Hindawi Publishing Corporation International Journal of Analysis Volume 2016, Article ID 6958098, 5 pages http://dx.doi.org/10.1155/2016/6958098
Research Article Coefficient Estimates for Certain Subclasses of Biunivalent Functions Defined by Convolution R. Vijaya,1 T. V. Sudharsan,2 and S. Sivasubramanian3 1
Department of Mathematics, SDNB Vaishnav College for Women, Chromepet, Chennai, Tamil Nadu 600044, India Department of Mathematics, SIVET College, Gowrivakkam, Chennai, Tamil Nadu 600073, India 3 Department of Mathematics, University College of Engineering Tindivanam, Anna University, Tindivanam, Tamil Nadu 604001, India 2
Correspondence should be addressed to S. Sivasubramanian;
[email protected] Received 7 July 2016; Accepted 25 October 2016 Academic Editor: Julien Salomon Copyright © 2016 R. Vijaya et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. We introduce two new subclasses of the function class Σ of biunivalent functions in the open disc defined by convolution. Estimates on the coefficients |𝑎2 | and |𝑎3 | for the two subclasses are obtained. Moreover, we verify Brannan and Clunie’s conjecture |𝑎2 | ≤ √2 for our subclasses.
1. Introduction
Indeed, the inverse function may have an analytic continuation to Δ, with
Let A denote the class of functions of the form ∞
𝑗
𝑓 (𝑧) = 𝑧 + ∑ 𝑎𝑗 𝑧 , 𝑗=2
(𝑎𝑗 ≥ 0)
𝑓−1 (𝑤) = 𝑤 − 𝑎2 𝑤2 + (2𝑎22 − 𝑎3 ) 𝑤3 (1)
which are analytic in the open disc Δ = {𝑧 : |𝑧| < 1} and normalized by 𝑓(0) = 0, 𝑓 (0) = 1. Let S be the subclass of A consisting of univalent functions 𝑓(𝑧) of form (1). For 𝑓(𝑧) defined by (1) and ℎ(𝑧) defined by ∞
ℎ (𝑧) = 𝑧 + ∑ ℎ𝑗 𝑧𝑗 , 𝑗=2
(ℎ𝑗 ≥ 0) ,
(2)
the Hadamard product (or convolution) of 𝑓 and ℎ is defined by ∞
(𝑓 ∗ ℎ) (𝑧) = 𝑧 + ∑ 𝑎𝑗 ℎ𝑗 𝑧𝑗 = (ℎ ∗ 𝑓) (𝑧) . 𝑗=2
(3)
It is well known that every function 𝑓 ∈ 𝑆 has an inverse 𝑓−1 defined by 𝑓−1 (𝑓 (𝑧)) = 𝑧 𝑓 (𝑓−1 (𝑤)) = 𝑤
(𝑧 ∈ Δ) , 1 (|𝑤| < 𝑟0 (𝑓) : 𝑟0 (𝑓) ≥ ) . 4
(4)
− (5𝑎23 − 5𝑎2 𝑎3 + 𝑎4 ) 𝑤4 + ⋅ ⋅ ⋅ .
