SOME COEFFICIENT PROBLEMS FOR A CLASS OF STARLIKE FUNCTIONS

Abstract

This investigation explores the analytic and geometric properties of complex func tions. Specifically, we focus on a novel starlike function that is analytic, denoted as S ∗ nc, which is uniquely associated with a non-convex domain. The class on which we are going to work is defined as : S ∗ nc =

f ∈ A : z f ′ (z) f(z) = 1 + z cos z ≺ φnc(z), z ∈ D

. Here, A represents the set of functions analytic in D that satisfy f(0) = 0 and f ′ (0) = 1. The subordination condition involves a specific non-convex function φnc(z) = (1 + z)/cos z, which characterizes the geometric properties of functions belonging to this class. Our primary objective is to determine the sharp second-order of Hankel and Toeplitz determinats for the logarithmic coefficients of functions f belonging to this newly de fined class S ∗ nc. Furthermore, the study extends to finding these precise bounds for the logarithmic coefficients of z their inverse functions, f −1 . The determination of sharp bounds for these determinants and coefficients provides crucial insights into the in tricate behavior and structural properties of these analytic functions within the speci fied non-convex domain, contributing significantly to the understanding of coefficient problems in geometric function theory.