Recording date: 01/09/2011
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A study of the volume of the unit ball in Euclidean space based on complex methods

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.

Henrik L. Pedersen
Henrik L. Pedersen