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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
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