(5)
A function 𝑓 ∈ A is said to be biunivalent in Δ if 𝑓(𝑧) and 𝑓−1 (𝑧) are univalent in Δ. Let Σ denote the class of biunivalent functions in Δ given by (1). In 1967, Lewin [1] investigated the biunivalent function class Σ and showed that |𝑎2 | < 1.51. Brannan and Clunie [2] conjectured that |𝑎2 | ≤ √2. Netanyahu [3] introduced certain subclasses of biunivalent function class Σ similar to the familiar subclasses S∗ (𝛼) and K(𝛼) of starlike and convex functions of order 𝛼 (0 < 𝛼 ≤ 1). Brannan and Taha [4] defined 𝑓 ∈ A in the class SΣ (𝛼) of strongly bistarlike functions of order 𝛼 (0 < 𝛼 ≤ 1) if each of the following conditions is satisfied: 𝑓 ∈ Σ, 𝑧𝑓 (𝑧) arg ( < ) 𝑓 (𝑧) 𝑤𝑔 (𝑤) arg ( < ) 𝑔 (𝑤)
𝛼𝜋 2
(0 < 𝛼 ≤ 1, 𝑧 ∈ Δ) ,
𝛼𝜋 2
(0 < 𝛼 ≤ 1, 𝑤 ∈ Δ) ,
(6)
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International Journal of Analysis
where 𝑔 is as defined by (5). They also introduced the class of all bistarlike functions of order 𝛽 defined as a function 𝑓 ∈ A, which is said to be in the class S∗Σ (𝛽) if the following conditions are satisfied: 𝑓 ∈ Σ, 𝑧𝑓 (𝑧) ) > 𝛽, R( 𝑓 (𝑧)
(𝑧 ∈ Δ; 0 ≤ 𝛽 < 1) ,
𝑤𝑔 (𝑤) ) > 𝛽, 𝑔 (𝑤)
(𝑤 ∈ Δ; 0 ≤ 𝛽 < 1) ,
R(
(8)
Definition 2. A function 𝑓(𝑧) given by (1) is said to be in the class 𝑆Σ (ℎ, 𝛼, 𝜆), if the following conditions are satisfied: 𝑓 ∈ Σ,
𝛼𝜋 < , 0 < 𝛼 ≤ 1, 𝜆 ≥ 0, 𝑤 ∈ Δ, 2
(9)
(11)
− ⋅⋅⋅ . Clearly, 𝑆Σ (𝑧/(1 − 𝑧), 𝛼, 0) ≡ 𝑆Σ (𝛼), the class of all strong bistarlike functions of order 𝛼 introduced by Brannan and Taha [2]. Definition 3. A function 𝑓(𝑧) given by (1) is said to be in the class 𝑆Σ∗ (ℎ, 𝛽, 𝜆), if the following conditions are satisfied: 𝑓 ∈ Σ,
R[
𝑧 (𝑓 ∗ ℎ) (𝑧) + 𝜆𝑧2 (𝑓 ∗ ℎ) (𝑧) ]>𝛽 (𝑓 ∗ ℎ) (𝑧) 0 ≤ 𝛽 < 1, 𝜆 ≥ 0, 𝑧 ∈ Δ, (12) −1
R[
−1
𝑤 ((𝑓 ∗ ℎ) ) (𝑤) + 𝜆𝑤2 ((𝑓 ∗ ℎ) ) (𝑤) −1
((𝑓 ∗ ℎ) ) (𝑤)
[ > 𝛽,
] ]
0 ≤ 𝛽 < 1, 𝜆 ≥ 0, 𝑤 ∈ Δ,
where ℎ(𝑧) and (𝑓 ∗ ℎ)−1 (𝑤) are defined, respectively, as in (2) and (10). Clearly, 𝑆Σ (𝑧/(1 − 𝑧), 𝛽, 0) ≡ 𝑆Σ∗ (𝛽), the class of all strong bistarlike functions of order 𝛽 introduced by Brannan and Taha [2]. Theorem 4. Let 𝑓(𝑧) given by (1) be in the class 𝑆Σ (ℎ, 𝛼, 𝜆), 0 < 𝛼 ≤ 1 and 𝜆 ≥ 0. Then 𝑎2 ≤ 𝑎3 ≤
2 arg ( 𝑧 (𝑓 ∗ ℎ) (𝑧) + 𝜆𝑧 (𝑓 ∗ ℎ) (𝑧) ) < 𝛼𝜋 , 2 (𝑓 ∗ ℎ) (𝑧) −1 −1 𝑤 ((𝑓 ∗ ℎ) ) (𝑤) + 𝜆𝑤2 ((𝑓 ∗ ℎ) ) (𝑧) ) arg ( −1 ((𝑓 ∗ ℎ) ) (𝑤)
−1
((𝑓 ∗ ℎ) (𝑤)) = 1 − 2𝑎2 ℎ2 𝑤 + 3 (2𝑎22 ℎ22 − 𝑎3 ℎ3 ) 𝑤2
2. Coefficient Bounds for the Classes 𝑆Σ (ℎ, 𝛼, 𝜆) and 𝑆Σ∗ (ℎ, 𝛽, 𝜆)
0 < 𝛼 ≤ 1, 𝜆 ≥ 0, 𝑧 ∈ Δ,
+ ⋅⋅⋅
(7)
Lemma 1 (see [12]). If 𝑝 ∈ P, then |𝑝𝑘 | ≤ 2 for each 𝑘, where P is the family of all functions 𝑝(𝑧) analytic in Δ for which Re{𝑝(𝑧)} > 0; ∀𝑧 ∈ Δ.
