Inequalities concerning the rate of growth of polynomials involving the polar derivative

Abdullah Mir,Adil Hussain Malik

doi:10.17951/a.2020.74.1.67-75

Abstract

This paper contains some results for algebraic polynomials in the complex plane involving the polar derivative that are inspired by some classical results of Bernstein. Obtained results yield the polar derivative analogues of some inequalities giving estimates for the growth of derivative of lacunary polynomials.

Highlights

Let P (z) =n v=0 av zv be a polynomial of degree n in the complex plane and P (z) be its derivative
The Bernstein inequality that relates the norm of a polynomial to that of its derivative and its various versions form a classical topic in analysis
Over the last four decades many different authors produced a large number of results and established various Bernstein type inequalities for polynomials involving the polar derivative DαP (z) with various choices of P (z), α and other parameters

Summary

Introduction

Let P (z) =n v=0 av zv be a polynomial of degree n in the complex plane and P (z) be its derivative. The inequality (1.1) is a famous result due to Bernstein [3], who proved it in 1912 and is best possible with the equality holding for polynomials P (z) = λzn, λ being a complex number. In 1930 (see [4]), Bernstein revisited his inequality (1.1) and established a more general result: for two polynomials P (z) and Q(z) with the degree of P (z) not exceeding that of Q(z) and Q(z) = 0 for |z| > 1, the inequality |P (z)| ≤ |Q(z)| on the unit circle |z| = 1 implies the inequality of their derivatives |P (z)| ≤ |Q (z)| on

Objectives

Results

Conclusion