On the Analytical Determination of Geometric Characterizations of Analytic Functions

Abstract

As it is known, there are many sufficient conditions for the classification complex functions of one variable f(z), which are analytic and univalent in the open unit disc U = {z ∈ C ∶ SzS < 1}, and are also normalized with f(0) = 1 − f′(0) = 0 which are also known as normalization conditions. In this sense, the main goal of present article is to derive some special sufficient conditions for f(z) to be starlike of order 2−r and convex of order 2−r in U , with r is a positive integer.