−1
(𝑓 ∗ ℎ) (𝑤) = 𝑤 − 𝑎2 ℎ2 𝑤2 + (2𝑎22 ℎ22 − 𝑎3 ℎ3 ) 𝑤3 − (5𝑎23 ℎ23 − 5𝑎2 ℎ2 𝑎3 ℎ3 + 𝑎4 ℎ4 ) 𝑤4 (10)
where the function 𝑔 is as defined in (5). The classes S∗Σ (𝛼) and KΣ (𝛼) of bistarlike functions of order 𝛽 and biconvex functions of order 𝛽, corresponding to the function classes S∗ (𝛽) and K(𝛽), were introduced analogously. For each of the function classes S∗Σ (𝛽) and K∗Σ (𝛽), they found nonsharp estimates on the first two Taylor-Maclaurin coefficients |𝑎2 | and |𝑎3 | (see [2, 5]). Some examples of biunivalent functions are 𝑧/(1 − 𝑧), (1/2) log((1 + 𝑧)/(1 − 𝑧)), and − log(1 − 𝑧) (see [6]). The coefficient estimate problem for each of the following Taylor-Maclaurin coefficients, |𝑎𝑛 | (𝑛 ∈ N, 𝑛 ≥ 3), is still open ([6]). Various subclasses of biunivalent function class Σ were introduced and nonsharp estimates on the first two coefficients |𝑎2 | and |𝑎3 | in the Taylor-Maclaurin series (1) were found in several investigations (see [7–11]). In this present investigation, motivated by the works of Brannan and Taha [2] and Srivastava et al. [6], we introduce two new subclasses of biunivalent functions involving convolution. The first two initial coefficients of each of these two new subclasses are obtained. Further, we prove that Brannan and Clunie’s conjecture is true for our subclasses. In order to derive our main results, we have to recall the following lemma.
𝑝 (𝑧) = 1 + 𝑝1 𝑧 + 𝑝2 𝑧2 + 𝑝3 𝑧3 + ⋅ ⋅ ⋅
where the function ℎ(𝑧) is defined by (2) and (𝑓 ∗ ℎ)−1 (𝑤) is defined by
2𝛼 ℎ2 √(𝛼 + 1) (4𝜆 + 1) + 4𝜆2 (𝛼 − 1) 4𝛼2 𝛼 . + 2 ℎ3 (1 + 3𝜆) ℎ3 (1 + 2𝜆)
, (13)
𝑛 Further, for the choice of ℎ(𝑧) = 𝑧/(1 − 𝑧)2 = 𝑧 + ∑∞ 𝑛=2 𝑛𝑧 , one gets
𝛼 , 𝑎2 ≤ √(𝛼 + 1) (4𝜆 + 1) + 4𝜆2 (𝛼 − 1) 𝑎3 ≤
4𝛼2 𝛼 + . 2 3 (1 + 3𝜆) 3 (1 + 2𝜆)
(14)
International Journal of Analysis
3 Next, in order to find the bound on |𝑎3 |, by subtracting (20) from (22), we get
Proof. It follows from (9) that
𝑧 (𝑓 ∗ ℎ) (𝑧) + 𝜆𝑧2 (𝑓 ∗ ℎ) (𝑧) 𝛼 = [𝑝 (𝑧)] , (𝑓 ∗ ℎ) (𝑧) −1
−1
𝑤 ((𝑓 ∗ ℎ) ) (𝑤) + 𝜆𝑤2 ((𝑓 ∗ ℎ) ) (𝑤)
4 (1 + 3𝜆) (𝑎3 ℎ3 − 𝑎22 ℎ22 ) (15)
−1
((𝑓 ∗ ℎ) ) (𝑤) 𝛼
= 𝛼 (𝑝2 − 𝑞2 ) + +
= [𝑞 (𝑤)] , where 𝑝(𝑧) and 𝑞(𝑤) satisfy the following inequalities: Re {𝑝 (𝑧)} > 0
(𝑧 ∈ Δ) ,
Re {𝑞 (𝑤)} > 0
(𝑤 ∈ Δ) .
