Equality condition

An equality condition states the circumstances under which the two sides of an inequality take the same value. It identifies the cases where the inequality is sharp.

Inequality: A comparison stating that one quantity is less than, greater than, or otherwise ordered relative to another. An equality condition identifies when an inequality's comparison becomes exact.
Equality case in the triangle inequality: The characterization of vectors or complex numbers for which the triangle inequality holds with equality. It shows how geometric alignment turns a general bound into an exact equality criterion.
Arithmetic–geometric mean inequality: An inequality stating that the arithmetic mean of nonnegative numbers is at least their geometric mean. Its equality condition requires all numbers to be equal, a common symmetric extremal pattern.
Strict inequality: An inequality in which one quantity is strictly less than or greater than another. A strict inequality has no equality case wherever it is asserted.
Equality (mathematics): A relation stating that two mathematical expressions have the same value. The condition describes precisely when this relation holds between an inequality's sides.
Cauchy–Schwarz inequality: An inequality bounding the inner product of two vectors by the product of their lengths. Its equality condition is linear dependence of the vectors, illustrating a structural criterion.
Hölder's inequality: An inequality bounding the sum or integral of products using conjugate-power norms. Its equality criterion identifies when the norm bound is attained.
Sharp inequality: An inequality whose constant or bound cannot be improved within its stated setting. A sharp bound may be attained, and its equality condition names the attaining cases.
Domain of a function: The set of inputs for which a function is defined. Equality cases must be drawn from inputs where every expression in the inequality exists.
Young's inequality: An inequality bounding a product by a sum of powers of its factors. Its equality condition links the factors through a specific power relation.

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