Absolutely and completely monotonic functions and sequences


In mathematics, the notions of an absolutely monotonic function and a completely monotonic function are two very closely related concepts. Both imply very strong monotonicity properties. Both types of functions have derivatives of all orders. In the case of an absolutely monotonic function, the function as well as its derivatives of all orders must be non-negative in its domain of definition which would imply that the function as well as its derivatives of all orders are monotonically increasing functions in the domain of definition. In the case of a completely monotonic function, the function and its derivatives must be alternately non-negative and non-positive in its domain of definition which would imply that function and its derivatives are alternately monotonically increasing and monotonically decreasing functions.
Such functions were first studied by S. Bernstein in 1914 and the terminology is also due to him. There are several other related notions like the concepts of almost completely monotonic function, logarithmically completely monotonic function, strongly logarithmically completely monotonic function, strongly completely monotonic function and almost strongly completely monotonic function. Another related concept is that of a completely/absolutely monotonic sequence. This notion was introduced by Hausdorff in 1921.
The notions of completely and absolutely monotonic function/sequence play an important role in several areas of mathematics. For example, in classical analysis they occur in the proof of the positivity of integrals involving Bessel functions or the positivity of Cesàro means of certain Jacobi series. Such functions occur in other areas of mathematics such as probability theory, numerical analysis, and elasticity.

Absolutely and completely monotonic functions

Definitions

A real valued function defined over an interval in the real line is called an absolutely monotonic function if it has derivatives of all orders and for all in. The function is called a completely monotonic function if for all in.
The two notions are mutually related. The function is completely monotonic if and only if is absolutely monotonic on where the interval obtained by reflecting with respect to the origin.
In applications, the interval on the real line that is usually considered is the closed-open right half of the real line, that is, the interval.

Examples

The following functions are absolutely monotonic in the specified regions.
  1. , where a non-negative constant, in the region
  2. , where for all, in the region
  3. in the region
  4. in the region
  5. in the region
The following functions are completely monotonic on
  1. for
  2. for
  3. for
  4. because it is the Laplace transform of
The following combinations of completely and absolutely monotonic functions are completely monotonic:
  • non-negative linear combination of completely monotonic functions
  • product of completely monotonic functions
  • where is absolutely monotonic and is completely monotonic
  • where is completely monotonic and is a Bernstein function
The forward difference for of a completely monotonic function is completely monotonic.

Representation

Bernstein's theorem on monotone functions: A function that is absolutely monotonic on can be represented there as a Laplace integral in the form
where is non-decreasing and bounded on.

Some important properties

Bernstein's little theorem: A function that is absolutely monotonic on a closed interval can be extended to an analytic function on the interval defined by.
A function that is absolutely monotonic on can be extended to a function that is not only analytic on the real line but is even the restriction of an entire function to the real line.

Bernstein functions

Definition

Related to the above, Bernstein functions are defined as those that are non-negative and whose derivative is completely monotonic.

Examples

The following functions are Bernstein functions
  1. for
  4. for

Representation

Every Bernstein function has the representation:
where and is a measure on the positive real half-line such that

Absolutely and completely monotonic sequences

Definition

A sequence is called an absolutely monotonic sequence if its elements are non-negative and its successive differences are all non-negative, that is, if
where.
A sequence is called a completely monotonic sequence if its elements are non-negative and its successive differences are alternately non-positive and non-negative, that is, if

Examples

The sequences and for are completely monotonic sequences.

Representation

A sequence is completely monotonic if and only if there exists an increasing function on such that
The determination of this function from the sequence is referred to as the Hausdorff moment problem.

Logarithmically completely monotonic functions

A positive function is said to be logarithmically completely monotonic if is completely monotonic.
Every logarithmically completely monotonic function is completely monotonic. Writing with, Faà di Bruno’s formula expresses the -th derivative of as
where denotes the -th Bell polynomial. Each Bell polynomial is a finite sum of monomials of the form with the exponents satisfying and all coefficients strictly positive. Since logarithmic complete monotonicity implies, we get
so that.
As, it follows that. As this true for all, we get that is completely monotonic.
Note that this is a special case of being completely monotonic when is absolutely monotonic and is completely monotonic, for the case of being the exponential function. This can be proved as above with the more general version of Faà di Bruno’s formula.
The converse implication is false in general, and logarithmically completely monotonic functions form a proper subclass of completely monotonic functions.