  In 1922, Harald Bohr and Johannes Mollerup established a remarkable
characterization of the Euler gamma function using its log-convexity property.
A decade later, Emil Artin investigated this result and used it to derive the
basic properties of the gamma function using elementary methods of the
calculus. Bohr-Mollerup's theorem was then adopted by Nicolas Bourbaki as the
starting point for his exposition of the gamma function.
  This open access book develops a far-reaching generalization of
Bohr-Mollerup's theorem to higher order convex functions, along lines initiated
by Wolfgang Krull, Roger Webster, and some others but going considerably
further than past work. In particular, this generalization shows using
elementary techniques that a very rich spectrum of functions satisfy analogues
of several classical properties of the gamma function, including
Bohr-Mollerup's theorem itself, Euler's reflection formula, Gauss'
multiplication theorem, Stirling's formula, and Weierstrass' canonical
factorization.
  The scope of the theory developed in this work is illustrated through various
examples, ranging from the gamma function itself and its variants and
generalizations (q-gamma, polygamma, multiple gamma functions) to important
special functions such as the Hurwitz zeta function and the generalized
Stieltjes constants.
  This volume is also an opportunity to honor the 100th anniversary of
Bohr-Mollerup's theorem and to spark the interest of a large number of
researchers in this beautiful theory.
