submultipli

Submultipli is a term used in mathematics to denote objects that satisfy a submultiplicative inequality. It is not a universally standardized term, but appears in various texts as an informal label for sequences, norms, or measures that do not grow faster than the product of their parts. The core idea is that the whole is bounded by the product of its pieces, in a sense compatible with multiplication.

A common formalization appears in two related forms. For a nonnegative sequence (a_n), submultipli means a_{m+n} ≤ a_m a_n for all nonnegative integers m and n. For a norm ||·|| on a semigroup or a multiplicative structure, a norm is submultipli if ||xy|| ≤ ||x|| · ||y|| for all elements x and y in the structure. In both contexts, taking logs turns the submultiplicative condition into a subadditive one, which connects submultipli to broader growth and bounding techniques.

Notable examples include the family of exponential sequences a_n = c^n with c ≥ 0, which satisfy a_{m+n} = a_m a_n and hence are submultipli, as well as submultiplicative norms on Banach algebras. Submultipli properties arise naturally in analyses of growth rates, complexity bounds, and stability questions, where controlling a quantity by the product of its components is convenient.

Key consequences include that if a_n is submultipli, then the limit lim_{n→∞} a_n^{1/n} exists and equals inf_{n} a_n^{1/n}. This mirrors the standard result for subadditive sequences via the logarithm transform and Fekete-type arguments. Submultipli thus provides a unifying lens for bounding and understanding multiplicative growth across different mathematical settings. See also submultiplicative function, subadditive sequences, and Fekete's lemma.