Properties of the volume Wn of the unit ball in Rn have been investigated by many authors. In terms of
Euler's gamma function this volume can be expressed as
Wn =
pn=2
G(1+n=2)
:
Often results are formulated via the quantity
vn = W1=nlogn
n :
50 CHAPTER 6. THURSDAY, SEPTEMBER 1
We prove that fvn+2g is a Hausdor moment sequence and in particular decreasing and logarithmically
convex.
The proof is based on properties of the functions
Fa(x) =
logG(x+1)
x log(ax)
; a > 0:
These functions are extended to the complex plane cut along the negative real axis. We obtain that Fa
maps the upper half plane into itself (and hence is a so-called Pick function) when a  1.
Other results concern the ratio vn=vn+1. Alzer found the best constants c and d such that for n  2,
ec=n(logn)2
 vn=vn+1  ed=n(logn)2
;
and he proved the estimates 2=3.
** This is a joint work with: C. Berg.