On Zygmund-Type Inequalities Concerning Polar Derivative of Polynomials

Abstract

Let P(z) be a polynomial of degree n, then concerning the estimate for maximum of |P′(z)| on the unit circle, it was proved by S. Bernstein that ∥P′∥∞≤n∥P∥∞. Later, Zygmund obtained an Lp-norm extension of this inequality. The polar derivative Dα[P](z) of P(z), with respect to a point α∈C, generalizes the ordinary derivative in the sense that limα→∞Dα[P](z)/α=P′(z). Recently, for polynomials of the form P(z)=a0+∑nj=μajzj, 1≤μ≤n and having no zero in |z|<k where k>1, the following Zygmund-type inequality for polar derivative of P(z) was obtained: ∥Dα[P]∥p≤n(|α|+kμ∥kμ+z∥p)∥P∥p,where|α|≥1,p>0. In this paper, we obtained a refinement of this inequality by involving minimum modulus of |P(z)| on |z|=k, which also includes improvements of some inequalities, for the derivative of a polynomial with restricted zeros as well.