Integral mean estimates for polynomials whose zeros are within a circle
type: Item
instance of: scholarly article
zbMATH Open document ID: 0874.30001
author: A. Aziz
author: N. A. Rather
publication date: 1997-11-05T00:00:00Z
Mathematics Subject Classification ID: 30A10
zbMATH DE Number: 981741
zbMATH Keywords: integral mean estimates
MaRDI profile type: Publication
title: Integral mean estimates for polynomials whose zeros are within a circle
published in: Glasnik Matematički. Serija III
review text: The authors prove four theorems concerning inequalities of \(L_p\) norms on the unit circle of a polynomial with restricted zeros and its derivative. In all theorems the numbers \(p\) and \(q\) satisfy \(p>1\), \(q> 1\), \(p^{-1} +q^{-1} =1\) and \(P(z)\) is a polynomial with complex coefficients of degree \(n\). Theorem 1. If all the zeros of \(P(z)\) lie in the closed disk \(|z|\leq K\), \(K\leq 1\), then for each \(r>0\) the inequality NEWLINE\[NEWLINEn\left \{\int_0^{2\pi} \bigl|P (e^{i\theta}) \bigr|^r d\theta \right\}^{1 \over r} \leq\left \{\int^{2\pi}_0 |1+Ke^{i\theta} |^{ qr} d\theta \right\}^{1 \over qr} \left\{\int_0^{2\pi} \bigl|P' (e^{i\theta}) \bigr|^{pr} d \theta \right\}^{1\over pr}NEWLINE\]NEWLINE holds. Theorem 2. If all the zeros of \(P(z)\) lie in the closed unit disk, then for each \(r>0\) and every \(\alpha\) such that \(|\alpha |\leq 1\) the inequality NEWLINE\[NEWLINEn \left\{\int_0^{2 \pi} \bigl|P(e^{i\theta}) +\alpha m\bigr|^r d\theta\right\}^{1 \over r} \leq\left \{\int_0^{2\pi} |1+e^{i\theta} |^{qr} d\theta \right\}^{1 \over qr} \left\{\int_0^{2\pi} \bigl|P'(e^{i \theta}) \bigr|^{pr} d\theta \right\}^{1 \over pr}NEWLINE\]NEWLINE holds, where \(m=\min_{|z |=1} |P(z) |\). Theorem 3. If all the zeros of \(P(z)\) lie in the closed disk \(|z|\leq K\) where \(K\geq 1\) then for each \(r\geq 1\) the inequality NEWLINE\[NEWLINEn\left \{\int_0^{2\pi} \bigl|P(e^{i\theta}) \bigr|^r d\theta \right\}^{1\over r} \leq C_r \left\{\int_0^{2\pi} |1+e^{i\theta} |^{qr} d\theta \right\}^{1 \over qr} \left\{\int_0^{2\pi} \bigl|P'(e^{i\theta}) \bigr|^{pr} d\theta \right\}^{1\over pr}NEWLINE\]NEWLINE holds, where the constant \(C_r\) is given by NEWLINE\[NEWLINEC_r= {\{\int_0^{2\pi} |1+ K^ne^{i\theta} |^r d\theta\}^{1\over r} \over\{\int_0^{2\pi} |1+ e^{i\theta} |^rd \theta\}^{1\over r}}.NEWLINE\]NEWLINE Theorem 4. If all the zeros of \(P(z)\) lie in the closed disk \(|z|\leq K\), \(K\geq 1\), then for each \(r\geq 1\) and every \(\alpha\) such that \(|\alpha |\…
reviewed by: Zalman Rubinstein
description: scientific article; zbMATH DE number 981741