## Carlson-Levin Constant

N.B. A detailed on-line essay by S. Finch was the starting point for this entry.

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
 (5)

where
 (6) (7) (8)

and is the Gamma Function.

References

Beckenbach, E. F.; and Bellman, R. Inequalities. New York: Springer-Verlag, 1983.

Boas, R. P. Jr. Review of Levin, V. I. Exact Constants in Inequalities of the Carlson Type.'' Math. Rev. 9, 415, 1948.

Finch, S. Favorite Mathematical Constants.'' http://www.mathsoft.com/asolve/constant/crlslvn/crlslvn.html

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. Kluwer, 1991.