On starlikeness of $p$-valent analytic functions

TL;DR

This paper extends Ozaki's condition to provide new sufficient criteria for p-valent functions to be starlike of a certain order, enhancing understanding of their geometric properties.

Contribution

It introduces an extended Ozaki's condition and new criteria for p-valent functions to be starlike of a given order, advancing geometric function theory.

Findings

Extended Ozaki's condition for p-valent functions.

New sufficient conditions for p-valent starlike functions.

Characterization of p-valent starlikeness of order alpha.

Abstract

The known Ozaki's condition says that $\mathfrak{Re}\left\{f^{(p)}(z)\right\}>0$ for $|z|<1$ implies that $f(z)=z^p+a_{p+1}z^{p+1}+\cdots$ is at most $p$-valent in $\mathbb D$. In this paper prove an extension of Ozaki's condition. Also, we shall determine the new sufficient conditions for functions to be in the class of $p$-valent starlike of order $\alpha$.