ON ZYGMUND–TYPE INEQUALITIES CONCERNING POLAR DERIVATIVE OF POLYNOMIALS

Let \(P(z)\) be a polynomial of degree \(n\), then concerning the estimate for maximum \(|P'(z)|\) on unit circle, it was proved by S. Bernstein that \(\| P'\|_{\infty}\leq n\| P\|_{\infty}\). Later, Zygmund obtained an \(L_p\)-norm extension this inequality. The polar derivative \(D_{\alpha}[P](z)\) \(P(z)\), with respect to point \(\alpha \in \mathbb{C}\), generalizes ordinary in sense \(\lim_{\alpha\to\infty} D_{\alpha}[P](z)/{\alpha} = P'(z).\) Recently, polynomials form \(P(z) a_0 + \sum_{j=\mu}^n a_jz^j,\) \(1\leq\mu\leq n\) and having no zero \(|z| < k\) where \(k > 1\), following Zygmund-type inequality 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 paper, we refinement involving minimum modulus \(|P(z)|\) k\), which also includes improvements some inequalities, restricted zeros as well.