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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Stolarsky's inequality with general weights


Authors: Lech Maligranda, Josip E. Pečarić and Lars Erik Persson
Journal: Proc. Amer. Math. Soc. 123 (1995), 2113-2118
MSC: Primary 26D10
MathSciNet review: 1243171
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Abstract: Recently Stolarsky proved that the inquality

$\displaystyle \int_0^1 {g({x^{1/(a + b)}})\,dx \geq \int_0^1 {g({x^{1/a}})\,dx\int_0^1 {g({x^{1/b}})\,dx} } }$ ($ \ast$)

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.

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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1995-1243171-8
PII: S 0002-9939(1995)1243171-8
Keywords: Inequalities, Chebyshev inequality, Stolarsky inequality, functions of bounded variation, gamma function, weights
Article copyright: © Copyright 1995 American Mathematical Society