Remote Access Proceedings of the American Mathematical Society
Green Open Access

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
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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.

References [Enhancements On Off] (What's this?)

  • [1] A. M. Fink and M. Jodeit, Jr., Chebyshev inequalities and functions with higher monotonicities, Tech. Rep. 1980.
  • [2] A. M. Fink and M. Jodeit, Jr., On Chebyshev's other inequality, Inequalities in Statistics and Probability, Lecture Notes IMS, vol. 5, Inst. Math. Statist., Hayward, CA, 1984, pp. 115-120. MR 789242 (86m:26017)
  • [3] G. H. Hardy, J. E. Littlewood, and G. Pólya, Inequalities, Cambridge Univ. Press, Cambridge and London, 1934.
  • [4] D. S. Mitrinović, Analytic inequalities, Springer, New York, 1970. MR 0274686 (43:448)
  • [5] J. E. Pečarić, On the Ostrowski generalization of Chebyshev's inequality, J. Math. Anal. Appl. 102 (1984), 479-487. MR 755978 (86a:26030)
  • [6] K. B. Stolarsky, From Wythoff's Nim to Chebyshev's inequality, Amer. Math. Monthly 98 (1991), 889-900. MR 1137536 (93b:90132)
  • [7] P. M. Vasić, L. R. Stanković, and J. E. Pečarić, Notes on the Cebysev inequality, Numerical Methods and Approximation Theory (Novi Sad, Sept. 4-6, 1985), Inst. Math., Univ. Novi Sad, 1985, pp. 115-120. MR 822492 (87b:26039)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 26D10

Retrieve articles in all journals with MSC: 26D10

Additional Information

Keywords: Inequalities, Chebyshev inequality, Stolarsky inequality, functions of bounded variation, gamma function, weights
Article copyright: © Copyright 1995 American Mathematical Society

American Mathematical Society