Volume 32, Issue 1
Some Inequalities for the Polynomial with S-Fold Zeros at the Origin

A. Zireh and M. Bidkham

10.4208/ata.2016.v32.n1.3

Anal. Theory Appl., 32 (2016), pp. 27-37

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  • Abstract

Let p(z) be a polynomial of degree $n$, which has no zeros in |z|< 1, Dewan et al. [K. K. Dewan and Sunil Hans, Generalization of certain well known polynomial inequalities, J. Math. Anal. Appl., 363 (2010), pp. 38--41] established$$\Big|zp'(z)+\frac{n\beta}{2}p(z)\Big|\leq &\frac{n}{2}\Big\{\Big(\Big|\frac{\beta}{2}\Big|+\Big|1+\frac{\beta}{2}\Big|\Big)\max_{|z|=1}|p(z)|-\Big(\Big|1+\frac{\beta}{2}\Big|-\Big|\frac{\beta}{2}\Big|\Big)\min_{|z|=1}|p(z)|\Big\},$$for any $|\beta|\leq 1$ and |z|=1. In this paper we improve theabove inequality for the polynomial which has no zeros in $|z|< k, $ $ k\geq 1$, except $s$-fold zeros at the origin. Our resultsgeneralize certain well known polynomial inequalities.

  • History

Published online: 2016-01

  • AMS Subject Headings

30A10, 30C10, 30D15

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