[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":


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.


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