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Carlson-Levin Constant

Assume that f is a nonnegative real function on [0,infty) and that the two integrals

int_0^inftyx^(p-1-lambda)[f(x)]^pdx
(1)
int_0^inftyx^(q-1+mu)[f(x)]^qdx
(2)

exist and are finite. If p=q=2 and lambda=mu=1, Carlson (1934) determined

int_0^inftyf(x)dx<=sqrt(pi)(int_0^infty[f(x)]^2dx)^(1/4)(int_0^inftyx^2[f(x)]^2dx)^(1/4)
(3)

and showed that sqrt(pi) 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

int_0^inftyf(x)dx<=C(int_0^inftyx^(p-1-lambda)[f(x)]^pdx)^s(int_0^inftyx^(q-1+mu)[f(x)]^qdx)^t,
(4)

and Levin (1948) showed that the best constant is

C=1/((ps)^s(qt)^t)[(Gamma(s/alpha)Gamma(t/alpha))/((lambda+mu)Gamma((s+t)/alpha))]^alpha,
(5)

where

s=mu/(pmu+qlambda)
(6)
t=lambda/(pmu+qlambda)
(7)
alpha=1-s-t
(8)

and Gamma(z) 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

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