Estimates for coefficients of certain analytic functions

Abstract

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 ′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 = 2β − 1 (β > 1) and B = 1. As an application, for F := f −1 , A = 2β − 1 (β > 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) (|z| < 1, −1 ≤ B < 1).

Certain Estimates of Normalized Analytic Functions

“…In [12,15], authors obtained the initial inverse coefficients for the well known classes C and S * (α) (0 ≤ α ≤ 1). Recently, Ravichandran and Verma [29] determined the bounds on inverse coefficient for the Janowski starlike functions.…”