[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
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