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.

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.