Editing Bohr–Mollerup theorem

{{Short description|Theorem in complex analysis}}
In [[mathematical analysis]], the '''Bohr–Mollerup theorem'''<ref>{{springer|title=Bohr–Mollerup theorem|id=p/b120330}}</ref><ref>{{MathWorld|urlname=Bohr-MollerupTheorem|title=Bohr–Mollerup Theorem}}</ref> is a theorem proved by the Danish mathematicians [[Harald Bohr]] and [[Johannes Mollerup]].<ref name="BM">{{cite book|first= Bohr, H.|last= Mollerup, J.|title= Lærebog i Kompleks Analyse vol. III, Copenhagen|year=1922}}</ref> The theorem [[characterization (mathematics)|characterizes]] the [[gamma function]], defined for {{math|''x'' > 0}} by

:<math>\Gamma(x)=\int_0^\infty t^{x-1} e^{-t}\,\mathrm{d}t</math>

as the ''only'' positive function {{mvar|&nbsp;f&nbsp;}}, with domain on the interval {{math|''x'' > 0}}, that simultaneously has the following three properties:

* {{math|&nbsp;''f''&nbsp;(1) {{=}} 1}}, and
* {{math|&nbsp;''f''&nbsp;(''x'' + 1) {{=}} ''x''&nbsp;''f''&nbsp;(''x'')}} for {{math|''x'' > 0}} and
* {{mvar|&nbsp;f&nbsp;}} is [[logarithmic convexity|logarithmically convex]].

A treatment of this theorem is in [[Emil Artin|Artin]]'s book ''The Gamma Function'',<ref>{{cite book|last=Artin|first= Emil|title=The Gamma Function|url= https://archive.org/details/gammafunction0000arti|url-access= registration|year= 1964|publisher= Holt, Rinehart, Winston}}</ref> which has been reprinted by the AMS in a collection of Artin's writings.<ref>{{cite book|last= Rosen |first= Michael |title=  Exposition by Emil Artin: A Selection|year= 2006 |publisher= American Mathematical Society}}</ref>

The theorem was first published in a textbook on [[complex analysis]], as Bohr and Mollerup thought it had already been proved.<ref name="BM"/>

The theorem admits a far-reaching generalization to a wide variety of functions (that have convexity or concavity properties of any order).<ref>{{cite book|author1=J.-L. Marichal|author2=N. Zenaïdi|url=https://link.springer.com/book/10.1007/978-3-030-95088-0|title=A Generalization of Bohr-Mollerup's Theorem for Higher Order Convex Functions|series=Developments in Mathematics |publisher=Developments in Mathematics, Vol. 70. Springer, Cham, Switzerland|date=2022|volume=70 |doi=10.1007/978-3-030-95088-0 |isbn=978-3-030-95087-3 }}</ref>

==Statement==
:'''Bohr–Mollerup Theorem.''' &nbsp; &nbsp; {{math|Γ(''x'')}} is the only function that satisfies {{math|&nbsp;''f''&nbsp;(''x'' + 1) {{=}} ''x''&nbsp;''f''&nbsp;(''x'')}} with {{math|log(&nbsp;''f''&nbsp;(''x''))}} convex and also with {{math|&nbsp;''f''&nbsp;(1) {{=}} 1}}.

==Proof==
Let {{math|Γ(''x'')}} be a function with the assumed properties established above: {{math|Γ(''x'' + 1) {{=}} ''x''Γ(''x'')}} and {{math|log(Γ(''x''))}} is convex, and {{math|Γ(1) {{=}} 1}}. From {{math|Γ(''x'' + 1) {{=}} ''x''Γ(''x'')}} we can establish

:<math>\Gamma(x+n)=(x+n-1)(x+n-2)(x+n-3)\cdots(x+1)x\Gamma(x)</math>

The purpose of the stipulation that {{math|Γ(1) {{=}} 1}} forces the {{math|Γ(''x'' + 1) {{=}} ''x''Γ(''x'')}} property to duplicate the factorials of the integers so we can conclude now that {{math|Γ(''n'') {{=}} (''n'' − 1)!}} if {{math|''n'' ∈ '''N'''}} and if {{math|Γ(''x'')}} exists at all. Because of our relation for {{math|Γ(''x'' + ''n'')}}, if we can fully understand {{math|Γ(''x'')}} for {{math|0 < ''x'' ≤ 1}} then we understand {{math|Γ(''x'')}} for all values of {{mvar|x}}.

