## 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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