On Zygmund-type inequalities concerning polar derivative of polynomials

Nisar Ahmad Rather ; Suhail Gulzar ; Aijaz Bhat

Ural mathematical journal, Tome 7 (2021) no. 1, pp. 87-95

Résumé

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.

Détail

Keywords: $L^{p}$-inequalities, polar derivative, polynomials.