A new (probabilistic) proof of the Diaz–Metcalf and Pólya–Szegő inequalities and some applications
Author:
Tibor K. Pogány
Translated by:
The author
Journal:
Theor. Probability and Math. Statist. 70 (2005), 113-122
MSC (2000):
Primary 26D15, 60E15
DOI:
https://doi.org/10.1090/S0094-9000-05-00635-6
Published electronically:
August 12, 2005
MathSciNet review:
2109828
Full-text PDF Free Access
Abstract |
References |
Similar Articles |
Additional Information
Abstract: The Diaz–Metcalf and Pólya–Szegő 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
- Villő Csiszár and Tamás F. Móri, The convexity method of proving moment-type inequalities, Statist. Probab. Lett. 66 (2004), no. 3, 303–313. MR 2045475, DOI https://doi.org/10.1016/j.spl.2003.11.007
- J. B. Diaz and F. T. Metcalf, Stronger forms of a class of inequalities of G. Pólya-G. Szegö, and L. V. Kantorovich, Bull. Amer. Math. Soc. 69 (1963), 415–418. MR 146324, DOI https://doi.org/10.1090/S0002-9904-1963-10953-2
- Werner Greub and Werner Rheinboldt, On a generalization of an inequality of L. V. Kantorovich, Proc. Amer. Math. Soc. 10 (1959), 407–415. MR 105028, DOI https://doi.org/10.1090/S0002-9939-1959-0105028-3
- L. V. Kantorovič, Functional analysis and applied mathematics, Uspehi Matem. Nauk (N.S.) 3 (1948), no. 6(28), 89–185 (Russian). MR 0027947
- O. I. Klesov, Letter to the author (2003). (Unpublished)
- Dragoslav S. Mitrinović, Analitičke nejednakosti, University of Belgrade Monographs, vol. 2, Gradevinska Knjiga, Belgrade, 1970 (Serbo-Croatian). With the collaboration of Petar M. Vasić. MR 0279261
- D. S. Mitrinović and J. E. Pečarić, Mean Values in Mathematics, Matematički problemi i ekspozicije, vol. 14, Naučna Knjiga, Beograd, 1989.
- George Pólya and Gabor Szegő, Problems and theorems in analysis. II, Classics in Mathematics, Springer-Verlag, Berlin, 1998. Theory of functions, zeros, polynomials, determinants, number theory, geometry; Translated from the German by C. E. Billigheimer; Reprint of the 1976 English translation. MR 1492448
- B. C. Rennie, On a class of inequalities, J. Austral. Math. Soc. 3 (1963), 442–448. MR 0166313
- P. Schweitzer, An inequality concerning the arithmetic mean, Math. Phys. Lapok 23 (1914), 257–261. (Hungarian)
References
- 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)
- J. B. Diaz and F. T. Metcalf, Stronger forms of a class of inequalities of G. Pólya–G. Szegő and L. V. Kantorovich, Bull. Amer. Math. Soc. 69 (1963), 415–418. MR 0146324 (26:3846)
- 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)
- 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)
- O. I. Klesov, Letter to the author (2003). (Unpublished)
- D. S. Mitrinović, Analitičke nejednakosti, Gra\Dd evinska Knjiga, Beograd, 1970. MR 0279261 (43:4984)
- D. S. Mitrinović and J. E. Pečarić, Mean Values in Mathematics, Matematički problemi i ekspozicije, vol. 14, Naučna Knjiga, Beograd, 1989.
- G. Pólya and G. Szegő, Problems and Theorems in Analysis, Classics in Mathematics Series, vol. I, Springer-Verlag, New York, 1976. MR 1492448
- B. C. Rennie, On a class of inequalities, J. Austral. Math. Soc. 3 (1963), 442–448. MR 0166313 (29:3590)
- P. Schweitzer, An inequality concerning the arithmetic mean, Math. Phys. Lapok 23 (1914), 257–261. (Hungarian)
Similar Articles
Retrieve articles in Theory of Probability and Mathematical Statistics
with MSC (2000):
26D15,
60E15
Retrieve articles in all journals
with MSC (2000):
26D15,
60E15
Additional Information
Tibor K. Pogány
Affiliation:
Faculty of Maritime Studies, University of Rijeka, Studentska 2, 51000 Rijeka, Croatia
Email:
poganj@brod.pfri.hr
Keywords:
Almost surely bounded random variable,
Diaz–Metcalf inequality,
discrete inequality,
integral inequality,
Kantorovich inequality,
mathematical expectation,
Pólya–Szegő inequality,
Rennie inequality,
Schweitzer inequality
Received by editor(s):
March 20, 2002
Published electronically:
August 12, 2005
Article copyright:
© Copyright 2005
American Mathematical Society
