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# Carlson-Levin Constant

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) (7) (8)

and is the gamma function.

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## 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.

## Referenced on Wolfram|Alpha

Carlson-Levin Constant

## Cite this as:

Weisstein, Eric W. "Carlson-Levin Constant." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Carlson-LevinConstant.html