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Date: 21-11-2020
602
Date: 13-8-2020
501
Date: 11-11-2019
501
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Assume that is a nonnegative real function on and that the two integrals
(1) |
(2) |
exist and are finite. If and , Carlson (1934) determined
(3) |
and showed that is the best constant (in the sense that counterexamples can be constructed for any stricter inequality which uses a smaller constant). For the general case
(4) |
and Levin (1948) showed that the best constant is
(5) |
where
(6) |
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(7) |
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(8) |
and is the gamma function.
REFERENCES:
Beckenbach, E. F.; and Bellman, R. "Carlson's Inequality" and "Generalizations of Carlson's Inequality." §5.8 and 5.9 in Inequalities, 2nd rev. printing. New York: Springer-Verlag, pp. 175-177, 1965.
Boas, R. P. Jr. Review of Levin, V. I. "Exact Constants in Inequalities of the Carlson Type." Math. Rev. 9, 415, 1948.
Carlson, F. "Une inégalité." Arkiv för Mat., Astron. och Fys. 25B, 1-5, 1934.
Finch, S. R. "Carlson-Levin Constant." §3.2 in Mathematical Constants. Cambridge, England: Cambridge University Press, pp. 211-212, 2003.
Levin, V. I. "Exact Constants in Inequalities of the Carlson Type." Doklady Akad. Nauk. SSSR (N. S.) 59, 635-638, 1948. English review in Boas (1948).
Mitrinovic, D. S.; Pecaric, J. E.; and Fink, A. M. Inequalities Involving Functions and Their Integrals and Derivatives. Amsterdam, Netherlands: Kluwer, 1991.
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