A generalization of Bohr–Mollerup’s theorem for higher order convex functions: a tutorial

Abstract

In its additive version, Bohr–Mollerup’s remarkable theorem states that the unique (up to an additive constant) convex solution f(x) to the equation $\Delta f(x)=\ln x$ on the open half-line $(0,\infty )$ is the log-gamma function $f(x)=\ln \Gamma (x)$, where $\Delta$ denotes the classical difference operator and $\Gamma (x)$ denotes the Euler gamma function. In a recently published open access book, the authors provided and illustrated a far-reaching generalization of Bohr–Mollerup’s theorem by considering the functional equation $\Delta f(x)=g(x)$, where g can be chosen from a wide and rich class of functions that have convexity or concavity properties of any order. They also showed that the solutions f(x) arising from this generalization satisfy counterparts of many properties of the log-gamma function (or equivalently, the gamma function), including analogues of Bohr–Mollerup’s theorem itself, Burnside’s formula, Euler’s infinite product, Euler’s reflection formula, Gauss’ limit, Gauss’ multiplication formula, Gautschi’s inequality, Legendre’s duplication formula, Raabe’s formula, Stirling’s formula, Wallis’s product formula, Weierstrass’ infinite product, and Wendel’s inequality for the gamma function. In this paper, we review the main results of this new and intriguing theory and provide an illustrative application.