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

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