[FOM] 605: Integer and Real Functions
Mitchell Spector
spector at alum.mit.edu
Mon Aug 31 02:53:42 EDT 2015
Timothy Y. Chow wrote:
> ...
> Do you know (or does anyone else know) if we get other examples if, in the Bohr-Mollerup theorem,
> "logarithmic convexity" is weakened to "convexity"?
I had written:
> BOHR-MOLLERUP THEOREM:
>
> The Gamma function on the positive reals is the unique function f mapping the positive reals to the positive reals such that:
>
> (1) f(x+1) = x f(x);
>
> (2) f is logarithmically convex (meaning that log f(x) is concave upward);
>
> and (3) f(1) = 1.
I think the following is a counterexample to a hypothetical weakening of the Bohr-Mollerup theorem
with "convex" instead of "logarithmically convex":
Take
f(x) = 1, for 1 < x <= 2,
and then extend f to all the positive reals via the recurrence relation f(x+1) = x f(x).
Explicitly:
f(x) = 1/x, for 0 < x <= 1,
and
f(x) = the product from k = 1 to [x] - 1 of (x - k), for x > 1
(where square brackets denote the greatest integer function).
Then f is a convex function which satisfies f(x+1) = x f(x) for all positive real numbers x, and
f(1) = 1, but f is not the Gamma function.
Mitchell
>
> Tim
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