Some Lp Inequalities for Polynomials Not Vanishing inside a Circle

Abdullah Mir,Bilal Dar

doi:10.1155/2014/272405

Abstract

If P(z) is a polynomial of degree n having no zeros in |z|<1, then it is known that, for all α,β∈𝒞 with |α|≤1, |β|≤1, R>r≥1, and p>0, P(Rz)-αP(rz)+β{((R+1)/(r+1))n-|α|}P(rz)p≤([Rn-αrn+β{((R+1)/(r+1))n-|α|}]z+[1-α+β{((R+1)/(r+1))n-α}]p/1+zp)P(z)p. In this paper, we will prove a result which not only generalizes the above inequality but also generalize and refines the various results pertaining to the Lp norm of P(z)∀p>0. We will also prove a result which extends and refines a result of Boas Jr. and Rahman (1962). Also we will see that our results lead to some striking conclusions giving refinements and generalizations of other well-known results.

Highlights

For an nth degree polynomial P(z), define ‖P‖p := { ∫󵄨󵄨󵄨󵄨󵄨P󵄨󵄨󵄨󵄨󵄨p 1/p }, p > 0, (1) ‖P‖∞
If P(z) is a polynomial of degree n which does not vanish in |z| < 1, for all α, β ∈ C with |α| ≤ 1, |β| ≤ 1, R > r ≥ 1 and |z| ≥ 1
If P(z) is a polynomial of degree n which does not vanish in |z| < 1, for all α, β ∈ C with |α| ≤ 1, |β| ≤ 1, R > r ≥ 1 and p > 0

Summary

Introduction

If P(z) is a polynomial of degree n which does not vanish in |z| < 1, for every α, β, δ ∈ C with |α| ≤ 1, |β| ≤ 1, |δ| ≤ 1, R > r ≥ 1, and p > 0, Further the following result given in [8] provides a refinement of Theorem A which among other results provides a compact generalization of inequalities (6) and (7) as well. As an application of Theorem 1, we prove the following generalization and refinement of a result of Boas Jr. and Rahman [14] for p ≥ 1. For α = r = 1, Theorem 4 reduces to the following corollary which is a compact generalization of inequality (7) due to Aziz and Dawood [5, Theorem 2] to Lp norm.

Lemmas

Proof of Theorems