How is evaluated iteratively?
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.