Minkowski Sum Inequality

If $p>1$ and $a_k$, $b_k > 0$, then

$$\left[{\,\sum_{k=1}^n (a_k+b_k)^p}\right]^{1/p} \leq \left({... ...}^p}\right)^{1/p}+\left({\,\sum_{k=1}^n {b_k}^p}\right)^{1/p}.$$

Equality holds Iff the sequences $a_1$, $a_2$, ... and $b_1$, $b_2$, ... are proportional.

See also Minkowski Integral 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. 24-26, 1988.