Fekete-Szegö Inequalities for Certain Classes of Biunivalent Functions

We obtain the Fekete-Szegö inequalities for the classes S S,Σ *(α, ϕ) and £ S,Σ(α, ϕ) of biunivalent functions denoted by subordination. The results presented in this paper improve the recent work of Crisan (2013).

A function ( ) ∈ is said to be biunivalent in if both ( ) and −1 ( ) are univalent in . Let Σ denote the class of biunivalent functions defined in the unit disk .
Let be an analytic and univalent function with positive real part in with (0) = 1, (0) > 0, and maps the unit disk onto a region starlike with respect to 1 and symmetric 2 International Scholarly Research Notices with respect to the real axis. Taylor's series expansion of such function is of the form where all coefficients are real and 1 > 0.
The classes * ( ) and ( ) are the extensions of classical sets of starlike and convex functions and in such a form were defined and studied by Ma and Minda [13]. They investigated growth and distortion properties of functions in * ( ) and ( ) as well as Fekete-Szegö inequalities for * ( ) and ( ). Their proof of Fekete-Szegö inequalities requires the univalence of . Ali et al. [14] have investigated Fekete-Szegö problems for various other classes and their proof does not require the univalence or starlikeness of . So in this paper, we assume that has series expansion ( ) = 1 + 1 + 2 2 + ⋅ ⋅ ⋅ , 1 , 2 are real, and 1 > 0. A function is bistarlike of Ma-Minda type or biconvex of Ma-Minda type if both and −1 are, respectively, Ma-Minda starlike or convex. These classes are denoted, respectively, by * Σ ( ) and Σ ( ) (see [15]).
and in the class In this paper, motivated by the earlier work of Zaprawa [19], we obtain the Fekete-Szegö inequalities for the classes * ,Σ ( , ) and m ,Σ ( , ). These inequalities will result in bounds of the third coefficient which are, in some cases, better than these obtained in [7].
In order to derive our main results, we require the following lemma.