AMGM

AMGM commonly refers to the Arithmetic Mean-Geometric Mean inequality, a fundamental result in mathematics. It states that for any set of nonnegative real numbers x1, x2, ..., xn, the arithmetic mean is at least as large as the geometric mean: (x1 + x2 + ... + xn)/n ≥ (x1 x2 ... xn)^(1/n). Equality holds if and only if x1 = x2 = ... = xn. In the two-variable case, this specializes to (a + b)/2 ≥ sqrt(ab), with equality when a = b.

The inequality is widely used in analysis, optimization, probability, number theory, and computer science. It has numerous proofs, including a direct proof for n = 2 using (a − b)^2 ≥ 0, an inductive proof to extend to n numbers, and proofs based on convexity or Jensen’s inequality. A weighted version also exists: for positive numbers x_i with weights w_i ≥ 0 summing to 1, ∑ w_i x_i ≥ ∏ x_i^{w_i}.

Variations include generalizations to different means, such as power means and other forms of weighted inequality. The AM-GM inequality is closely related to other classical inequalities and serves as a tool for establishing bounds and proving additional results across mathematics.

The term AMGM is commonly used in mathematical literature as shorthand for this inequality, with AM-GM often appearing as the more standard notation. While the basic statement is simple, its proofs, generalizations, and applications have a long history and broad relevance across disciplines.