Geometrically Convex Solutions of Certain Difference Equations and Generalized Bohr-Mollerup Type Theorems

Let G: (0, ∞) → (0, ∞) be logarithmically concave on a neighbourhood of ∞ and suppose lim_x→∞ G(x + δ)/G(x) = 1 for some δ > 0. Then, the functional equation $$g(x+1)=G(x)\cdot g(x),\ \ \ x\in (0,\infty),$$ admits, up to a multiplicative constant, at most one solution g: (0, ∞) → (0, ∞), geometrically convex on a neighbourhood of ∞. Sufficient conditions on G are given, for which also such a unique geometrically convex solution of (D) exists. This result improves the classical theorems of Bohr-Mollerup type and gives a new characterization of the gamma function and the q-gamma function for q ∈ (0, 1).