Generalizations of coefficient estimates for certain classes of analytic functions

Abstract

Let \[ E_r= \Biggl\{z\in\mathbb{C}: {x^2\over (1+{1\over r^2})^2}+ {y^2\over(1- {1\over r^2})^2}< 1,\;z= x+iy\Biggr\}, \] where \(r> 1\). Let \(S(E_r)\) denote the class of functions \(F\) analytic and univalent in \(E_r\) with \(F(0)= 0= F'(0)- 1\). Let \(C(E_r)= \{F\in S(E_r): F(E_r)\) is convex\}. Also let \(T(E_r)\) denote the class of functions \(F\) analytic in \(E_r\) with \(F(0)= 0= F'(0)- 1\), having real values if and only if \(-1-{1\over r^2}< z< 1+{1\over r^2}\). Finally, let \(P(E_r)\) denote the class of functions \(P\) analytic in \(E_r\) with \(\text{Re }P(z)> 0\), with a condition on \(P(0)\). Sharp bounds for the Faber coefficients for \(f\) in \(C(E_r)\), \(P(E_r)\) and \(T(E_r)\) are obtained. These results generalize earlier results obtained by the author for \(r= 2\).