How is evaluated iteratively?
Certainly. But, my original proof didn't work out. So I found an alternate...but it's fairly technical. You might find 's suggestion more useful. Also, I did find an alternate proof on page 22 of this book (a very good book by the wayP may be found for a more general result. Regardless...
For define . Note firstly then that . Moreover,
Finally, note that . So that where is the digamma function. But, a quick check (using digamma's series expression) shows then that and thus is convex, and so is log convex. The result then follows from the Bohr-Mollerup theorem.