Second Hankel determinant for certain subclasses ofbi-univalent Second Hankel determinant for certain subclasses ofbi-univalent functions functions

: In the present paper, we obtain the upper bounds for the second Hankel determinant for certain subclasses of analytic and bi-univalent functions. Moreover, several interesting applications of the results presented here are also discussed.

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Introduction and definitions
Let S denote the class of all functions in A that are univalent in U. The Koebe one-quarter theorem (see, for example, [9]) ensures that the image of U under every f ∈ S contains a disk of radius 1⧸4. Clearly, every f ∈ S has an inverse function f −1 satisfying f −1 (f (z)) = z (z ∈ U) and f (f −1 (w)) = w (|w| < r 0 (f ); r 0 (f ) ≧ 1⧸4) , A function f ∈ A is said to be bi-univalent in U if both f (z) and f −1 (z) are univalent in U. Let σ denote the class of bi-univalent functions in U given by (1.1).
known estimate for functions in the class σ. Brannan and Taha [3] obtained estimates on the initial coefficients |a 2 | and |a 3 | for functions in the classes of bi-starlike functions of order β and bi-convex functions of order β .
The study of bi-univalent functions was revived in recent years by Srivastava et al. [30] and a considerably large number of sequels to the work of Srivastava et al. [30] have appeared in the literature since then. In particular, several results on coefficient estimates for the initial coefficients |a 2 |, |a 3 |, and |a 4 | were proved for various subclasses of σ (see, for example, [1,4,5,10,12,14,16,25,28,29,32,33]).
Recently, Deniz [7] and Kumar et al. [19] both extended and improved the results of Brannan and Taha [3] by generalizing their classes by means of the principle of subordination between analytic functions. The problem of estimating the coefficients |a n | (n ≧ 2) is still open (see also [29] in this connection).
Among the important tools in the theory of univalent functions are Hankel determinants, which are used, for example, in showing that a function of bounded characteristic in U, that is, a function that is a ratio of two bounded analytic functions, with its Laurent series around the origin having integral coefficients, is rational [6].
The Hankel determinants H q (n) (n = 1, 2, 3, · · · , q = 1, 2, 3, · · · ) of the function f are defined by (see [23]) This determinant was discussed by several authors with q = 2 . For example, we know that the functional is known as the Fekete-Szegö functional and one usually considers the further generalized functional a 3 − µa 2 2 where µ is some real number (see [11]). Estimating for the upper bound of a 3 − µa 2 2 is known as the Fekete-Szegö problem. In 1969, Keogh and Merkes [18] solved the Fekete-Szegö problem for the classes of starlike and convex functions. One can see the Fekete-Szegö problem for the classes of starlike functions of order β and convex functions of order β in special cases in the paper of Orhan et al. [24]. On the other hand, quite recently, Zaprawa (see [34,35]) studied the Fekete-Szegö problem for some classes of bi-univalent functions. In special cases, he gave the Fekete-Szegö problem for the classes of bi-starlike functions of order β and bi-convex functions of order β .
The second Hankel determinant H 2 (2) is given by H 2 (2) = a 2 a 4 − a 2 3 . The bounds for the second Hankel determinant H 2 (2) were obtained for the classes of starlike and convex functions in [15]. Lee et al. [20] established the sharp bound for |H 2 (2)| by generalizing their classes by means of the principle of subordination between analytic functionds. In their paper [20], one can find the sharp bound for |H 2 (2)| for the functions in the classes of starlike functions of order β and convex functions of order β . Recently, Deniz et al. [8] and Orhan et al. [26] found the upper bound for the functional H 2 (2) = a 2 a 4 − a 2 3 for the subclasses of bi-univalent functions.
The object of the present paper is to seek the upper bound for the functional a 2 a 4 − a 2 3 for f ∈ N σ (β) and f ∈ N α σ , which are defined as follows.
Definition 1 (see [30]) A function f (z) given by (1.1) is said to be in the class f ∈ N σ (β) (0 ≦ β < 1) if the following conditions are satisfied: where the function g is given by Definition 2 (see [30]) A function f (z) given by (1.1) is said to be in the class f ∈ N α σ (0 < α ≦ 1) if the following conditions are satisfied: and where the function g is defined by (1.4).
For special values of the parameters α and β , we have Let P be the class of functions with positive real part consisting of all analytic functions P : U → C satisfying p(0) = 1 and ℜ (p(z)) > 0.
To establish our main results, we shall require the following lemmas.
Lemma 1 (see, for example, [27]) If the function p ∈ P is given by the following series: then the sharp estimate given by holds true.
Lemma 2 (see [13]) If the function p ∈ P is given by the series (1.7), then for some x and z with |x| ≦ 1 and |z| ≦ 1.

