Certain inequalities for the polar derivatives of polynomials

Abstract

Let p(z) be a polynomial of degree n , having no zeros in |z| < k , k ≥ 1 , then Aziz [In this paper we first generalize as well as improves upon the above inequality which in turns also gives generalization and improvement of some well known inequalities. Besides,we also generalize as well as improves upon a result due to Govil [ J. Approx. Theory, Vol. 66 (1991), pp. 29-35] by extending it to Polar derivative.

Extensions of some polynomial inequalities to the polar derivative

“…It is easy to see that Theorem 3 also provides a refinement of the following result due to Dewan, Singh and Lal [8].…”

Refinements of some inequalities concerning the polar derivative of a polynomial

“…Dewan et al [7] (see also [16]) extended inequality (1.6) to the polar derivative and they proved that if P (z) = a n z n + n j=µ a n−j z n−j , 1 µ n, is a polynomial of degree n having all its zeros in |z| k, k 1, then for every complex number α with |α| k µ ,…”

Note on an inequality of M.A. Malik

“…The corresponding polar derivative analogue of ( 4) and a refinement of (5), was given by Dewan et al [3]. They proved that if P (z) = n v=0 a v z v is a polynomial of degree n having all its zeros in |z| ≤ k, k ≤ 1, then for every complex number α with |α| ≥ k,…”