Mittelwertungleichungen für Lösungen gewisser Differenzengleichungen. (Mean value inequalities for solutions of certain difference equations) (Q917826)

Let n be a positive integer and let \(a_ 1,...,a_ n\) be positive reals with \(a_ 1+...+a_ n=1\). Sufficient conditions on \(x_ 1,...,x_ n\) are given which guarantee that \(\Gamma (x_ 1^{a_ 1}\cdot...\cdot x_ n^{a_ n})\leq \Gamma^{a_ 1}(x_ 1)\cdot...\cdot \Gamma^{a_ n}(x_ n)\) (or \(\Gamma (x_ 1^{a_ 1}\cdot...\cdot x_ n^{a_ n})\geq \Gamma^{a_ 1}(x_ 1)\cdot...\cdot \Gamma^{a_ n}(x_ n),\) respectively). Moreover, for convex solutions of difference equations of the form \(f(x+1)-f(x)=\phi (x)\) \((x>0)\) for certain functions \(\phi\) the inequality \(a_ 1f(x_ 1)+...+a_ nf(x_ n)\geq f(x_ 1^{a_ 1}\cdot...\cdot x_ n^{a_ n})\) is proved under certain assumptions on \(x_ 1,...,x_ n\).