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

$$\underset{|z|=1}{max}|P^{\prime }(z)|\ge \{{\displaystyle \frac{n}{2}}+{\displaystyle \frac{1}{2}}{\displaystyle \frac{|c_n|-|c_0|}{|c_n|+|c_0|}}\}\underset{|z|=1}{max}|P(z)|.$$

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].