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Mittelwertungleichungen für Lösungen gewisser Differenzengleichungen. (Mean value inequalities for solutions of certain difference equations). (German) Zbl 0705.39002

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 \(x_ 1,...,x_ n\).
Reviewer: H.Länger

MSC:

39A10 Additive difference equations
26D20 Other analytical inequalities
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References:

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