# Differential Inequalities and Univalent Functions

## TL;DR

This paper studies a class of univalent functions defined by a differential inequality, proving that the harmonic mean of two such functions remains in the class, and explores their geometric properties like starlikeness.

## Contribution

It introduces a new class of univalent functions defined via a differential inequality and proves the harmonic mean of two functions in this class also belongs to it.

## Key findings

- Harmonic mean of two functions in the class remains in the class.
- Several functions in the class are shown to be starlike.
- Conjecture that not all functions in the class are starlike.

## Abstract

Let ${\mathcal M}$ be the class of analytic functions in the unit disk $\ID$ with the normalization $f(0)=f'(0)-1=0$, and satisfying the condition $$\left |z^2\left (\frac{z}{f(z)}\right )''+ f'(z)\left(\frac{z}{f(z)} \right)^{2}-1\right |\leq 1, \quad z\in \ID. $$ Functions in $\mathcal{M}$ are known to be univalent in $\ID$. In this paper, it is shown that the harmonic mean of two functions in ${\mathcal M}$ are closed, that is, it belongs again to ${\mathcal M}$. This result also holds for other related classes of normalized univalent functions. A number of new examples of functions in $\mathcal{M}$ are shown to be starlike in $\ID$. However we conjecture that functions in $\mathcal{M}$ are not necessarily starlike, as apparently supported by other examples.