A Note on Sharp Multivariate Bernstein- and Markov-Type Inequalities

TL;DR

This paper establishes sharp Bernstein- and Markov-type inequalities for entire functions and algebraic polynomials related to convex bodies, providing precise bounds and properties of extremal functions.

Contribution

It introduces new sharp inequalities for entire functions and polynomials associated with convex bodies, extending classical results to more general settings.

Findings

Proved sharp Bernstein-type inequalities for entire functions with spectra in convex bodies.

Established sharp Markov-type inequalities for algebraic polynomials on convex bodies.

Analyzed properties of extremal functions related to these inequalities.

Abstract

Let $V$ be a symmetric convex body in $\R^m$. We prove sharp Bernstein-type inequalities for entire functions of exponential type with the spectrum in $V$ and discuss certain properties of the extremal functions. Markov-type inequalities with sharp constants for algebraic polynomials on $V$ and certain non-symmetric convex bodies are proved as well.