Date: Mon, 6 Nov 95 15:35:07 CST
From: rusin (Dave Rusin)
To: hil@cs.brown.edu
Subject: Re: Q: bounds on Gamma(k+0.5)/Gamma(k)?
Newsgroups: sci.math

In article <47gp38$61j@cocoa.brown.edu> you write:
>
>I would like to know if the following bound is written anywhere.
>Would you please drop me a line if you have any information about it?
>Thank you very much.
>
>   Let k = n/2 where n is a positive integer. I would like
>   to know a upper bound for
>
>        Gamma(k+0.5)
>       -------------
>        Gamma(k)

Call this  f(z). Then f(z)*f(z+0.5)=Gamma(z+1)/gamma(z)=z. On the other
hand, log gamma  is convex, so in particular  log gamma(z+0.5)  is
at most  [log gamma(z) + log gamma(z+1)]/2  which unscrambles to say
f(z) < f(z+0.5). Thus  f(z) < sqrt(z). You don't need integrality or
anything.

Conway's Complex Analysis (like many others, I'm sure) has a nice section
on the gamma function. Try also a text on analytic number theory.

dave