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 \(\| P'\|_{\infty}\leq n\| P\|_{\infty}\). Later, Zygmund obtained an \(L_p\)-norm extension of this inequality. The polar derivative \(D_{\alpha}[P](z)\) of \(P(z)\), with respect to a point \(\alpha \in \mathbb{C}\), generalizes the ordinary derivative in the sense that \(\lim_{\alpha\to\infty} D_{\alpha}[P](z)/{\alpha} = P'(z).\) Recently, for polynomials of the form \(P(z) = a_0 + \sum_{j=\mu}^n a_jz^j,\) \(1\leq\mu\leq n\) and having no zero in \(|z| < k\) where \(k > 1\), the following Zygmund-type inequality for polar derivative of \(P(z)\) was obtained: $$\|D_{\alpha}[P]\|_p\leq n \Big(\dfrac{|\alpha|+k^{\mu}}{\|k^{\mu}+z\|_p}\Big)\|P\|_p, \quad \text{where}\quad |\alpha|\geq1,\quad p>0.$$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.