$(s,p)$-Valent Functions

TL;DR

This paper introduces and characterizes $(s,p)$-valent functions, a generalization of $p$-valent functions, providing new inequalities and distortion theorems that extend classical polynomial bounds to this broader class.

Contribution

The paper defines $(s,p)$-valent functions, characterizes them via Taylor coefficients and recurrences, and establishes new distortion and Remez-type inequalities for these functions.

Findings

Characterization of $(s,p)$-valent functions through Taylor domination.

A distortion theorem comparing $(s,p)$-valent functions with polynomials.

An essentially sharp Remez-type inequality for $(s,p)$-valent functions.

Abstract

We introduce the notion of $(\mathcal F,p)$-valent functions. We concentrate in our investigation on the case, where $\mathcal F$ is the class of polynomials of degree at most $s$. These functions, which we call $(s,p)$-valent functions, provide a natural generalization of $p$-valent functions (see~\cite{Ha}). We provide a rather accurate characterizing of $(s,p)$-valent functions in terms of their Taylor coefficients, through "Taylor domination", and through linear non-stationary recurrences with uniformly bounded coefficients. We prove a "distortion theorem" for such functions, comparing them with polynomials sharing their zeroes, and obtain an essentially sharp Remez-type inequality in the spirit of~\cite{Y3} for complex polynomials of one variable. Finally, based on these results, we present a Remez-type inequality for $(s,p)$-valent functions.