On harmonic combination of univalent functions

Abstract

Let ${\mathcal S}$ be the class of all functions $f$ that are analytic and univalent in the unit disk $\mathbb D$ 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 $$\left |f'(z)\left (\frac{z}{f(z)} \right )^{2}-1\right | <\lambda ~\mbox{ for $z\in \mathbb D$}, $$ and 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 $$\frac{z}{F(z)}=\frac{1}{2}\left( \frac{z}{f(z)}+\frac{z}{g(z)} \right), $$ 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 $F$.