Some theorems of Jensen type for generalized logarithmic means. (English) Zbl 0863.26021

Pearce, C. E. M.; Pečarić, J.

Rev. Roum. Math. Pures Appl. 40, No. 9-10, 789-795 (1995).

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 \(\ln g\).

Reviewer: J.Aczél (Waterloo/Ontario)

MSC:

26D15

Inequalities for sums, series and integrals

Keywords:

integral inequalities; mean values; convexity; probability measure