是一個 
，並且以下兩個積分![int_0^inftyx^(p-1-lambda)[f(x)]^pdx](/images/equations/Carlson-LevinConstant/NumberedEquation1.svg) |
(1)
|
![int_0^inftyx^(q-1+mu)[f(x)]^qdx](/images/equations/Carlson-LevinConstant/NumberedEquation2.svg) |
(2)
|
存在且是有限的。如果 
且 
, Carlson (1934) 確定了
![int_0^inftyf(x)dx<=sqrt(pi)(int_0^infty[f(x)]^2dx)^(1/4)(int_0^inftyx^2[f(x)]^2dx)^(1/4)](/images/equations/Carlson-LevinConstant/NumberedEquation3.svg) |
(3)
|
並表明 
是最佳常數（就反例可以被構建為任何使用更小常數的更嚴格不等式的意義而言）。對於一般情況
![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,](/images/equations/Carlson-LevinConstant/NumberedEquation4.svg) |
(4)
|
Levin (1948) 表明最佳常數是
![C=1/((ps)^s(qt)^t)[(Gamma(s/alpha)Gamma(t/alpha))/((lambda+mu)Gamma((s+t)/alpha))]^alpha,](/images/equations/Carlson-LevinConstant/NumberedEquation5.svg) |
(5)
|
其中
 |  |  |
(6)
|
 |  |  |
(7)
|
 |  |  |
(8)
|
且 
是伽瑪函式。
Carlson-Levin 常數
使用 探索
參考文獻
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.在 上引用
Carlson-Levin 常數引用為
Weisstein, Eric W. "Carlson-Levin Constant." 來自 —— 資源。 https://mathworld.tw/Carlson-LevinConstant.html