Bernstein and Markov-type inequalities for polynomials on Lp(μ) spaces

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 Frechet 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.