AM–GM inequality

The arithmetic mean of nonnegative real numbers is at least their geometric mean. Equality holds exactly when all the numbers are equal.

Arithmetic mean: The sum of a finite set of numbers divided by their count. AM–GM compares this ordinary average directly with the geometric mean.
Optimization: The study of finding maximum or minimum values subject to specified constraints. AM–GM often identifies extrema when a sum is fixed and a product is optimized.
Hölder's inequality: An inequality bounding sums of products using conjugate power means. Hölder generalizes product-sum bounds beyond AM–GM's equal-weight mean comparison.
Euclid: An ancient Greek mathematician traditionally regarded as the author of Elements. A geometric construction in Elements VI.13 establishes a two-number mean comparison equivalent to AM–GM.
Geometric mean: The nth root of the product of n nonnegative numbers. This is the multiplicative average that AM–GM bounds from above.
Nesbitt's inequality: For positive a, b, and c, the sum of a/(b+c), b/(c+a), and c/(a+b) is at least 3/2. AM–GM supplies a standard lower-bound step in proofs of this inequality.
Minkowski inequality: The triangle inequality for norms defined by sums of powers. Minkowski compares sums of powered quantities, unlike AM–GM's comparison of two means.
Elements: Euclid's treatise presenting geometry and number theory in a systematic form. Book VI contains an early geometric result underlying the two-variable inequality.
Weighted AM–GM inequality: An extension comparing weighted arithmetic and geometric means for nonnegative numbers. Unequal weights preserve the inequality when variables have unequal roles.
Product of distances: A geometric quantity formed by multiplying distances from a point to specified objects. AM–GM can bound such products when the corresponding distances have a known sum.

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