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## Useful inequalities cheat sheet

Useful Inequalities:**Download PDF** **Download zipped PostScript**

##### References

*Behind every great theorem lies a great inequality.* (paraphrasing A. N. Kolmogorov and H. Balzac - a variant of the quote appears in Marshall et al.: Inequalities... - the other variant is in the movie The Godfather.)

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. 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, but I tried to resist the temptation of including only the most general form (which may not be the most easily applicable).

Useful Inequalities:

- G. H. Hardy, J. E. Littlewood, G. Pólya: Inequalities.
- J. M. Steele: The Cauchy-Schwarz Master Class: An Introduction to the Art of Mathematical Inequalities.
- D. S. Bernstein: Matrix Mathematics: Theory, Facts, and Formulas.
- D. E. Knuth: The Art of Computer Programming (Volume 4)
- A. W. Marshall, I. Olkin, B. C. Arnold: Inequalities: Theory of Majorization and Its Applications.
- D. S. Mitrinovic: Analytic Inequalities.
- D. S. Mitrinovic, J. E. Pecaric, A. M. Fink: Classical and New Inequalities in Analysis.
- P. S. Bullen: A Dictionary of Inequalities.
- P. S. Bullen: Handbook of Means and Their Inequalities.
- J. Herman, R. Kucera, J. Simsa: Equations and Inequalities.
- A. Lohwater: Introduction to Inequalities.
- D. Dubhashi, A. Panconesi: Concentration of Measure for the Analysis of Randomized Algorithms.
- S. Jukna: Extremal Combinatorics.
- J. H. Spencer: Ten Lectures on the Probabilistic Method.
**Other sources:**- Wikipedia: List of inequalities
- G. J. Woeginger: When Cauchy and Hölder Met Minkowski: A Tour through Well-Known Inequalities.
- Zhi-Hong Sun: Inequalities for binomial coefficients.
- functions.wolfram.com: Elementary Functions
- http://rjlipton.wordpress.com/2014/01/15/bounds-on-binomial-coefficents/
- LaTeX template: http://www.stdout.org/~winston/latex/
- Suggestions, observations: Sebastian Ziesche (product diff. ineq.), ...

2011- László Kozma. Please send corrections, completions, suggestions to Lkozma@gmail.com. I will keep uploading the newest version to this page. CC Attribution-ShareAlike 3.0.