Inequalities for trigonometric polynomials

Abstract

Let t_n(x) be any real trigonometric polynomial of degreen n such that ∥t_m∥∞⩽1. Here we are concerned with obtaining the best possible upper estimate of

$$\int_0^{2k} {|t_m^{(h)} (x)|^q } dx/\int_0^{2k} {|t_n^{(h)} (x)|^{q - 2} } dx,$$

where q>2. In addition, we shall obtain the estimate of$||t_m^{(k)} ||_q$ in terms of ∥t_m∥_q and ∥t ^(r)_n .