On harmonic combination of univalent functions

TL;DR

This paper investigates the harmonic mean of two univalent functions, analyzing its properties, univalence radius, and proposing conjectures about its univalence within the class of analytic and univalent functions.

Contribution

It introduces a new harmonic mean operation on univalent functions and studies its univalence properties, including radius and conjectures.

Findings

Determined the radius of univalence for the harmonic mean function.

Proposed two conjectures on the univalence of the harmonic mean.

Analyzed properties of the harmonic mean within specific classes of univalent functions.

Abstract

Let ${\mathcal S}$ be the class of all functions $f$ that are analytic and univalent in the unit disk $\ID$ with the normalization $f(0)=f'(0)-1=0$. Let $\mathcal{U} (\lambda)$ denote the set of all $f\in {\mathcal S}$ satisfying the condition $$|f'(z)(\frac{z}{f(z)})^{2}-1| <\lambda ~for $z\in \ID$, $$ for some $\lambda \in (0,1]$. In this paper, among other things, we study a "harmonic mean" of two univalent analytic functions. More precisely, we discuss the properties of the class of functions $F$ of the form $$\frac{z}{F(z)}=1/2(\frac{z}{f(z)}+\frac{z}{g(z)}), $$ where $f,g\in \mathcal{S}$ or $f,g\in \mathcal{U}(1)$. In particular, we determine the radius of univalency of $F$, and propose two conjectures concerning the univalency of $F$.