### 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.

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