Some inequalities for trigonometric polynomials and their derivatives

In trigonometric approximation one can show that smoothness of a function is equivalent to a quick decrease to zero of its error of approximation by trigonometric polynomials. The key steps to show this are the Jackson- and the Bernstein-inequality. In this paper we will investigate some Bernstein-type inequalities concerning conjugate functions and Laplacian. In a forthcoming paper we will show that these inequalities imply the equivalence of the order of approximation by the classical Jackson operator (in higher dimension) and a corresponding measure for the smoothness of the function.