Estimates for Coefficients of Certain Analytic Functions

For -1 <= B <= 1 and A > B, let S*[A, B] denote the class of generalized Janowski starlike functions consisting of all normalized analytic functions f defined by the subordination zf'(z)/f (z) < (1 + Az)/(1 + Bz) (vertical bar z vertical bar < 1). For -1 <= B <= 1 < A, we investigate the inverse coefficient problem for functions in the class S*[A, B] and its meromorphic counter part. Also, for -1 <= B <= 1 < A, the sharp bounds for first five coefficients for inverse functions of generalized Janowski convex functions are determined. A simple and precise proof for inverse coefficient estimations for generalized Janowski convex functions is provided for the case A = 2 beta-1 (beta > 1) and B = 1. As an application, for F : = f(-1), A = 2 beta-1 (beta > 1) and B = 1, the sharp coefficient bounds of F/F' are obtained when f is a generalized Janowski starlike or generalized Janowski convex function. Further, we provide the sharp coefficient estimates for inverse functions of normalized analytic functions f satisfying f'(z) < (1 + z)/(1 + Bz) (vertical bar z vertical bar < 1, -1 <= B < 1).