Eine allgemeine logarithmische Ungleichung. (A general logarithmic inequality)
The purpose of this paper is to prove the following. Let \(f:[a,b]\to {\mathbb{R}}(0<a<b)\) be positive and continuous on \([a,b]\) and have a positive derivative on \(]a,b[.\) Then \[ (f(a)f(b))^{1/2}<\frac{f(a)-f(b)}{\ln (f(a)/f(b))}<((f(a)^{1/3}+f(b)^{1/3})/2)^ 3. \] \{The reviewer does not understand what the difference is between this and the case \(f=id\) which, as the editors note in a postscript, \textit{T. P. Lin} [Am. Math. Mon. 81, 879-883 (1974; Zbl 0292.26015)] has proved. He also thinks there are errors or extensive misprints in the present proof.\}
chains of inequalities between mean values
scientific article; zbMATH DE number 3981470