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Power means and the reverse Hölder inequality

Volume 207 / 2011

Victor D. Didenko, Anatolii A. Korenovskyi Studia Mathematica 207 (2011), 85-95 MSC: Primary 42B25, 42B35; Secondary 46E30. DOI: 10.4064/sm207-1-6

Abstract

Let $w$ be a non-negative measurable function defined on the positive semi-axis and satisfying the reverse Hölder inequality with exponents $0<\alpha<\beta$. In the present paper, sharp estimates of the compositions of the power means $\mathcal{P}_\alpha w(x):=((1/x)\int_0^x w^\alpha(t)\,dt)^{1/\alpha}$, $x>0$, are obtained for various exponents $\alpha$. As a result, for the function $w$ a property of self-improvement of summability exponents is established.

Authors

  • Victor D. DidenkoUniversity of Brunei Darussalam
    BE1410 Bandar Seri Begawan, Brunei
    e-mail
  • Anatolii A. KorenovskyiOdessa I. I. Mechnikov National University
    Dvoryanskaya 2
    65026 Odessa, Ukraine
    e-mail

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