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Journal of Mathematical Sciences

, Volume 183, Issue 6, pp 762–771 | Cite as

The reverse Hölder inequality for power means

  • Viktor D. Didenko
  • Anatolii A. Korenovskyi
Article
  • 75 Downloads

Abstract

For a function φ non-negative on the interval [0, 1], the power mean of order α ≠ 0 is defined by the equality \( \mathcal{M}_{\alpha \varphi} (t) = {\left( {\frac{1}{t}\int_0^t {{\varphi^\alpha }(u)du} } \right)^{1/\alpha }},\,0 < t \leqslant 1 \). We consider the class \( {\widetilde{{RH}}^{\alpha, \beta }}(B) \)of functions φ satisfying the reverse Hölder inequality
$$ {\mathcal{M}_\beta }_\varphi \leqslant B \cdot {\mathcal{M}_\alpha }_\varphi $$
at some α < β,α·β ≠ 0,β > 1. The sharp estimates for the summability exponents of the compositions of power means are established. As a result, we determine the properties of self-improvement of the summability exponents of functions from \( {\widetilde{{RH}}^{\alpha, \beta }}(B) \).

Keywords

Power means property of self-improvement of the summability exponents reverse Hölder inequality 

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Copyright information

© Springer Science+Business Media, Inc. 2012

Authors and Affiliations

  1. 1.University of Brunei Darussalam, Faculty of ScienceBandar Seri BegawanBrunei
  2. 2.Institute of Mathematics, Economics, and MechanicsI. I. Mechnikov Odessa National University 2OdessaUkraine

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