Assume that
is a nonnegativereal
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
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.