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A new (probabilistic) proof of the Diaz-Metcalf and Pólya-Szego inequalities and some applications
Author(s):
Tibor
K.
Pogány
Translated by:
The author
Original publication:
Teoriya Imovirnostei ta Matematichna Statistika,
vipusk 70
(2004).
Journal:
Theor. Probability and Math. Statist.
No. 70
(2005),
113-122.
MSC (2000):
Primary 26D15, 60E15
Posted:
August 12, 2005
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Additional information
Abstract:
The Diaz-Metcalf and Pólya-Szego inequalities are proved in the probabilistic setting. These results generalize the classical case for both sums and integrals. Using these results we obtain some other well-known inequalities in the probabilistic setting, namely the Kantorovich, Rennie, and Schweitzer inequalities.
References:
-
- 1.
- V. Csiszár and T. F. Móri, The convexity method of proving moment type inequalities, Stat. Probab. Lett. 66 (2004), no. 3, 303-313. MR 2045475 (2005g:60032)
- 2.
- J. B. Diaz and F. T. Metcalf, Stronger forms of a class of inequalities of G. Pólya-G. Szego and L. V. Kantorovich, Bull. Amer. Math. Soc. 69 (1963), 415-418. MR 0146324 (26:3846)
- 3.
- W. Greub and W. Rheinboldt, On a generalization of an inequality of L. V. Kantorovich, Proc. Amer. Math. Soc. 10 (1959), 407-415. MR 0105028 (21:3774)
- 4.
- L. V. Kantorovich, Functional analysis and applied mathematics, Uspekhi Matem. Nauk (N.S.) 3 (1948), no. 6(28), 89-185. (Russian) MR 0027947 (10:380a)
- 5.
- O. I. Klesov, Letter to the author (2003). (Unpublished)
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- D. S. Mitrinovic, Analiticke nejednakosti, Gradevinska Knjiga, Beograd, 1970. MR 0279261 (43:4984)
- 7.
- D. S. Mitrinovic and J. E. Pecaric, Mean Values in Mathematics, Matematicki problemi i ekspozicije, vol. 14, Naucna Knjiga, Beograd, 1989.
- 8.
- G. Pólya and G. Szego, Problems and Theorems in Analysis, Classics in Mathematics Series, vol. I, Springer-Verlag, New York, 1976. MR 1492448
- 9.
- B. C. Rennie, On a class of inequalities, J. Austral. Math. Soc. 3 (1963), 442-448. MR 0166313 (29:3590)
- 10.
- P. Schweitzer, An inequality concerning the arithmetic mean, Math. Phys. Lapok 23 (1914), 257-261. (Hungarian)
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Additional Information:
Tibor
K.
Pogány
Affiliation:
Faculty of Maritime Studies, University of Rijeka, Studentska 2, 51000 Rijeka, Croatia
Email:
poganj@brod.pfri.hr
DOI:
10.1090/S0094-9000-05-00635-6
PII:
S 0094-9000(05)00635-6
Keywords:
Almost surely bounded random variable,
Diaz--Metcalf inequality,
discrete inequality,
integral inequality,
Kantorovich inequality,
mathematical expectation,
P\'olya--Szeg\H o inequality,
Rennie inequality,
Schweitzer inequality
Received by editor(s):
20/MAR/2002
Posted:
August 12, 2005
Copyright of article:
Copyright
2005,
American Mathematical Society
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