Dolomites Research Notes on Approximation

In this work we discuss generalizations of the classical Bernstein and Markov type inequalities for polynomials and we present some new inequalities for the kth Fréchet derivative of homogeneous polynomials on real and complex Lp(μ) spaces. We also give applications to homogeneous polynomials and symmetric multilinear mappings in Lp(μ) spaces. Finally, Bernstein’s inequality for homogeneous polynomials on both real and complex Hilbert spaces has been discussed.