### Inequalities for the polar derivative of a complex polynomial

#### Abstract

Let P(z) :=

Pn

v=0

avzv be a univariate complex coecient polynomial of degree n.

Then as a generalization of a well-known classical inequality of Turan [23], it was shown

by Govil [6] that if P(z) has all its zeros in jzj k; k 1, then

max

jzj=1

jP0(z)j

n

1 + kn max

jzj=1

jP(z)j;

whereas, if P(z) 6= 0 in jzj < k; k 1, it was again Govil [5] who gave an extension of

the classical Erdos-Lax inequality [12], by obtaining

max

jzj=1

jP0(z)j

n

1 + kn max

jzj=1

jP(z)j;

provided jP0(z)j and jQ0(z)j attain maximum at the same point on jzj = 1, where

Q(z) = znP

????1

z

. In this paper, we obtain several generalizations and renements of

the above inequalities and related results while taking into account the placement of

the zeros and the extremal coecients of the polynomial. Moreover, some concrete numerical

examples are presented, showing that in some situations, the bounds obtained

by our results can be considerably sharper than the ones previously known.

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