SOME INTEGRAL INEQUALITIES FOR THE POLAR DERIVATIVE OF A POLYNOMIAL

If P（z） is a polynomial of degree n which does not vanish in ｜z｜ 〈 1, then it is recently proved by Rather [Jour. Ineq. Pure andAppl. Math., 9 （2008）, Issue 4, Art. 103] that for every γ 〉 0 and every real or complex number a with ｜ α ｜ ≥ 1, {∫ 2π 0｜DαP（e^iθ）｜γdθ｜}^1/γ≤n（｜α｜＋1）Cγ{∫2π0｜P（e^iθ）｜γ^dθ}^...

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Bibliographic Details
Published inAnalysis in theory & applications Vol. 27; no. 4; pp. 340 - 350
Main Authors Mir, Abdullah, Baba, Sajad Amin
Format Journal Article
LanguageEnglish
Published Heidelberg Editorial Board of Analysis in Theory and Applications 01.12.2011
Subjects
Analysis
Approximations and Expansions
DAP
Mathematics
Mathematics and Statistics
复数
多项式
数学
极地
积分不等式
衍生
证明
polar derivative
30C10
polynomial
30A10
Zygmund inequality
zeros
30D15
41A17
Online AccessGet full text
ISSN1672-4070
1573-8175
DOI10.1007/s10496-011-0340-z

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Summary:If P（z） is a polynomial of degree n which does not vanish in ｜z｜ 〈 1, then it is recently proved by Rather [Jour. Ineq. Pure andAppl. Math., 9 （2008）, Issue 4, Art. 103] that for every γ 〉 0 and every real or complex number a with ｜ α ｜ ≥ 1, {∫ 2π 0｜DαP（e^iθ）｜γdθ｜}^1/γ≤n（｜α｜＋1）Cγ{∫2π0｜P（e^iθ）｜γ^dθ}^1/γ, Cγ={1/2π∫2π 0｜1＋e^iβ｜^γdβ}^-1/γ, where DaP（z） denotes the polar derivative of P（z） with respect to α. In this paper we prove a result which not only provides a refinement of the above inequality but also gives a result of Aziz and Dawood [J. Approx. Theory, 54 （1988）, 306-313] as a special case.
Bibliography:32-1631/O1
polar derivative, polynomial, Zygmund inequality, zeros
If P（z） is a polynomial of degree n which does not vanish in ｜z｜ 〈 1, then it is recently proved by Rather [Jour. Ineq. Pure andAppl. Math., 9 （2008）, Issue 4, Art. 103] that for every γ 〉 0 and every real or complex number a with ｜ α ｜ ≥ 1, {∫ 2π 0｜DαP（e^iθ）｜γdθ｜}^1/γ≤n（｜α｜＋1）Cγ{∫2π0｜P（e^iθ）｜γ^dθ}^1/γ, Cγ={1/2π∫2π 0｜1＋e^iβ｜^γdβ}^-1/γ, where DaP（z） denotes the polar derivative of P（z） with respect to α. In this paper we prove a result which not only provides a refinement of the above inequality but also gives a result of Aziz and Dawood [J. Approx. Theory, 54 （1988）, 306-313] as a special case.
ISSN:1672-4070
1573-8175
DOI:10.1007/s10496-011-0340-z