For {{math|''x''<sub>1</sub>}}, {{math|''x''<sub>2</sub>}}, the slope {{math|''S''(''x''<sub>1</sub>, ''x''<sub>2</sub>)}} of the [[line segment]] connecting the points {{math|(''x''<sub>1</sub>, log(Γ&nbsp;(''x''<sub>1</sub>)))}} and {{math|(''x''<sub>2</sub>, log(Γ&nbsp;(''x''<sub>2</sub>)))}} is monotonically increasing in each argument with {{math|''x''<sub>1</sub> < ''x''<sub>2</sub>}} since we have stipulated that {{math|log(Γ(''x''))}} is convex. Thus, we know that
:<math>S(n-1,n) \leq S(n,n+x)\leq S(n,n+1)\quad\text{for all }x\in(0,1].</math>
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
:<math>(n-1)^x(n-1)! \leq \Gamma(n+x)\leq n^x(n-1)!.</math>
From previous work this expands to
:<math>(n-1)^x(n-1)! \leq (x+n-1)(x+n-2)\cdots(x+1)x\Gamma(x)\leq n^x(n-1)!,</math>
and so
:<math>\frac{(n-1)^x(n-1)!}{(x+n-1)(x+n-2)\cdots(x+1)x} \leq \Gamma(x) \leq \frac{n^xn!}{(x+n)(x+n-1)\cdots(x+1)x}\left(\frac{n+x}{n}\right).</math>
The last line is a strong statement. In particular, ''it is true for all values of'' {{mvar|n}}. That is {{math|Γ(''x'')}} is not greater than the right hand side for any choice of {{mvar|n}} and likewise, {{math|Γ(''x'')}} is not less than the left hand side for any other choice of {{mvar|n}}. 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 {{mvar|n}} for the RHS and the LHS. In particular, if we keep {{mvar|n}} for the RHS and choose {{math|''n'' + 1}} for the LHS we get:

:<math>\begin{align}
\frac{((n+1)-1)^x((n+1)-1)!}{(x+(n+1)-1)(x+(n+1)-2)\cdots(x+1)x}&\leq \Gamma(x)\leq\frac{n^xn!}{(x+n)(x+n-1)\cdots(x+1)x}\left(\frac{n+x}{n}\right)\\
\frac{n^xn!}{(x+n)(x+n-1)\cdots(x+1)x}&\leq \Gamma(x)\leq\frac{n^xn!}{(x+n)(x+n-1)\cdots(x+1)x}\left(\frac{n+x}{n}\right)
\end{align}</math>

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 {{math|''n'' → ∞}}:

:<math>\lim_{n\to\infty} \frac{n+x}{n} = 1</math>

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

:<math>\frac{n^xn!}{(x+n)(x+n-1)\cdots(x+1)x}</math>

is sandwiched in between. This can only mean that

:<math>\lim_{n\to\infty}\frac{n^xn!}{(x+n)(x+n-1)\cdots(x+1)x} = \Gamma (x).</math>

In the context of this proof this means that

:<math>\lim_{n\to\infty}\frac{n^xn!}{(x+n)(x+n-1)\cdots(x+1)x}</math>

has the three specified properties belonging to {{math|Γ(''x'')}}. Also, the proof provides a specific expression for {{math|Γ(''x'')}}.  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 {{math|0 < ''x'' ≤ 1}} only one possible number {{math|Γ(''x'')}} can exist.  Therefore, there is no other function with all the properties assigned to {{math|Γ(''x'')}}.

The remaining loose end is the question of proving that {{math|Γ(''x'')}} makes sense for all {{mvar|x}} where

:<math>\lim_{n\to\infty}\frac{n^xn!}{(x+n)(x+n-1)\cdots(x+1)x}</math>

exists.  The problem is that our first double inequality

:<math>S(n-1,n)\leq S(n+x,n)\leq S(n+1,n)</math>

was constructed with the constraint {{math|0 < ''x'' ≤ 1}}. If, say, {{math|''x'' > 1}} then the fact that {{mvar|S}} is monotonically increasing would make {{math|''S''(''n'' + 1, ''n'') < ''S''(''n'' + ''x'', ''n'')}}, contradicting the inequality upon which the entire proof is constructed. However,

:<math>\begin{align}
\Gamma(x+1)&= \lim_{n\to\infty}x\cdot\left(\frac{n^xn!}{(x+n)(x+n-1)\cdots(x+1)x}\right)\frac{n}{n+x+1}\\
\Gamma(x)&=\left(\frac{1}{x}\right)\Gamma(x+1)
\end{align}</math>

which demonstrates how to bootstrap {{math|Γ(''x'')}} to all values of {{mvar|x}} where the limit is defined.

== See also ==
* [[Wielandt theorem]]

==References==
{{reflist}}

{{DEFAULTSORT:Bohr-Mollerup theorem}}
[[Category:Gamma and related functions]]
[[Category:Theorems in complex analysis]]