(16)
Furthermore, the functions 𝑝(𝑧) and 𝑞(𝑤) have the forms 𝑝 (𝑧) = 1 + 𝑝1 𝑧 + 𝑝2 𝑧2 + 𝑝3 𝑧3 + ⋅ ⋅ ⋅ ,
(17)
𝑞 (𝑤) = 1 + 𝑞1 𝑤 + 𝑞2 𝑤2 + 𝑞3 𝑤3 + ⋅ ⋅ ⋅ .
(18)
𝛼 (𝛼 − 1) 2 (𝑝1 − 𝑞12 ) 2
𝛼2 (𝑝12 − 𝑞12 ) (1 + 2𝜆)
(27)
.
Upon substituting the value of 𝑎22 from (24) and observing that 𝑝12 = 𝑞12 , it follows that 𝑎3 = 𝑎22 +
𝛼 (𝑝2 − 𝑞2 ) 4ℎ3 (1 + 3𝜆)
𝛼2 (𝑝12 + 𝑞12 )
𝛼 (𝑝2 − 𝑞2 ) . + = 2 4ℎ3 (1 + 3𝜆) 2ℎ3 (1 + 2𝜆)
(28)
Applying Lemma 1 once again for the coefficients 𝑝1 , 𝑝2 , 𝑞1 , and 𝑞2 , we get
Now, equating the coefficients in (15), we get (1 + 2𝜆) 𝑎2 ℎ2 = 𝛼𝑝1
(19) 𝑎3 ≤
4𝛼2 𝛼 . + ℎ3 (1 + 2𝜆)2 ℎ3 (1 + 3𝜆)
𝛼2 𝑝12 𝛼 (𝛼 − 1) 2 2 (1 + 3𝜆) 𝑎3 ℎ3 = 𝑝2 𝛼 + 𝑝1 + 2 (1 + 2𝜆)
(20)
− (1 + 2𝜆) 𝑎2 ℎ2 = 𝛼𝑞1 ,
(21)
This completes the proof.
(22)
Remark 5. When ℎ(𝑧) = 𝑧/(1 − 𝑧) and 𝜆 = 0, in (13), we get the results obtained due to [4].
2 (1 + 3𝜆) (2𝑎22 ℎ22 − 𝑎3 ℎ3 ) = 𝑞2 𝛼 +
𝛼2 𝑞12 𝛼 (𝛼 − 1) 2 . 𝑞1 + 2 (1 + 2𝜆)
Remark 6. When 𝜆 = 0, 𝛼 = 1, and ℎ2 = 1, we obtain Brannan and Clunie’s [2] conjecture |𝑎2 | ≤ √2.
From (19) and (21), we get 𝑝1 = −𝑞1 ,
(23)
2 (1 + 2𝜆)2 𝑎22 ℎ22 = 𝛼2 (𝑝12 + 𝑞12 ) .
(24)
Now, from (20), (22), and (24), we obtain 4 (1 + 3𝜆) 𝑎22 ℎ22 = 𝛼 (𝑝2 + 𝑞2 ) + +
𝛼 (𝛼 − 1) 2 (𝑝1 + 𝑞12 ) 2
𝛼2 (𝑝12 + 𝑞12 ) (1 + 2𝜆)
ℎ2 √(𝛼 + 1) (4𝜆 + 1) + 4𝜆2 (1 − 𝛼)
This gives the bound on |𝑎2 |.
