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.