Hölder-type inequalities for quasiarithmetic means
In generalization of a theorem of \textit{L. Losonczi} [General inequalities 3, 3rd int. Conf., Oberwolfach 1981, ISNM 64, 91-106 (1983; Zbl 0573.26011)], and in a more elementary manner, the following is proved: \[ \frac{1}{n}\sum^{n}_{i=1}\prod^{k}_{j=1}x_{ij} \leq \prod^{k}_{j=1}\phi_ j^{-1}(\frac{1}{n}\sum^{n}_{i=1}\phi_ j(x_{ij}))\quad (x>0;\quad n=1,2,...) \] iff there exist \(p_ j>1\) \((j=1,...,k)\) with \(\sum^{k}_{j=1}(1/p_ j)=1\) such that \[ (\frac{1}{n}\sum^{n}_{i=1}x_ i^{p_ j})^{1/p_ j} \leq \phi_ j^{-1}(\frac{1}{n}\sum^{n}_{i=1}\phi_ j(x_ i))\quad (x_ i>0;\quad j=1,2,...,k;\quad n=1,2,...). \]
scientific article; zbMATH DE number 3977324