Some generalizations involving the polar derivative for an inequality of Paul Turán

N. Govil; G. McTume

doi:10.1023/b:amhu.0000034366.66170.01

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Acta Mathematica Hungarica

Volume/Issue:: Volume 104: Issue 1-2

Some generalizations involving the polar derivative for an inequality of Paul Turán

Authors: N. Govil¹ and G. McTume²

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¹ Auburn University Department of Mathematics Auburn Al 36849-5310 U.S.A Auburn Al 36849-5310 U.S.A
² Ferris State University Department of Mathematics Big Rapids Mi 49307-2225 U.S.A Big Rapids Mi 49307-2225 U.S.A

DOI:: https://doi.org/10.1023/b:amhu.0000034366.66170.01

Page Count:: 115–126

Publication Date:: 01 Jul 2004

Online Publication Date:: 02 Nov 2004

Article Category:: Research Article

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Abstract

Let p(z) be a polynomial of degree n and for a complex number α, let D_αp(z) = np(z) + (α-z)p'(z) denote the polar derivative of the polynomial p(z) with respect to α. In this paper, we obtain inequalities for the polar derivative of a polynomial having all its zeros in |z| ≤ K. Our results generalize and sharpen a famous inequality of Turn and some other known results in this direction.