Bounds for the faber coefficients of certain classes of functions analytic in an ellipse

Abstract

Let Omega be a bounded, simply connected domain in C with 0 is an element of Omega and aOmega analytic. Let S(Omega) denote the class of functions F(z) which are analytic and univalent in Omega with F(0) = 0 and F'(0) = 1. Let {Phi(n)(z)} infinity n=0 be the Faber polynomials associated with Omega. If F(z) is an element of S(Omega), then F(z) can be expanded in a series of the form where r > 1. In this paper we obtain sharp bounds for the Faber coefficients A(0), A(1) and A(2) of functions F(z) in S(E-r) and in certain related classes.