### 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].**Published:**2021-05-24

**How to Cite this Article:**Barchand Chanam, Maisnam Triveni Devi, Kshetrimayum Krishnadas, N. Reingachan, Some integral mean inequalities concerning polar derivative of a polynomial, J. Math. Comput. Sci., 11 (2021), 4032-4041 Copyright © 2021 Barchand Chanam, Maisnam Triveni Devi, Kshetrimayum Krishnadas, N. Reingachan. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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