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Copson's Inequality

Let $\{a_n\}$ be a Nonnegative Sequence and $f(x)$ a Nonnegative integrable function. Define

$\displaystyle A_n$ $\textstyle =$ $\displaystyle \sum_{k=1}^n a_k$ (1)
$\displaystyle B_n$ $\textstyle =$ $\displaystyle \sum_{k=n}^\infty a_k$ (2)

$\displaystyle F(x)$ $\textstyle =$ $\displaystyle \int_0^x f(t)\,dt$ (3)
$\displaystyle G(x)$ $\textstyle =$ $\displaystyle \int_x^\infty f(t)\,dt,$ (4)

and take $0<p<1$. For integrals,
\int_0^\infty \left[{G(x)\over x}\right]^p\,dx > \left({p\over p-1}\right)^p\int_0^\infty [f(x)]^p\,dx
\end{displaymath} (5)

(unless $f$ is identically 0). For sums,
\left({1+{1\over p-1}}\right){B_1}^p+\sum_{n=2}^\infty \left...
...t)^p >
\left({p\over p-1}\right)^p \sum_{n=1}^\infty {a_n}^p
\end{displaymath} (6)

(unless all $a_n=0$).


Beesack, P. R. ``On Some Integral Inequalities of E. T. Copson.'' In General Inequalities 2 (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.

© 1996-9 Eric W. Weisstein