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Results in Mathematics

, Volume 26, Issue 3–4, pp 290–297 | Cite as

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

  • Detlef Gronau
  • Janusz Matkowski
Article

Abstract

Let G: (0, ∞) → (0, ∞) be logarithmically concave on a neighbourhood of ∞ and suppose limx→∞ 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).

AMS Subject Classification

39B12 39B22 33B15 

Key words and phrases

Linear difference equations Bohr-Mollerup type theorem, geometrically logarithmically, and Jensen convex functions gamma function and q-gamma function 

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Copyright information

© Birkhäuser Verlag, Basel 1994

Authors and Affiliations

  • Detlef Gronau
    • 1
  • Janusz Matkowski
    • 2
  1. 1.Institut für MathematikUniversität GrazGrazAustria
  2. 2.Department of MathematicsTechnical UniversityBielsko-BiałaPoland

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