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 Free Access
Abstract: Recently Stolarsky proved that the inquality
holds for every and every nonincreasing function on [0, 1] satisfying . 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 holds for every function g of bounded variation. We also generalize another inequality by Stolarsky concerning the -function.
-  A. M. Fink and M. Jodeit, Jr., Chebyshev inequalities and functions with higher monotonicities, Tech. Rep. 1980.
-  A. M. Fink and Max Jodeit Jr., On Chebyshev’s other inequality, Inequalities in statistics and probability (Lincoln, Neb., 1982) IMS Lecture Notes Monogr. Ser., vol. 5, Inst. Math. Statist., Hayward, CA, 1984, pp. 115–120. MR 789242, https://doi.org/10.1214/lnms/1215465637
-  G. H. Hardy, J. E. Littlewood, and G. Pólya, Inequalities, Cambridge Univ. Press, Cambridge and London, 1934.
-  D. S. Mitrinović, Analytic inequalities, Springer-Verlag, New York-Berlin, 1970. In cooperation with P. M. Vasić. Die Grundlehren der mathematischen Wissenschaften, Band 165. MR 0274686
-  Josip E. Pečarić, On the Ostrowski generalization of Čebyšev’s inequality, J. Math. Anal. Appl. 102 (1984), no. 2, 479–487. MR 755978, https://doi.org/10.1016/0022-247X(84)90187-2
-  Kenneth B. Stolarsky, From Wythoff’s Nim to Chebyshev’s inequality, Amer. Math. Monthly 98 (1991), no. 10, 889–900. MR 1137536, https://doi.org/10.2307/2324146
-  Petar M. Vasić, Ljubomir R. Stanković, and Josip E. Pečarić, Notes on the Čebyšev inequality, Numerical methods and approximation theory, II (Novi Sad, 1985) Univ. Novi Sad, Novi Sad, 1985, pp. 115–119 (English, with Serbo-Croatian summary). MR 822492
- A. M. Fink and M. Jodeit, Jr., Chebyshev inequalities and functions with higher monotonicities, Tech. Rep. 1980.
- 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)
- G. H. Hardy, J. E. Littlewood, and G. Pólya, Inequalities, Cambridge Univ. Press, Cambridge and London, 1934.
- D. S. Mitrinović, Analytic inequalities, Springer, New York, 1970. MR 0274686 (43:448)
- J. E. Pečarić, On the Ostrowski generalization of Chebyshev's inequality, J. Math. Anal. Appl. 102 (1984), 479-487. MR 755978 (86a:26030)
- K. B. Stolarsky, From Wythoff's Nim to Chebyshev's inequality, Amer. Math. Monthly 98 (1991), 889-900. MR 1137536 (93b:90132)
- 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)
Retrieve articles in Proceedings of the American Mathematical Society with MSC: 26D10
Retrieve articles in all journals with MSC: 26D10
Keywords: Inequalities, Chebyshev inequality, Stolarsky inequality, functions of bounded variation, gamma function, weights
Article copyright: © Copyright 1995 American Mathematical Society