Hölder Sum Inequality

If

$${1\over p}+{1\over q}=1$$

with $p$, $q>1$, then

$$\sum_{k=1}^n \vert a_kb_k\vert\leq \left({\,\sum_{k=1}^n \ve... ...t)^{1/p} \left({\,\sum_{k=1}^n \vert b_k\vert^q}\right)^{1/q},$$

with equality when $\vert b_k\vert=c\vert a_k\vert^{p-1}$. If $p=q=2$, this becomes the Cauchy Inequality.

References

Abramowitz, M. and Stegun, C. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, p. 11, 1972.

Gradshteyn, I. S. and Ryzhik, I. M. Tables of Integrals, Series, and Products, 5th ed. San Diego, CA: Academic Press, p. 1092, 1979.

Hardy, G. H.; Littlewood, J. E.; and Pólya, G. Inequalities, 2nd ed. Cambridge, England: Cambridge University Press, pp. 10-15, 1988.