Some Zygmund type Lq inequalities for polynomials

Let p(z) be a polynomial of degree n which does not vanish in |z| < k. It is known that for each q >0 and k 1, P(s) q n(n−1) . . . (n− s + 1) ks + z q P q . In this paper, we present a refinement of this inequality which besides yielding some interesting results as corollaries, includes some well-known results as special cases. We also consider an analogous problem for the class of polynomials p(z) = anzn + nν =m an−ν zn−ν not vanishing outside the disk |z| k, where k 1 and obtain a sharp result.  2003 Elsevier Inc. All rights reserved.