þ

Also, from (3.8) and (3.10), we find that

References

2ð2k þ 1Þa22 h22 ¼ ð1  bÞðp2 þ q2 Þ: Thus, we have ð1  bÞ 2ð1  bÞ ðjp2 j þ jq2 jÞ ¼ ; 2ð2k þ 1Þh22 ð2k þ 1Þh22

which is the bound on Œa2Œ as given in (3.3). Next, in order to find the bound on Œa3Œ, by subtracting (3.8) form (3.10), we get 2ð2k þ 1Þa3 h3  2ð2k þ 1Þa22 h22 ¼ ð1  bÞðp2  q2 Þ or, equivalently,   1 ð1  bÞðp2  q2 Þ : a3 ¼ a22 h22 þ h3 2ð2k þ 1Þ Upon substituting the value of a22 from (3.12), we obtain !  2 1 ð1  bÞ p21 þ q21 ð1  bÞðp2  q2 Þ a3 ¼ þ : h3 2ð2k þ 1Þ 2ðk þ 1Þ2 Applying Lemma 1 for the coefficients p1, p2, q1 and q2, we readily get ! 1 4ð1  bÞ2 2ð1  bÞ ja3 j 6 þ ; h3 ðk þ 1Þ2 ð2k þ 1Þ which is the bound on Œa3Œ as asserted in (3.4).

1þ‘þ2c 1þ‘

im

and

p1 ¼ q1

ja22 j 6

1

h

Remark 2 z (i) Putting k = 1 and hðzÞ ¼ 1z in Theorem 1, we obtain the results obtained by Srivastava et al. [5, Theorem 2].

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Please cite this article in press as: R.M. El-Ashwah, Subclasses of bi-univalent functions defined by convolution, Journal of the Egyptian Mathematical Society (2013), http://dx.doi.org/10.1016/j.joems.2013.06.017