Integral Inequalities for the Polar derivative and the generalized Polar derivative of complex Polynomials

Abstract

For a polynomial P(z) of degree n , having all zeros in |z|≤1, Malik [11] proved that for each q>0, \(n [∫_0^{2π} |P(e^{iθ})|^q dθ] ^{1⁄q} ≤[∫_0^{2π} | 1 + e^{iθ} |^q dθ]^{1⁄q} \underset{|z|=1}{max} |P^ʹ (z)|\). In this paper we generalize the above inequality to polar derivative and generalized polar derivative, which as special cases include several known results in this area.