1 √ 2 (1 − 𝛽) , 𝑎2 ≤ ℎ2 (1 + 4𝜆) 2
(30)
(25) 𝑛 Further, for the choice of ℎ(𝑧) = 𝑧/(1 − 𝑧)2 = 𝑧 + ∑∞ 𝑛=2 𝑛𝑧 , we get
.
2𝛼
Theorem 7. Let 𝑓(𝑧) given by (1) be in the class 𝑆Σ∗ (ℎ, 𝛽, 𝜆), 0 ≤ 𝛽 < 1 and 𝜆 ≥ 0. Then
(1 − 𝛽) 1 4 (1 − 𝛽) ]. [ + 𝑎3 ≤ 2 ℎ3 (1 + 2𝜆) (1 + 3𝜆)
Applying Lemma 1 for the coefficients 𝑝2 and 𝑞2 , we immediately have 𝑎2 ≤
(29)
.
(26)
√ (1 − 𝛽) , 𝑎2 ≤ (1 + 4𝜆)
(31) 2
(1 − 𝛽) 1 4 (1 − 𝛽) ]. + 𝑎3 ≤ [ 3 (1 + 2𝜆)2 (1 + 3𝜆)
(32)
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International Journal of Analysis
Proof. It follows from (12) that there exist 𝑝(𝑧) and 𝑞(𝑤), such that
Applying Lemma 1 for the coefficients 𝑝1 , 𝑝2 , 𝑞1 , and 𝑞2 , we readily get 2
𝑧 (𝑓 ∗ ℎ) (𝑧) + 𝜆𝑧2 (𝑓 ∗ ℎ) (𝑧) (𝑓 ∗ ℎ) (𝑧)
(1 − 𝛽) 1 4 (1 − 𝛽) ]. [ + 𝑎3 ≤ 2 ℎ3 (1 + 2𝜆) (1 + 3𝜆)
(44)
= 𝛽 + (1 − 𝛽) 𝑝 (𝑧) , −1
(33)
−1
𝑤 ((𝑓 ∗ ℎ) ) (𝑤) + 𝜆𝑤2 ((𝑓 ∗ ℎ) ) (𝑤) −1
((𝑓 ∗ ℎ) ) (𝑤)
Remark 8. When ℎ(𝑧) = 𝑧/(1 − 𝑧) and 𝜆 = 0 in (30), we have the following result due to [4]. The bounds are √ 𝑎2 ≤ 2 (1 − 𝛽),
= 𝛽 + (1 − 𝑞) 𝑞 (𝑤) where 𝑝(𝑧) and 𝑞(𝑤) have forms (17) and (18), respectively. Equating coefficients in (33) we obtain (1 + 2𝜆) 𝑎2 ℎ2 = 𝑝1 (1 − 𝛽)
(34)
2 𝑎3 ≤ (1 − 𝛽) + 4 (1 − 𝛽) .
Remark 9. When ℎ(𝑧) = 𝑧/(1 − 𝑧) and 𝜆 = 0 in (30), we have the following result due to [4]. The bounds are √ 𝑎2 ≤ 2 (1 − 𝛽),
2
𝑝2 (1 − 𝛽) 2 (1 + 3𝜆) 𝑎3 ℎ3 = 𝑝2 (1 − 𝛽) + 1 (1 + 2𝜆)
(35)
− (1 + 2𝜆) 𝑎2 ℎ2 = 𝑞1 (1 − 𝛽) ,
(36) 2
𝑞2 (1 − 𝛽) 2 (1 + 3𝜆) (2𝑎22 ℎ22 − 𝑎3 ℎ3 ) = 𝑞2 (1 − 𝛽) + 1 . (37) (1 + 2𝜆) From (34) and (36), we get
(46)
Remark 10. When 𝜆 = 0, 𝛽 = 0, and ℎ2 = 1 we obtain Brannan and Clunie’s [2] conjecture |𝑎2 | ≤ √2.
Competing Interests
(38) 2
2 (1 + 2𝜆)2 𝑎22 ℎ22 = (1 − 𝛽) (𝑝12 + 𝑞12 ) .
