L^q inequalities for polynomials with restricted zeros concerning the growth

Abstract

If P_n denotes the class of polynomials of degree at most n, then for P ∈ P_n, a consequence of Maximum Modulus theorem yields for R > 1, || P(R, )||_∞ < Rⁿ || P ||_∞. Various generalizations and refinements of this result are available in literature. In this paper, we consider a general class of polynomials P_{n, μ}, 1 < μ < n, with restriction on zeros in a specific way and obtain Zygmund-type inequalities concerning the growth of polynomials. Besides obtaining a refinement of a result due to Aziz and Shah, we improve a result of Aziz and Rather.