Abstract
In this paper, we investigate a normalized analytic (symmetric under rotation) function, f, in an open unit disk that satisfies the condition ℜfzgz>0, for some analytic function, g, with ℜz+1−2nzgz>0,∀n∈N. We calculate the radius constants for different classes of analytic functions, including, for example, for the class of star-like functions connected with the exponential functions, i.e., the lemniscate of Bernoulli, the sine function, cardioid functions, the sine hyperbolic inverse function, the Nephroid function, cosine function and parabolic star-like functions. The results obtained are sharp.
Highlights
Introduction and MotivationsLet by Dr, we denote the open unit disk with radius r, given by Academic Editor: Stanisława KanasReceived: 16 November 2021Dr = {z : z ∈ C and |z| < r }.It can be seen thatAccepted: 14 December 2021 D = D1 .Published: 19 December 2021Let H be the family of analytic functions in Publisher’s Note: MDPI stays neutralD = {z : z ∈ C and |z| < 1}with regard to jurisdictional claims in published maps and institutional affiliations.and An ⊂ H such that f ∈ An has the series representation:
The radius of star-likeness for functions in the class S has been obtained by Grunsky [1] and is given by π
Between 1916 and 1985, many good scholars of the time attempted to prove or reject this theory. They discovered multiple subfamilies of a class S of univalent functions that are associated with different image domains
Summary
Let by Dr , we denote the open unit disk with radius r, given by Academic Editor: Stanisława Kanas. The radius of star-likeness for functions in the class S has been obtained by Grunsky [1] and is given by π. Between 1916 and 1985, many good scholars of the time attempted to prove or reject this theory As a result, they discovered multiple subfamilies of a class S of univalent functions that are associated with different image domains. Mac-Gregor [11,12] obtained the radius of star-likeness for the class of functions f ∈ Υ satisfying. Many well-known mathematicians found radius constants for several classes of functions f ∈ Υ that, defined by their ratio to a certain function g, meet one of the following conditions: I. By assigning different values to some perpetrators we obtain some interesting special cases of our main results
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