Research Article | Open Access
Coefficient Estimates for Certain Subclasses of Biunivalent Functions Defined by Convolution
We introduce two new subclasses of the function class of biunivalent functions in the open disc defined by convolution. Estimates on the coefficients and for the two subclasses are obtained. Moreover, we verify Brannan and Clunie’s conjecture for our subclasses.
Let denote the class of functions of the formwhich are analytic in the open disc and normalized by , . Let be the subclass of consisting of univalent functions of form (1).
For defined by (1) and defined bythe Hadamard product (or convolution) of and is defined by
It is well known that every function has an inverse defined by Indeed, the inverse function may have an analytic continuation to , with
A function is said to be biunivalent in if and are univalent in . Let denote the class of biunivalent functions in given by (1). In 1967, Lewin  investigated the biunivalent function class and showed that . Brannan and Clunie  conjectured that . Netanyahu  introduced certain subclasses of biunivalent function class similar to the familiar subclasses and of starlike and convex functions of order (). Brannan and Taha  defined in the class of strongly bistarlike functions of order () if each of the following conditions is satisfied:where is as defined by (5). They also introduced the class of all bistarlike functions of order defined as a function , which is said to be in the class if the following conditions are satisfied:where the function is as defined in (5). The classes and of bistarlike functions of order and biconvex functions of order , corresponding to the function classes and , were introduced analogously. For each of the function classes and , they found nonsharp estimates on the first two Taylor-Maclaurin coefficients and (see [2, 5]). Some examples of biunivalent functions are , , and (see ). The coefficient estimate problem for each of the following Taylor-Maclaurin coefficients, (, ), is still open (). Various subclasses of biunivalent function class were introduced and nonsharp estimates on the first two coefficients and 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  and Srivastava et al. , 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.
Lemma 1 (see ). If , then for each , where is the family of all functions analytic in for which ;
2. Coefficient Bounds for the Classes and
Definition 2. A function given by (1) is said to be in the class , if the following conditions are satisfied:where the function is defined by (2) and is defined byClearly, , the class of all strong bistarlike functions of order introduced by Brannan and Taha .
Definition 3. A function given by (1) is said to be in the class , if the following conditions are satisfied:where and are defined, respectively, as in (2) and (10).
Clearly, , the class of all strong bistarlike functions of order introduced by Brannan and Taha .
Theorem 4. Let given by (1) be in the class , and . ThenFurther, for the choice of , one gets
Proof. It follows from (9) thatwhere and satisfy the following inequalities:Furthermore, the functions and have the formsNow, equating the coefficients in (15), we getFrom (19) and (21), we getNow, from (20), (22), and (24), we obtainApplying Lemma 1 for the coefficients and , we immediately haveThis gives the bound on .
Next, in order to find the bound on , by subtracting (20) from (22), we get Upon substituting the value of from (24) and observing that , it follows that Applying Lemma 1 once again for the coefficients , and , we get This completes the proof.
Remark 6. When , , and , we obtain Brannan and Clunie’s  conjecture .
Theorem 7. Let given by (1) be in the class , and . ThenFurther, for the choice of , we get
Proof. It follows from (12) that there exist and , such thatwhere and have forms (17) and (18), respectively.
Equating coefficients in (33) we obtainFrom (34) and (36), we get Now from (35), (37), and (39), we obtain Therefore, we haveApplying Lemma 1, for the coefficients and , we immediately haveNext, in order to find the bound on , by subtracting (35) from (37), we get Applying Lemma 1 for the coefficients , and , we readily get
Remark 10. When , , and we obtain Brannan and Clunie’s  conjecture .
The authors declare that they have no competing interests.
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.
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