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

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