# Geometric properties of the shifted hypergeometric functions

## TL;DR

This paper establishes conditions on the parameters of the shifted hypergeometric function to ensure it belongs to certain subclasses of starlike functions, leading to univalence in the unit disk.

## Contribution

It provides new sufficient conditions for the geometric properties of shifted hypergeometric functions, specifically their starlikeness and univalence.

## Key findings

- Conditions for starlikeness of order α
- Criteria for λ-spirallikeness of order α
- Results on strong starlikeness and univalence

## Abstract

We will provide sufficient conditions for the shifted hypergeometric function $z_2F_1(a,b;c;z)$ to be a member of a specific subclass of starlike functions in terms of the complex parameters $a,b$ and $c.$ For example, we study starlikeness of order $\alpha,$ $\lambda$-spirallikeness of order $\alpha$ and strong starlikeness of order $\alpha.$ In particular, those properties lead to univalence of the shifted hypergeometric functions on the unit disk.