Bernstein and Markov-type inequalities for polynomials on $L_{p}(\mu)$ spaces

TL;DR

This paper extends classical Bernstein and Markov inequalities to polynomials on $L_{p}(u)$ spaces, providing new bounds for derivatives and applications to multilinear mappings and Hilbert spaces.

Contribution

It introduces new inequalities for derivatives of polynomials on $L_{p}(u)$ spaces and explores their applications, including Bernstein's inequality on Hilbert spaces.

Findings

New inequalities for the $k$th Fre9chet derivative of polynomials

Applications to multilinear mappings in $L_{p}(u)$ spaces

Bernstein's inequality established for polynomials on Hilbert spaces

Abstract

In this work, we discuss generalizations of the classical Bernstein and Markov type inequalities for polynomials and we present some new inequalities for the $k$th Fr\'echet derivative of homogeneous polynomials on real and complex $L_{p}(\mu)$ spaces. We also give applications to homogeneous polynomials and symmetric multilinear mappings in $L_{p}(\mu)$ spaces. Finally, Bernstein's inequality for homogeneous polynomials on both real and complex Hilbert spaces has been discussed.