Main results
Our first main result for the class f ∈ N σ (β) is stated as follows: Then , .
where the functions p(z) and q(z) given by are in class P . Comparing the coefficients in (2.2), we have 3) and (2.6), we find that and Now, from (2.4), (2.7) and (2.10), we get Also, from (2.5) and (2.8), we find that Thus, we can easily establish that According to Lemma 2 and (2.9), we write and Moreover, we have for some x, y and z, w with |x| ≦ 1, |y| ≦ 1, |z| ≦ 1 and |w| ≦ 1. Using (2.14) and (2.15) in (2.13), and applying the triangle inequality, we have Since p ∈ P, we have |c 1 | ≦ 2. Letting c 1 = c, we may assume without loss of generality that c ∈ [0, 2].
Thus, for λ = |x| ≦ 1 and µ = |y| ≦ 1, we obtain We must investigate the maximum of F (λ, µ) according to c = (0, 2), c = 0 and c = 2 , keeping in view the First, let c ∈ (0, 2). Since T 3 < 0 and T 3 + 2T 4 > 0 for c ∈ (0, 2) , we conclude that Thus, the function F cannot have a local maximum in the interior of the square S. Now we investigate the maximum of F on the boundary of the square S.
For λ = 1 and 0 ≦ µ ≦ 1, we obtain Thus, from the above Case 1 and Case 2 for T 3 + T 4 , we get Similarly, for µ = 1 and 0 ≦ λ ≦ 1, we have on the boundary of the square S. Thus, clearly, the maximum of the function F (λ, µ) occurs when λ = 1 and µ = 1 in the closed square S and for c ∈ (0, 2) .
Let K : (0, 2) → R be given by Substituting the values of T 1 , T 2 , T 3 , and T 4 into the function K(c) defined by (2.17) yields We now investigate the maximum value of K(c) in the interval (0, 2). By elementary calculation, we find that As a result of some calculations, we can accomplish the following results.

Result 1.
Let Then K ′ (c) > 0 for every c ∈ (0, 2). Furthermore, since K(c) is an increasing function in the interval (0, 2), it has no maximum value in this interval.

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Then K ′ (c) = 0 implies the real critical point given by In the case when , then c 01 ≧ 2, that is , c 01 lies outside of the interval (0, 2). In the case when ) .
Thus, clearly, it is observed that .
Secondly, let c = 2 and (λ, µ) ∈ S. We then obtain a constant function of the dependent variables λ and µ as follows: for every 0 ≦ β < 1.
We thus obtain the second inequality of (2.1) for .
On the other hand, since the following inequality: is satisfied for every β (0 ≦ β < 1), we obtain the first inequality of (2.1) for This completes the proof of Theorem 1. Our second main result for the class N α σ is given by Theorem 2 below.
In order to determine the maximum of Ψ(λ, µ), we can analogously follow the derivation of the maximum of where P, Q , and R are given by (2.36).
This completes the proof of Theorem 2. 2 For β = 0 in Theorem 1 or for α = 1 in Theorem 2, we obtain the coefficient estimate given by the corollary below.