Fekete-Szegö Problem for a New Class of Analytic Functions

which are analytic in the open unit disk U {z : z ∈ C and |z| < 1} and S denote the subclass ofA that are univalent in U. A function f z inA is said to be in class S∗ of starlike functions of order zero in U, if R zf ′ z /f z > 0 for z ∈ U. Let K denote the class of all functions f ∈ A that are convex. Further, f is convex if and only if zf ′ z is star-like. A function f ∈ A is said to be close-to-convex with respect to a fixed star-like function g ∈ S∗ if and only if R zf ′ z /g z > 0 for z ∈ U. Let C denote of all such close-to-convex functions 1 . Fekete and Szegö proved a noticeable result that the estimate


Introduction and Preliminaries
Let A denote the class of functions of the form which are analytic in the open unit disk U {z : z ∈ C and |z| < 1} and S denote the subclass of A that are univalent in U. A function f z in A is said to be in class S * of starlike functions of order zero in U, if R zf z /f z > 0 for z ∈ U. Let K denote the class of all functions f ∈ A that are convex.Further, f is convex if and only if zf z is star-like.A function f ∈ A is said to be close-to-convex with respect to a fixed star-like function g ∈ S * if and only if R zf z /g z > 0 for z ∈ U. Let C denote of all such close-to-convex functions 1 .
Fekete and Szeg ö proved a noticeable result that the estimate Let φ z be an analytic function with positive real part on U with φ 0 1, φ 0 > 0 which maps the unit disk U onto a star-like region with respect to 1 which is symmetric with respect to the real axis.Let S * φ be the class of functions in f ∈ S for which and C φ be the class of functions in f ∈ S for which where ≺ denotes the subordination between analytic functions.These classes were introduced and studied by Ma and Minda 6 .They have obtained the Fekete-Szeg ö inequality for the functions in the class C φ .Motivated by the class R τ λ β in paper 7 , we introduce the following class.
where φ z is defined the same as above.
If we set which is again a new class.We list few particular cases of this class discussed in the literature and the result is sharp for the functions given by

Fekete-Szeg ö Problem
Our main result is the following theorem.

2.1
The result is sharp.
Proof.If f z ∈ R τ γ φ , then there exists a Schwarz function w z analytic in U with w 0 0 and Define the function p 1 z by

International Journal of Mathematics and Mathematical Sciences
Since w z is a Schwarz function, we see that Rp 1 z > 0 and p 1 0 1.Define the function p z by In view of 2.2 , 2.3 , 2.4 , we have From 2.4 , we obtain Therefore, we have where

2.9
Our result now is followed by an application of Lemma

2
International Journal of Mathematics and Mathematical Sciences holds for any normalized univalent function f z of the form 1.1 in the open unit disk U and for 0 λ 1.This inequality is sharp for each λ see 2 .The coefficient functional The problem of maximising the absolute value of the functional φ λ f is called the Fekete-Szeg ö problem; see 2 .In 3 , Koepf solved the Fekete-Szeg ö problem for close-to-convex functions and the largest real number λ for which φ λ f is maximised by the Koebe function z/ 1 − z 2 is λ 1/3, and later in 4 see also 5 , this result was generalized for functions that are close-to-convex of order β.
If p z 1 c 1 z c 2 z 2 c 3 z 3 • • • z ∈ U isa function with positive real part, then for any complex number μ, Az 1 Bz −1 z ∈ U 1 A − B z − AB − B 2 z 2 • • • .2.12Thus, putting B 1 A − B and B 2 −B A − B in Theorem 2.1, we get the following corollary. 2.2.13