A geometric investigation of a certain subclass of univalent functions

TL;DR

This paper investigates geometric properties and radius bounds of a subclass of univalent functions, including sharp radii for various subclasses and Bohr radii, expanding understanding of univalent function behavior.

Contribution

It introduces new radius results and sharp bounds for functions in the class al M(7), including specific subclasses and related radii such as Bohr and Bohr-Rogosinski.

Findings

Determined the largest disks with sharp radius for certain function relations.

Established sharp Bohr, Bohr-Rogosinski, and improved Bohr radii for subclasses of starlike functions.

Abstract

Let $\mathcal{H}$ be the space of all functions that are analytic in $\mathbb{D}$. Let $\mathcal{A}$ denote the family of all functions $f\in\mathcal{H}$ and normalized by the conditions $f(0)=0=f'(0)-1$. Obradovi\'{c} and Ponnusamy have introduced the class $\mathcal{M}(\lambda)$ such that the functions in $\mathcal{M}(\lambda)$ are univalent in $\mathbb{D}$ whenever $0<\lambda\leq 1$. In this paper, we address a radius property of the class $\mathcal{M}(\lambda)$ and a number of associated results pertaining to $\mathcal{M}$. The main objective of this paper is to examine the largest disks with sharp radius for which the functions $F$ defined by the relations $g(z)h(z)/z$, $z^2/g(z)$, and $z^2/\int_0^z (t/g(t))dt$ belong to the class $\mathcal{M}$, where $g$ and $h$ belong to some suitable subclasses of $\mathcal{S}$, the class of univalent functions from $\mathcal{A}$. In the final analysis, we obtain the sharp Bohr radius, Bohr-Rogosinski radius and improved Bohr radius for a certain subclass of starlike functions.