Gruber P. Convex and Discrete Geometry

1 Convex Functions of One Variable 17
The characterization of the gamma function by Bohr and Mollerup [136] is also
a result of this type. It will be presented below, together with Artin’s [39] elegant
proof.
For more information on functional equations and convex functions the reader
may consult the books of Kuczma [620], Castillo and Ruiz Cobo [196] and Czerwik
[234], to which we add the nice little treatise of Smítal [944].
The Gamma Function
Ɣ is defined by
Ɣ(x) =
�
+∞
0
t x−1 e −t dt for x > 0
and then extended, by analytic continuation, to the whole complex plane C, except
the points 0, −1, −2,..., where it has poles of first-order. There are other ways to
define Ɣ. See, e.g. formula (6) at the end of this section (Fig. 1.6).
We first collect some properties of Ɣ.
Theorem 1.10. Ɣ has the following properties:
(i) Ɣ(1) = 1.
(ii) Ɣ(x + 1) = xƔ(x) for x > 0, i.e. Ɣ satisfies Euler’s functional equation.
(iii) Ɣis logarithmic convex, i.e. log Ɣ is convex for x > 0.
Proof. (i) Ɣ(1) =
�
+∞
0
e −t dt = lim
�s
s→+∞
0
1
−1 0 1
e −t � −t
dt = lim − e
s→+∞
� �s �
= 1. 0
Fig. 1.6. Gamma and log-gamma function
Ɣ
log Ɣ