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.


Introduction
Let A denote the class of functions of the form which are analytic in the open disc Δ = { : | | < 1} and normalized by (0) = 0, (0) = 1. Let S be the subclass of A consisting of univalent functions ( ) of form (1). For ( ) defined by (1) and ℎ( ) defined by the Hadamard product (or convolution) of and ℎ is defined by It is well known that every function ∈ has an inverse −1 defined by Indeed, the inverse function may have an analytic continuation to Δ, with A function ∈ A is said to be biunivalent in Δ if ( ) and −1 ( ) 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 | 2 | < 1.51. Brannan and Clunie [2] conjectured that | 2 | ≤ √ 2. Netanyahu [3] introduced certain subclasses of biunivalent function class Σ similar to the familiar subclasses S * ( ) and K( ) of starlike and convex functions of order (0 < ≤ 1). Brannan and Taha [4] defined ∈ A in the class S Σ ( ) of strongly bistarlike functions of order (0 < ≤ 1) if each of the following conditions is satisfied: 2 International Journal of Analysis where is as defined by (5). They also introduced the class of all bistarlike functions of order defined as a function ∈ A, which is said to be in the class S * Σ ( ) if the following conditions are satisfied: where the function is as defined in (5). The classes S * Σ ( ) and K Σ ( ) of bistarlike functions of order and biconvex functions of order , corresponding to the function classes S * ( ) and K( ), were introduced analogously. For each of the function classes S * Σ ( ) and K * Σ ( ), they found nonsharp estimates on the first two Taylor-Maclaurin coefficients | 2 | and | 3 | (see [2,5]). Some examples of biunivalent functions are /(1 − ), (1/2) log((1 + )/(1 − )), and − log(1 − ) (see [6]). The coefficient estimate problem for each of the following Taylor-Maclaurin coefficients, | | ( ∈ N, ≥ 3), is still open ( [6]). Various subclasses of biunivalent function class Σ were introduced and nonsharp estimates on the first two coefficients | 2 | and | 3 | in the Taylor-Maclaurin series (1) were found in several investigations (see [7][8][9][10][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.
This gives the bound on | 2 |.
Next, in order to find the bound on | 3 |, by subtracting (20) from (22), we get Upon substituting the value of 2 2 from (24) and observing that 2 1 = 2 1 , it follows that Applying Lemma 1 once again for the coefficients 1 , 2 , 1 , and 2 , we get This completes the proof.