Editing Bohr–Mollerup theorem (section)

{{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>