Eine Integralungleichung für streng monotone Funktionen mit logarithmisch konvexer Umkehrfunktion. (An integral inequality for strictly monotonic functions with a logarithmically convex inverse function). (German) Zbl 0721.26010

Seiffert, H.-J.

Elem. Math. 44, No. 1, 16-18 (1989).

The author proves the following result: If f is a strictly increasing [resp. decreasing] continuous function on \([a,b],\quad 0<a<b,\) having a logarithmically convex inverse function, then \[ (1)\quad (b-a)^{- 1}\int^{b}_{a}f(x)dx\leq [resp.\quad \geq]f(I(a,b)), \] where I(a,b) (the so-called “identric-mean”) is defined by \(I(a,b)=e^{-1}\cdot (b^ b/a^ a)^{1/(b-a)}\) for \(a\neq b\), \(I(a,a)=a\). He also gives an interesting application for K. B. Stolarsky’s means [Math. Mag. 48, 87-92 (1975; Zbl 0302.26003)].

MSC:

26D15

Inequalities for sums, series and integrals

Keywords:

integral inequality; strictly monotone functions; logarithmically convex inverse function; identric-mean