Second Hankel Determinants for Some Subclasses of Biunivalent Functions Associated with Pseudo-Starlike Functions

Copyright © 2017 K. Rajya Laxmi and R. Bharavi Sharma. 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 second Hankel determinant of biunivalent analytic functions associated with λ-pseudo-starlike function in the open unit disc Δ subordinate to a starlike univalent function whose range is symmetric with respect to the real axis.

In fact, the inverse function is given by A function ∈ A is said to be biunivalent in Δ if both and −1 are univalent in Δ. Let Σ denote the class of all biunivalent functions defined in the unit disc Δ. We notice that Σ is nonempty. The behavior of the coefficients is unpredictable when the biunivalency condition is imposed on the function ∈ A. In 1967, Lewin [1] introduced the class Σ of biunivalent functions and investigated second coefficient in Taylor-Maclaurin series expansion for every ∈ Σ. Subsequently, in 1967, Brannan and Clunie [2] introduced bistarlike functions and biconvex functions similar to the familiar subclasses of univalent functions consisting of strongly starlike, strongly convex, starlike, and convex functions and so on and obtained estimates on the initial coefficients conjectured that | 2 | ≤ √2 for bistarlike functions and | 2 | ≤ 1 for biconvex functions. Only the last estimate is sharp; equality occurs only for ( ) = /(1 − ) or its rotation. Since then, various subclasses of biunivalent functions class Σ were introduced and nonsharp estimates on the first two coefficients | 2 | and | 3 | in Taylor-Maclaurin series expansion were found in several investigations. The coefficient estimate problem for each of | | is still an open problem. In 1976, Noonan and 2 Journal of Complex Analysis Thomas [3] defined th Hankel determinants of for ≥ 1 and ≥ 1 which is stated as follows: Easily one can observe that 2 (1) = | 3 − 2 2 | is a special case of the well known Fekete-Szegö functional | 3 − 2 2 | where is real, for = 1. Now for = 2, = 2, we get second Hankel determinant In particular, sharp upper bounds on 2 (2) were obtained by the authors of articles [4][5][6] for various subclasses of analytic and univalent functions. In 2013, Babalola [7] determined the second Hankel determinant with Fekete-Szegö parameter | 2 4 − 2 3 | for some subclasses of analytic functions. Let be an analytic function with positive real part in Δ such that (0) = 1, (0) > 0 which is symmetric with respect to the real axis. Such a function has a Maclaurin series expansion of the form ( ) = 1 + 1 + 2 2 + 3 3 + ⋅ ⋅ ⋅ ( 1 > 0). Researchers like Duren [8], Singh [9], and so on have studied various subclasses of usual known Bazilevic function of order denoted by ( ) which satisfy the geometric condition Re( ( ) −1 ( )/ −1 ) > 0, where is nonnegative real number, different ways of perspectives of convexity, radii of convexity and starlikeness, inclusion properties, and so on. The class ( ) reduces to the starlike function and bounded turning function whenever = 0 and = 1, respectively. This class is extended to ( , ) which satisfy the geometric condition Re( ( ) −1 ( )/ −1 ) > , where is nonnegative real number and 0 ≤ < 1. Recently, Babalola [7] defined new subclass -pseudo-starlike functions of order (0 ≤ < 1) which satisfy the condition Re( [ ( )] / ( )) > , ( ≥ 1 ∈ R, 0 ≤ < 1, ∈ Δ) and is denoted by L ( ). Babalola [7] proved that all pseudo-starlike functions are Bazilevic of type (1−1/ ), order (1/ ) , and univalent in the open unit disc Δ. For = 2 we note that functions in L 2 ( ) are defined by Re ( )( ( )/ ( )) > which is a product combination of geometric expressions for bounded turning and starlike functions. Note that the singleton subclass L ∞ ( ) of contains the identity map. In 2016, Joshi et al. [10] defined two new subclasses of biunivalent functions using pseudostarlike functions, one is L Σ ( ) class of strong -bi-pseudostarlike functions of order and other is L Σ ( , ) -bipseudo-starlike functions of order in the open unit disc. Many researchers [11][12][13][14][15] have estimated the second Hankel determinants for some subclasses of biunivalent functions. Motivated by the above-mentioned work, in this paper we have introduced -bi-pseudo-starlike functions subordinate to a starlike univalent function whose range is symmetric with respect to the real axis and estimated second Hankel determinants.

Definition 1. A function
∈ Σ is said to be in the class L Σ ( ), ≥ 1, if it satisfies the following conditions: where is an extension of −1 to Δ.
, then the class L Σ ( ) reduces to the class L Σ ( ), 0 < ≤ 1 and satisfies following conditions: where is an extension of −1 to Δ.
Journal of Complex Analysis 3 (4) If = 1, then the class L Σ ( ) reduces to the class of bistarlike functions Σ ( ) and satisfies the following conditions: where is an extension of −1 to Δ.
Several choices of would reduce the class Σ ( ) to some well known subclasses of Σ.
(2) For the function given by ( ) = (1 + )/(1 − ), the class Σ ( ) reduces to the class Σ and satisfies the following conditions: where is an extension of −1 to Δ and this class is called class of bistarlike function.

Preliminary Lemmas
Let denote the class of functions consisting of , such that which are analytic in the open unit disc Δ and satisfy R{ ( )} > 0 for any ∈ Δ.
Lemma 3 (see [16]). The power series for ( ) = 1 + ∑ ∞ =1 given in (14) converges in the open unit disc Δ to a function in if and only if the Toeplitz determinants We may assume without any restriction that 1 > 0, on using Lemma 3 for = 2 and = 3, respectively, we have 2 = 2 1 2 which is equivalent to If we consider the determinant 3 = 2 1 2 3 we get the following inequality: From (17) and (19), it is obtained that for some , | | ≤ 1.

Main Results
Theorem 4. If ∈ L Σ ( ) and is of the form (1) then we have the following.
Define two functions ( ), ( ) such that Then 2 ) Then (22) becomes Now equating the coefficients in (25) 2 ) 2 ) Now from (26) and (29) Now from (27) and with the help of the above Lemma 2, we get the required results.