On inverses of the Hölder inequality
type: Item
instance of: scholarly article
title: On inverses of the Hölder inequality
zbMATH Open document ID: 0752.26009
DOI: 10.1016/0022-247X(91)90351-Y
published in: Journal of Mathematical Analysis and Applications
publication date: 1992-09-27T00:00:00Z
review text: The author derives an inequality complementary to \[ x^{1/p} y^{1/q}\leq x/p+y/q. \] Using this inequality, then converse Hölder- type inequalities of the form \[ a\| f\|_ 1+b\| g\|_ 1\leq c\| f^{1/p} g^{1/q}\|_ 1\qquad\text{and}\qquad \| f\|_ p\| g\|_ q\leq d\| fg\|_ 1 \] are obtained, where the ranges of \(f\) and \(g\) are compact subintervals of \((0,\infty)\). The constants \(c\) and \(d\) then depend on \(a\), \(b\), \(p\), \(q\) and the endpoints of the intervals containing the ranges of \(f\) and \(g\). The results include a lot of earlier inequalities complementary to Hölder's one.
Mathematics Subject Classification ID: 26D15
zbMATH DE Number: 57731
zbMATH Keywords: complementary inequality
zbMATH Keywords: Hölder inequality
zbMATH Keywords: inverse inequality
author: Ya-Dong Zhuang
reviewed by: Zsolt Páles
MaRDI profile type: Publication
cites work: Equivalence of certain inequalities complementing those of Cauchy-Schwarz and Holder
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description: scientific article; zbMATH DE number 57731