mathematical analysis Analysis is the branch of mathematics dealing with continuous functions, limit (mathematics), limits, and related theories, such as Derivative, differentiation, Integral, integration, measure (mathematics), measure, infinite sequences, series (m ...

, the Bohr–Mollerup theorem is a theorem proved by the Danish mathematicians

Harald Bohr Harald August Bohr (22 April 1887 – 22 January 1951) was a Danish mathematician and footballer. After receiving his doctorate in 1910, Bohr became an eminent mathematician, founding the field of almost periodic functions. His brother was the No ...

and

Johannes Mollerup Johannes Mollerup (3 December 1872 – 27 June 1937) was a Danish mathematician.characterizes the

gamma function In mathematics, the gamma function (represented by , the capital letter gamma from the Greek alphabet) is one commonly used extension of the factorial function to complex numbers. The gamma function is defined for all complex numbers except ...

, defined for by :

\Gamma(x)=\int_0^\infty t^ e^\,dt

as the ''only'' positive function , with domain on the interval , that simultaneously has the following three properties: * , and * for and * is logarithmically convex. A treatment of this theorem is in

Artin Artin may refer to: * Artin (name), a surname and given name, including a list of people with the name ** Artin, a variant of Harutyun, an Armenian given name * 15378 Artin, a main-belt asteroid See also

{{disambiguation, surname ...

's book ''The Gamma Function'', which has been reprinted by the AMS in a collection of Artin's writings. The theorem was first published in a textbook on

complex analysis Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates Function (mathematics), functions of complex numbers. It is helpful in many branches of mathemati ...

, as Bohr and Mollerup thought it had already been proved.

Statement

:Bohr–Mollerup Theorem. is the only function that satisfies with convex and also with .

Proof

Let be a function with the assumed properties established above: and is convex, and . From we can establish :

\Gamma(x+n)=(x+n-1)(x+n-2)(x+n-3)\cdots(x+1)x\Gamma(x)

The purpose of the stipulation that forces the property to duplicate the factorials of the integers so we can conclude now that if and if exists at all. Because of our relation for , if we can fully understand for then we understand for all values of . For , , the slope of the line segment connecting the points and is monotonically increasing in each argument with since we have stipulated that is convex. Thus, we know that :

S(n-1,n) \leq S(n,n+x)\leq S(n,n+1)\quad\textx\in(0,1].

After simplifying using the various properties of the logarithm, and then exponentiating (which preserves the inequalities since the exponential function is monotonically increasing) we obtain :

(n-1)^x(n-1)! \leq \Gamma(n+x)\leq n^x(n-1)!.

From previous work this expands to :

(n-1)^x(n-1)! \leq (x+n-1)(x+n-2)\cdots(x+1)x\Gamma(x)\leq n^x(n-1)!,

and so :

\frac \leq \Gamma(x) \leq \frac\left(\frac\right).

The last line is a strong statement. In particular, ''it is true for all values of'' . That is is not greater than the right hand side for any choice of and likewise, is not less than the left hand side for any other choice of . Each single inequality stands alone and may be interpreted as an independent statement. Because of this fact, we are free to choose different values of for the RHS and the LHS. In particular, if we keep for the RHS and choose for the LHS we get: :

\begin
\frac&\leq \Gamma(x)\leq\frac\left(\frac\right)\\
\frac&\leq \Gamma(x)\leq\frac\left(\frac\right)
\end

It is evident from this last line that a function is being sandwiched between two expressions, a common analysis technique to prove various things such as the existence of a limit, or convergence. Let : :

\lim_ \frac = 1

so the left side of the last inequality is driven to equal the right side in the limit and :

\frac

is sandwiched in between. This can only mean that :

\lim_\frac = \Gamma (x).

In the context of this proof this means that :

\lim_\frac

has the three specified properties belonging to . Also, the proof provides a specific expression for . And the final critical part of the proof is to remember that the limit of a sequence is unique. This means that for any choice of only one possible number can exist. Therefore, there is no other function with all the properties assigned to . The remaining loose end is the question of proving that makes sense for all where :

\lim_\frac

exists. The problem is that our first double inequality :

S(n-1,n)\leq S(n+x,n)\leq S(n+1,n)

was constructed with the constraint . If, say, then the fact that is monotonically increasing would make , contradicting the inequality upon which the entire proof is constructed. However, :

\begin
\Gamma(x+1)&= \lim_x\cdot\left(\frac\right)\frac\\
\Gamma(x)&=\left(\frac\right)\Gamma(x+1)
\end

which demonstrates how to bootstrap to all values of where the limit is defined.

Generalizations

A far-reaching generalization of Bohr-Mollerup's theorem to a wide variety of functions is given in the following open access book: J.-L. Marichal and N. Zenaïdi
''A Generalization of Bohr-Mollerup's Theorem for Higher Order Convex Functions''
Developments in Mathematics, Vol. 70. Springer, Cham, Switzerland, 2022.

References

* * * * * * * (''Textbook in Complex Analysis'') {{DEFAULTSORT:Bohr-Mollerup theorem Gamma and related functions Theorems in complex analysis