Lq mean extension for the polar derivative of a polynomial

Bidkham, Mahmood; Motamednezhad, Ahmad

doi:10.22075/ijnaa.2025.37212.5426

Lq mean extension for the polar derivative of a polynomial

Document Type : Research Paper

Authors

Mahmood Bidkham ¹

Ahmad Motamednezhad ²

¹ Department of Mathematics, University of Semnan, Semnan, Iran

² Faculty of Mathematical Sciences, Shahrood University of Technology, Shahrood, Iran

10.22075/ijnaa.2025.37212.5426

Abstract

For a polynomial $p(z)$ of degree $n$, we consider an operator $D_{\alpha}$ which map a polynomial $p(z)$ into $D_{\alpha}p(z):=(\alpha-z)p'(z)+np(z)$ with respect to $\alpha$. It was proved by Liman et al [ A. Liman, R. N. Mohapatra and W. M. Shah, Inequalities for the polar derivative of a polynomial, Complex Anal. Oper. Theory, 2012] that if $p(z)$ has no zeros in $|z|<1$ then for all $\alpha,\ \beta\in \mathbb{C}$ with $|\alpha|\geq 1 , \ |\beta|\leq 1$ and $|z|=1$,
\begin{align*}
\begin{split}
|zD_{\alpha}p(z)+n\beta\frac{|\alpha|-1}{2}&p(z)|\leq \frac{n}{2}\{
[|\alpha+\beta\frac{|\alpha|-1}{2}|+|z+\beta\frac{|\alpha|-1}{2}|]
\max_{|z|=1}|p(z)|. \end{split}\end{align*} In this paper, we present the integral $L_q$ mean extension of the above inequality for the polar derivative of polynomials. Our result generalize certain well-known polynomial inequalities.

Keywords

Polynomial

Integral inequality

Polar derivative

Restricted zeros

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