The Fekete-Szego Theorem for a Certain Class of Analytic Function

Let J(rho) (alpha, beta) denote the subclass of nonnalized analytic functions f in the open unit disk D = {z : vertical bar z vertical bar < 1} satisfying the condition Re {alpha zf'(z) + beta z(2) f(n) (z)/alpha g(z)} > 0 for some 0 < alpha <= 1, 0 <= beta <1 where g(z) is an element of S* (rho), g (z) not equal 0 is a normalized starlike function of order rho for 0 <= rho < 1. For f is an element of J(rho) (alpha, beta), we obtain the sharp upper bounds for the Fekete - Szego functional vertical bar a(3) - mu a(2)(2)vertical bar where mu is real.