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Useful inequalities cheat sheet

This is a collection of some of the most important mathematical inequalities. I tried to include non-trivial inequalities that can be useful in solving problems or proving theorems, particularly in computer science. I omitted many details, in some cases even necessary conditions (hopefully only when they were obvious). If you are not sure whether an inequality can be applied in some context, try to find a more detailed source for the exact definition. For lack of space I omitted proofs and discussions on when equality holds.

I didn't include inequalities which require lengthy definitions, inequalities involving complex functions, number theory, advanced calculus (most integral inequalities) or inequalities with a pure geometric character. Some of the inequalities are special cases of others, and I tried to resist the temptation of going for the most general form (which may not be the most easily applicable).

Useful Inequalities:    Download PDF     Download zipped PostScript


Behind every great theorem lies a great inequality. (paraphrasing A. N. Kolmogorov and H. Balzac)

2011- László Kozma.  Please send corrections, completions, suggestions to I will upload the newest version to this page.   CC Attribution-ShareAlike 3.0.