In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, subadditivity is a property of a function that states, roughly, that evaluating the function for the sum of two
elements of the
domain
Domain may refer to:
Mathematics
*Domain of a function, the set of input values for which the (total) function is defined
** Domain of definition of a partial function
** Natural domain of a partial function
**Domain of holomorphy of a function
* ...
always returns something less than or equal to the sum of the function's values at each element. There are numerous examples of subadditive functions in various areas of mathematics, particularly
norms and
square roots.
Additive maps are special cases of subadditive functions.
Definitions
A subadditive function is a
function , having a
domain
Domain may refer to:
Mathematics
*Domain of a function, the set of input values for which the (total) function is defined
** Domain of definition of a partial function
** Natural domain of a partial function
**Domain of holomorphy of a function
* ...
''A'' and an
ordered codomain ''B'' that are both
closed
Closed may refer to:
Mathematics
* Closure (mathematics), a set, along with operations, for which applying those operations on members always results in a member of the set
* Closed set, a set which contains all its limit points
* Closed interval, ...
under addition, with the following property:
An example is the
square root
In mathematics, a square root of a number is a number such that ; in other words, a number whose '' square'' (the result of multiplying the number by itself, or ⋅ ) is . For example, 4 and −4 are square roots of 16, because .
...
function, having the
non-negative real number
In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every ...
s as domain and codomain,
since
we have:
A
sequence
In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is called ...
, is called subadditive if it satisfies the
inequality
for all ''m'' and ''n''. This is a special case of subadditive function, if a sequence is interpreted as a function on the set of natural numbers.
Note that while a concave sequence is subadditive, the converse is false. For example, randomly assign
with values in
, then the sequence is subadditive but not concave.
Properties
Sequences
A useful result pertaining to subadditive sequences is the following
lemma
Lemma may refer to:
Language and linguistics
* Lemma (morphology), the canonical, dictionary or citation form of a word
* Lemma (psycholinguistics), a mental abstraction of a word about to be uttered
Science and mathematics
* Lemma (botany), ...
due to
Michael Fekete
Michael (Mihály) Fekete ( he, מיכאל פקטה; 19 July 1886 – 13 May 1957) was a Hungarian-Israeli mathematician.
Biography
Fekete was born in 1886 in Zenta, Austria-Hungary (today Senta, Serbia). He received his PhD in 1909 fro ...
.
The analogue of Fekete's lemma holds for superadditive sequences as well, that is:
(The limit then may be positive infinity: consider the sequence
.)
There are extensions of Fekete's lemma that do not require the inequality
to hold for all ''m'' and ''n'', but only for ''m'' and ''n'' such that
Moreover, the condition
may be weakened as follows:
provided that
is an increasing function such that the integral
converges (near the infinity).
There are also results that allow one to deduce the rate of convergence to the limit whose existence is stated in Fekete's lemma if some kind of both
superadditivity and subadditivity is present.
Besides, analogues of Fekete's lemma have been proved for subadditive real maps (with additional assumptions) from finite subsets of an amenable group
,
and further, of a cancellative left-amenable semigroup.
Functions
If ''f'' is a subadditive function, and if 0 is in its domain, then ''f''(0) ≥ 0. To see this, take the inequality at the top.
. Hence
A
concave function
In mathematics, a concave function is the negative of a convex function. A concave function is also synonymously called concave downwards, concave down, convex upwards, convex cap, or upper convex.
Definition
A real-valued function f on an ...