[FOM] 605: Integer and Real Functions

Timothy Y. Chow tchow at alum.mit.edu
Mon Aug 31 10:34:06 EDT 2015

```On Sun, 30 Aug 2015, Mitchell Spector wrote:
> I think the following is a counterexample to a hypothetical weakening of
> the Bohr-Mollerup theorem with "convex" instead of "logarithmically
> convex":

Nice!

I just did what I probably should have done before, which is look on
MathSciNet.  I found the following paper, which is very relevant to the
question of weakening logarithmic convexity in the Bohr-Mollerup theorem.

Teresa Bermudez, Antonio Martinon, and Kishin Sadarangani, On
quasi-gamma functions, J. Convex Anal. 21 (2014), no. 3, 765-783.

Here is part of the Math Review.

Let C be a convex subset of R. A function f:C -> R is said to be
quasi-convex if

f(tx + (1-t)y) <= max{f(x),f(y)} for all x,y in C and t in [0,1].

In the paper, by replacing [logarithmic convexity in the Bohr-Mollerup
theorem] by

f is quasi-convex on (0,infinity),

the authors give the definition of a quasi-gamma function and denote by
Q the class of all such functions.

The authors investigate the properties of quasi-gamma functions. They
first show that quasi-gamma functions are continuous (Theorem 4.1). It
follows that every quasi-gamma function has a minimum point c in (1,2),
is decreasing on (0,c] and is increasing on [c,infinity).

I'll skip the rest of the review and mention only that there is a somewhat
surprising occurrence of the constant (1 + sqrt(5))/2 in the theory of
quasi-gamma functions.

Tim
```