Additive function

Additive , subadditive, and superadditive functions are mathematical objects. There are certain classes of functions . Linear maps are special additive functions.

definition

A function is called additive if it has the functional equation ${\displaystyle f}$

${\displaystyle f(x+y)=f(x)+f(y)}$

Fulfills. If the definition and target area are Abelian groups , one also speaks of - linearity . ${\displaystyle \mathbb {Z} }$

Sub and super additive functions

If there is a semigroup with the link , then a mapping is called subadditive if the following applies to all and from : ${\displaystyle M}$ ${\displaystyle +}$ ${\displaystyle f\colon M\to \mathbb {R} }$ ${\displaystyle x}$ ${\displaystyle y}$ ${\displaystyle M}$

${\displaystyle f(x+y)\leq f(x)+f(y)}$.

The figure is called superadditive if the following applies to all and off : ${\displaystyle x}$ ${\displaystyle y}$ ${\displaystyle M}$

${\displaystyle f(x+y)\geq f(x)+f(y)}$.

Examples

• According to the triangle inequality , norms and amounts are always subadditive.
• Sublinear functions are subadditive.
• Linear maps are additive.

properties

• A mapping is additive if and only if it is both subadditive and superadditive.
• If it is an additive function, then for every finite number of elements : ${\displaystyle f}$ ${\displaystyle x_{1},\dotsc ,x_{n}}$ ${\displaystyle M}$

${\displaystyle f(x_{1}+\dotsb +x_{n})=f(x_{1})+\dotsb +f(x_{n})}$

The same applies to sub- and superadditivity.

Definition in number theory

With number theoretic functions one regards a link to the multiplication. A number theoretic function is called additive if the equation ${\displaystyle f\colon \mathbb {N} \to \mathbb {C} }$ ${\displaystyle \mathbb {N} }$

${\displaystyle f(xy)=f(x)+f(y)}$

for all coprime and applies. If this even applies to all and , the function is called strictly additive . ${\displaystyle x}$ ${\displaystyle y\in \mathbb {N} }$ ${\displaystyle x}$ ${\displaystyle y}$

There is a similar restriction of additivity (to disjoint instead of arbitrary unions) in measure theory.

See also

• σ-subadditivity
• σ-additivity

Individual evidence

1. ↑ Prasanna Sahoo, Thomas Riedel: Mean Value Theorem and Functional Equations . 1998, ISBN 978-981-02-3544-4 , pp. 1 .
2. ↑ ^{a b} Josip E. Peajcariaac, YL Tong: Convex functions, partial orderings, and Statistical Applications . Academic Press, 1992, ISBN 978-0-12-549250-8 , pp. 8 .