  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.
