A convexity property and a new characterization of Euler's gamma function

作者：Alzer Horst^*; Matkowski Janusz

来源：Archiv der Mathematik, 2013, 100(2): 131-137.

DOI：10.1007/s00013-013-0487-2

• Gamma function
• Convex
• Concave
• Inequalities
• Functional equation
• Characterization

Our main results are:
(I) Let alpha not equal 0 be a real number. The Gamma circle exp)(alpha) is convex on R if and only if
alpha >= max(0<t<x0) (-1/t psi(t) - psi'(t)/psi(t)(2)) = 0.0258 ....
Here, x(0) = 1.4616 ... denotes the only positive zero of psi = Gamma'/Gamma.
(II) Assume that a function f : (0, infinity) -> (0, infinity) is bounded from above on a set of positive Lebesgue measure (or on a set of the second category with the Baire property) and satisfies
f(x + 1) = xf(x) for x > 0 and f(1) = 1.
If there are a number b and a sequence of positive real numbers (a(n)) (n is an element of N) with lim(n ->infinity) a(n) = 0 such that for every n the f circle exp)(an) is Jensen convex on (b, infinity), then f is the gamma function.

• 出版日期2013-2