Inequalities for the Polar Derivative of a Complex Polynomial

Let P(z) := Sigma(n)(v=0) a(v)z(v) be a univariate complex coefficient polynomial of degree n. Then as a generalization of a well-known classical inequality of Turan [25], it was shown by Govil [7] that if P(z) has all its zeros in |z| <= k, k >= 1, then max(|z|=1) |P '(z)| >= n/1+k(n) max(|z|=1) |P(z)|, whereas, if P(z) not equal 0 in |z| < k, k <= 1, it was again Govil [6] who gave an extension of the classical Erdos-Lax inequality [13], by obtaining max(|z|=1) |P '(z)| <= n/1+k(n) max(|z|=1) |P(z)|, provided |P '(z)| and |Q '(z)| attain maximum at the same point on |z| = 1, where Q(z) = z(n)<(P(1/<(z)over bar>))over bar>. In this paper, we obtain several generalizations and refinements of the above inequalities and related results while taking into account the placement of the zeros and extremal coefficients of the underlying 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.