Entropic means
With a somewhat unusual notation, the authors denote by \[ (E_{d_{\phi}})\quad \min \{\sum^{n}_{i=1}w_ ia_ i\phi (x/a_ i):\quad X\in {\mathbb{R}}_+\} \] the problem of determining that positive number x (as function of the positive \(a_ 1,...,a_ n)\) for which \(\sum^{n}_{i=1}w_ ia_ i\phi (x/a_ i)\) is minimum \((w_ 1,...,w_{n-1},w_ n=1-\sum^{n-1}_{j=1}w_ j\) are given positive numbers, \(\phi\) : \({\mathbb{R}}_+\to {\mathbb{R}}^ a \)given strictly convex differentiable function with \((0,1]\subset dom \phi\) and such that \(\phi (1)=\phi '(1)-0),\) with reference to \textit{I. Csiszár} [Stud. Sci. Math. Hung. 2, 299-318 (1967; Zbl 0157.25802)]. Then they define similarly \[ (E_{d_ h})\quad \min \{h(x)-x\sum^{n}_{i=1}w_ ih'(a_ i):\quad x\in {\mathbb{R}}_+\} \] (h again strictly convex differentiable). They call the first (or both?) an entropic mean (''entropic'' not in the sense of \textit{I. M. E. Etherington} [e.g. Proc. R. Soc. Edinb., Sect. A 65, 84-108 (1958; Zbl 0228.17002)]. Then, as examples of one or the other, the power means, the geometric, the Lehmer means [\textit{D. H. Lehmer}, J. Math. Anal. Appl. 36, 183-200 (1971; Zbl 0222.26018), but see \textit{E. F. Beckenbach}, Am. Math. Mon. 57, 1-6 (1950; Zbl 0035.15704) or even \textit{K. Goldziher}, Skand. Aktuarietidskr. 1920, 72-96 (1920)], the Gini means [\textit{C. Gini}, Metron 13, No. 2, 3-22 (1938; Zbl 0018.41404)] and the quasilinear means [here called ''HLP-means'' for \textit{G. H. Hardy, J. E. Littlewood} and \textit{G. Pólya}, Inequalities (1952; Zbl 0047.05302; 1st ed. 1934; Zbl 0010.10703), who quote \textit{B. de Finetti}, Giorn. Ist. Ital. Attuari 2, 369-296 (1931; Zbl 0002.27802)], among others, are enumerated. For the first, \(''(E_{d_{\phi}})''\) kind of means a sufficient, but not necessarily necessary, condition of comparability and an asymptotic property is offered, which generalizes that of \textit{L. Hoehn} and \textit{I. Niven} [Math. Mag. 58, 151-156 (1985; Zbl 0601.26011), cf. \textit{J. L. Brenner}, Pi Mu Epsilon J. 8, 160-163 (1985; Zbl 0601.26012), \textit{R. P. Boas} and \textit{J. L. Brenner}, J. Math. Anal. Appl. 123, 262-264 (1987; Zbl 0614.26001), \textit{J. Aczél} and \textit{Zs. Páles}, Am. Math. Mon. 95, No.9, 856-860 (1988; Zbl 0671.26008), among others]. Integral means are also introduced. \{Remarks: The logarithmic mean \((a_ 2-a_ 1)/(\log a_ 2-\log a_ 1)\) is \underbar{not} a quasilinear (''HLP'') mean \(\psi^{-1}(w_ 1\psi (a_ 1)+w_ 2\psi (a_ 2))\) (p. 547, l. 12-15) but it is a quasi- arithmetic integral mean as defined in lines 22-26 of p. 547 and lines 1- 2 of p. 548. On p. 542, l, 29, \(\phi_ q(t)\) should be \({\tilde\phi}_ q(t).\}\)
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