Constrained Radius Estimates Of Certain Analytic Functions

TL;DR

This paper introduces new classes of normalized analytic functions constrained by conditions involving Carathéodory functions and aims to determine their radii of starlikeness.

Contribution

It defines novel classes of analytic functions based on constraints involving positive real part functions and characterizes their geometric properties.

Findings

Derived radius of starlikeness for the new classes.

Characterized classes via functions with positive real part.

Established geometric properties of constrained functions.

Abstract

Let $\mathcal{P}$ denote the Carath\'{e}odory class accommodating all the analytic functions $p$ having positive real part and satisfying $p(0)=1$. In this paper, the second coefficient of the normalized analytic function $f$ defined on the open unit disc is constrained to define new classes of analytic functions. The classes are characterised by the functions $f/g$ having positive real part or satisfying the inequality $|(f(z)/g(z))-1|<1$ such that $f(z)(1-z^2)/z$ and $g(z)(1-z^2)/z$ are Carath\'{e}odory functions for some analytic function $g$. This paper aims at determining radius of starlikeness for the introduced classes.