majorant

Majorant is a term used in mathematics to denote an upper bound with respect to a given order. It can refer to elements, functions, or series that bound others from above in a precise sense. The notion appears in several contexts, often under the idea of domination or control.

One common usage is a majorant function. If a function f is defined on a domain D, a nonnegative function M on D is a majorant for f if the absolute value of f is bounded by M at every point: |f(x)| ≤ M(x) for all x in D. If M is a constant, then f is uniformly bounded by that constant. Majorant functions are frequently used to estimate errors, to prove convergence, or to apply comparison principles in analysis.

A related concept is a majorant series. For a power series ∑ a_n z^n, a majorant is another series ∑ b_n z^n with nonnegative coefficients and |a_n| ≤ b_n for all n. Majorant series are used to compare radii of convergence and to apply the Weierstrass M-test: if ∑ M_n converges and |f_n(x)| ≤ M_n for all x, then ∑ f_n(x) converges uniformly.

In order theory, a majorant of a subset S in a partially ordered set is an element m that satisfies s ≤ m for every s in S; such an m is an upper bound, and the least such element (when it exists) is the supremum.

See also: minorant, upper bound, supremum, Weierstrass M-test, majorant series.