Abstract. An asymptotically sharp Bernstein-type inequality is proven for trigonometric polynomials in integral metric. This extends Zygmund's classical inequality on the $L^p$ norm of the derivatives of trigonometric polynomials to the case when the set consists of several intervals. The result also contains a recent theorem of Nagy and Toókos, who proved a similar statement for algebraic polynomials.

AMS Subject Classification (1991): 31A15, 41A17

Keyword(s): Bernstein inequality, integral norm, sharp constants

Received April 4, 2013, and in revised form July 3, 2013. (Registered under 23/2013.)