Inequalities for the polar derivative of a complex polynomial

Abdullah Mir ; Imtiaz Hussain Dar

Filomat, Tome 36 (2022) no. 16, p. 5631

Résumé

Let P(z) := n∑ v=0 avzv be a univariate complex coefficient polynomial of degree n. Then as a generalization of a well-known classical inequality of Tura´n [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 + kn max |z|=1 |P(z)|, whereas, if P(z) , 0 in |z| k, k ≤ 1, it was again Govil [6] who gave an extension of the classical Erdo¨s-Lax inequality [13], by obtaining max |z|=1 |P′(z)| ≤ n 1 + kn max |z|=1 |P(z)|, provided |P′(z)| and |Q′(z)| attain maximum at the same point on |z| = 1, where Q(z) = znP ( 1 z ) . 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.

Détail

DOI : 10.2298/FIL2216631M

Classification : 30A10, 30C15, 30D15
Keywords: Polar derivative of a polynomial, Minimum Modulus Principle, Zeros