Inequality

A mathematical statement that one quantity is greater than or less than another. ``$a$ is less than $b$'' is denoted $a<b$, and ``$a$ is greater than $b$'' is denoted $a>b$. ``$a$ is less than or equal to $b$'' is denoted $a\leq b$, and ``$a$ is greater than or equal to $b$'' is denoted $a\geq b$. The symbols $a\ll b$ and $a\gg b$ are used to denote ``$a$ is much less than $b$'' and ``$a$ is much greater than $b$,'' respectively.

Solutions to the inequality $\vert x-a\vert < b$ consist of the set $\{x: -b < x-a < b\}$, or equivalently $\{x: a-b<x<a+b\}$. Solutions to the inequality $\vert x-a\vert > b$ consist of the set $\{x: x-a > b\} \cup \{x: x-a < -b\}$. If $a$ and $b$ are both Positive or both Negative and $a<b$, then $1/a > 1/b$.

See also abc Conjecture, Arithmetic-Logarithmic-Geometric Mean Inequality, Bernoulli Inequality, Bernstein's Inequality, Berry-Osseen Inequality, Bienaymé-Chebyshev Inequality, Bishop's Inequality, Bogomolov-Miyaoka-Yau Inequality, Bombieri's Inequality, Bonferroni's Inequality, Boole's Inequality, Carleman's Inequality, Cauchy Inequality, Chebyshev Inequality, Chi Inequality, Copson's Inequality, Erdös-Mordell Theorem, Exponential Inequality, Fisher's Block Design Inequality, Fisher's Estimator Inequality, Gårding's Inequality, Gauss's Inequality, Gram's Inequality, Hadamard's Inequality, Hardy's Inequality, Harnack's Inequality, Hölder Integral Inequality, Hölder's Sum Inequality, Isoperimetric Inequality, Jarnick's Inequality, Jensen's Inequality, Jordan's Inequality, Kantorovich Inequality, Markov's Inequality, Minkowski Integral Inequality, Minkowski Sum Inequality, Morse Inequalities, Napier's Inequality, Nosarzewska's Inequality, Ostrowski's Inequality, Ptolemy Inequality, Robbin's Inequality, Schröder-Bernstein Theorem, Schur's Inequalities, Schwarz's Inequality, Square Root Inequality, Steffensen's Inequality, Stolarsky's Inequality, Strong Subadditivity Inequality, Triangle Inequality, Turán's Inequalities, Weierstraß Product Inequality, Wirtinger's Inequality, Young Inequality

References

Inequalities

Abramowitz, M. and Stegun, C. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, p. 16, 1972.

Beckenbach, E. F. and Bellman, Richard E. An Introduction to Inequalities. New York: Random House, 1961.

Beckenbach, E. F. and Bellman, Richard E. Inequalities, 2nd rev. print. Berlin: Springer-Verlag, 1965.

Hardy, G. H.; Littlewood, J. E.; and Pólya, G. Inequalities, 2nd ed. Cambridge, England: Cambridge University Press, 1952.

Kazarinoff, N. D. Geometric Inequalities. New York: Random House, 1961.

Mitrinovic, D. S. Analytic Inequalities. New York: Springer-Verlag, 1970.

Mitrinovic, D. S.; Pecaric, J. E.; and Fink, A. M. Classical & New Inequalities in Analysis. Dordrecht, Netherlands: Kluwer, 1993.

Mitrinovic, D. S.; Pecaric, J. E.; Fink, A. M. Inequalities Involving Functions & Their Integrals & Derivatives. Dordrecht, Netherlands: Kluwer, 1991.

Mitrinovic, D. S.; Pecaric, J. E.; and Volenec, V. Recent Advances in Geometric Inequalities. Dordrecht, Netherlands: Kluwer, 1989.