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Power means and the reverse Hölder inequality

Victor D. Didenko; Anatolii A. Korenovskyi — 2011

Studia Mathematica

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

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