(39)
4 (1 + 3𝜆) 𝑎22 ℎ22 = (1 − 𝛽) (𝑝2 + 𝑞2 ) 2
(1 − 𝛽) (𝑝12 + 𝑞12 ) (1 + 2𝜆)
(40)
References
.
Therefore, we have (1 − 𝛽) (𝑝2 + 𝑞2 ) . 2ℎ22 (1 + 4𝜆)
(41)
Applying Lemma 1, for the coefficients 𝑝2 and 𝑞2 , we immediately have 1 √ 2 (1 − 𝛽) . 𝑎2 ≤ ℎ2 (1 + 4𝜆)
(42)
Next, in order to find the bound on |𝑎3 |, by subtracting (35) from (37), we get 4 (1 + 3𝜆) (𝑎3 ℎ3 − 𝑎22 ℎ22 ) = (1 − 𝛽) (𝑝2 − 𝑞2 ) 2
Acknowledgments The work of the third author is supported by a grant from Department of Science and Technology, Government of India; vide Ref SR/FTP/MS-022/2012 under fast track scheme.
Now from (35), (37), and (39), we obtain
𝑎22 =
2 𝑎3 ≤ (1 − 𝛽) + 4 (1 − 𝛽) .
The authors declare that they have no competing interests.
𝑝1 = −𝑞1 ,
+
(45)
2 2 2 (1 − 𝛽) (𝑝2 − 𝑞2 ) (1 − 𝛽) (𝑝1 + 𝑞1 ) 𝑎3 = + . 4 (1 + 3𝜆) ℎ3 2ℎ3 (1 + 2𝜆)2
(43)
[1] M. Lewin, “On a coefficient problem for bi-univalent functions,” Proceedings of the American Mathematical Society, vol. 18, pp. 63–68, 1967. [2] “Aspects of contemporary complex analysis,” in Proceedings of the NATO Advanced Study Institute, D. A. Brannan and J. G. Clunie, Eds., University of Durham, July 1979. [3] E. Netanyahu, “The minimal distance of the image boundary from the origin and the second coefficient of a univalent function in |𝑧| < 1,” Archive for Rational Mechanics and Analysis, vol. 32, pp. 100–112, 1969. [4] D. A. Brannan and T. S. Taha, “On some classes of bi-univalent,” Studia Universitatis Babes, -Bolyai Mathematica, vol. 31, no. 2, pp. 70–77, 1986. [5] T. S. Taha, Topics in univalent functions theory [Ph.D. thesis], University of London, 1981. [6] H. M. Srivastava, A. K. Mishra, and P. Gochhayat, “Certain subclasses of analytic and bi-univalent functions,” Applied Mathematics Letters, vol. 23, no. 10, pp. 1188–1192, 2010. [7] R. M. El-Ashwah, “Subclasses of bi-univalent functions defined by convolution,” Journal of the Egyptian Mathematical Society, vol. 22, no. 3, pp. 348–351, 2014.
International Journal of Analysis [8] B. A. Frasin and M. K. Aouf, “New subclasses of bi-univalent functions,” Applied Mathematics Letters, vol. 24, no. 9, pp. 1569– 1573, 2011. [9] T. Hayami and S. Owa, “Coefficient bounds for bi-univalent functions,” Panamerican Mathematical Journal, vol. 22, no. 4, pp. 15–26, 2012. [10] Q.-H. Xu, Y.-C. Gui, and H. M. Srivastava, “Coefficient estimates for a certain subclass of analytic and bi-univalent functions,” Applied Mathematics Letters, vol. 25, no. 6, pp. 990–994, 2012. [11] Q.-H. Xu, H.-G. Xiao, and H. M. Srivastava, “A certain general subclass of analytic and bi-univalent functions and associated coefficient estimate problems,” Applied Mathematics and Computation, vol. 218, no. 23, pp. 11461–11465, 2012. [12] C. Pommerenke, Univalent Functions, Vandenheock and Ruprecht, G¨ottingen, Germany, 1975.
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