On relations between the classes $\mathcal S$ and $\mathcal U$

TL;DR

This paper explores the relationship between two classes of univalent functions, showing that some properties of the broader class can be refined within a specific subclass defined by a particular inequality.

Contribution

The paper investigates the connections between classes nd nd demonstrates which properties of an be improved or are limited within ased on their defining conditions.

Findings

Some results about re improved for ut not all.

The subclass xhibits both similarities and differences with nd unctions.

The paper clarifies the structural relations between these classes of analytic functions.

Abstract

Let ${\mathcal A}$ denote the family of all functions $f$ analytic in the unit disk $\ID$ and satisfying the normalization $f(0)=0= f'(0)-1$. Let $\mathcal{S}$ denote the subclass of ${\mathcal A}$ consisting of univalent functions in $\ID$. We consider the subclass $\mathcal{U} $ of $\mathcal{S}$ that is defined by the condition that for its members $f$ the condition $$\left |\left (\frac{z}{f(z)} \right )^{2}f'(z)-1\right | < 1 ~\mbox{ for $z\in \ID$} $$ holds. To theses relations belong striking similarities and on the other hand big differences. We show that some results about $\mathcal{S}$ can be improved for $\mathcal{U}$, while others cannot.