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Some theorems of Jensen type for generalized logarithmic means. (English) Zbl 0863.26021
The following is offered as main result (at least that is how the reviewer understands it; there are several misprints). Let $$p$$ be a real number, $$X$$ a set equipped with a probability measure $$\mu$$, let $$I$$ be a proper real interval, $$f$$ a real valued function on $$I$$ and let $$g^p$$ map $$X$$ into $$I$$. For $$p\neq 0$$, $f\Biggl(\Biggl(\int_X g^pd\mu\Biggr)^{1/p}\Biggr)\leq \int_X f\circ g d\mu$ if $$x\mapsto f(x^{1/p})$$ is convex and the integrals exist. If $$p=0$$ then $$z^{1/p}$$ has to be replaced by $$e^z$$ and $$g^p$$ by $$\ln g$$.

##### MSC:
 26D15 Inequalities for sums, series and integrals