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Copson's Inequality

Let {a_n} be a nonnegative sequence and f(x) a nonnegative integrable function. Define

A_n=sum_(k=1)^(n)a_k
(1)
B_n=sum_(k=n)^(infty)a_k
(2)

and

F(x)=int_0^xf(t)dt
(3)
G(x)=int_x^inftyf(t)dt,
(4)

and take 0<p<1. For integrals,

int_0^infty[(G(x))/x]^pdx>(p/(p-1))^pint_0^infty[f(x)]^pdx
(5)

(unless f is identically 0). For sums,

(1+1/(p-1))B_1^p+sum_(n=2)^infty((B_n)/n)^p>(p/(p-1))^psum_(n=1)^inftya_n^p
(6)

(unless all a_n=0).

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References

Beesack, P. R. "On Some Integral Inequalities of E. T. Copson." In General Inequalities 2: Proceedings of the Second International Conference on General Inequalities, held in the Mathematical Research Institut at Oberwolfach, Black Forest, July 30-August 5, 1978 (Ed. E. F. Beckenbach). Basel: Birkhäuser, 1980.Copson, E. T. "Some Integral Inequalities." Proc. Royal Soc. Edinburgh 75A, 157-164, 1975-1976.Hardy, G. H.; Littlewood, J. E.; and Pólya, G. Theorems 326-327, 337-338, and 345 in Inequalities. Cambridge, England: Cambridge University Press, 1934.Mitrinovic, D. S.; Pecaric, J. E.; and Fink, A. M. Inequalities Involving Functions and Their Integrals and Derivatives. Dordrecht, Netherlands: Kluwer, 1991.

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Copson's Inequality

Cite this as:

Weisstein, Eric W. "Copson's Inequality." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/CopsonsInequality.html

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