Inequalities for trigonometric polynomials

Abstract

Let tn(x) be any real trigonometric polynomial of degreen n such that ∥tm∥∞⩽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 ∥tm∥q and ∥t n (r) .