On Carleman and Knopp's inequalities.
The authors establish a number of interesting results connected to the Carleman inequality \[ \sum_{k=1}^{\infty}(a_1\dots a_k)^{1/k}<e\sum_{k=1}^{\infty}a_k \] and it's continuous analogue \[ \int_0^{\infty}\exp\left(\frac1x\int_0^ x\log f(t)\,dt\right)\,dx < e \int_0^{\infty} f(x)\,dx, \] traditionally referred to as the Knopp inequality. First, they prove a~sharpened version of the Carleman inequality and point out several interesting corollaries and connections to other known inequalities (including the AG inequality). Next, they establish an~inequality (called the Carleman--Knopp inequality), which generalizes and unifies both the above inequalities, and illustrate it with an example. In the final section of the paper, a~simple change of variables is used to give a~new and elementary proof of the Knopp inequality. The idea is in turn used to establish a~new, fairly general inequality \[ \int_0^{\infty}\phi\left(\frac1x\int_0^ x\log f(t)\,dt\right)\frac{dx}x \leq \int_0^{\infty} \phi(f(x))\frac{dx}x. \] Final discussion reveals some connections to the classical Hardy inequality, possible improvements of best constants, etc.
scientific article; zbMATH DE number 1891487