Stolarsky’s inequality with general weights
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- by Lech Maligranda, Josip E. Pečarić and Lars Erik Persson
- Proc. Amer. Math. Soc. 123 (1995), 2113-2118
- DOI: https://doi.org/10.1090/S0002-9939-1995-1243171-8
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Abstract:
Recently Stolarsky proved that the inquality \begin{equation}\tag {$\ast $} \int _0^1 {g({x^{1/(a + b)}}) dx \geq \int _0^1 {g({x^{1/a}}) dx\int _0^1 {g({x^{1/b}}) dx} } } \end{equation} holds for every $a,b > 0$ and every nonincreasing function on [0, 1] satisfying $0 \leq g(u) \leq 1$. In this paper we prove a weighted version of this inequality. Our proof is based on a generalized Chebyshev inequality. In particular, our result shows that the inequality $( \ast )$ holds for every function g of bounded variation. We also generalize another inequality by Stolarsky concerning the $\Gamma$-function.References
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Bibliographic Information
- © Copyright 1995 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 123 (1995), 2113-2118
- MSC: Primary 26D10
- DOI: https://doi.org/10.1090/S0002-9939-1995-1243171-8
- MathSciNet review: 1243171