Some integral mean inequalities concerning polar derivative of a polynomial

Abstract

Let $P(z) = \sum\limits_{j=0}^{n}c_{j}z^{j}$ be a polynomial of degree $n$ having all its zeros in $|z|\leq 1$, then Dubinin [J. Math. Sci., 143(2007), 3069-3076.] proved $$\max\limits_{|z|=1}|P'(z)|\geq\left\{\dfrac{n}{2}+\dfrac{1}{2}\dfrac{|c_{n}|-|c_{0}|}{|c_{n}|+|c_{0}|}\right\}\max\limits_{|z|=1}|P(z)|.$$ In this paper, we shall first obtain an integral inequality for the polar derivative of the above inequality. As an application of this result, we prove another inequality which is the $L^{r}$ analogue of an inequality in polar derivative proved recently by Mir et al. [J. Interdisciplinary Math. 21(2018), 1387-1393].