Comparison of discrete mixed means containing symmetric functions
Let \(\mathcal{A}^{(k)}\) denote the \(k\)-dimensional arithmetic mean defined for data sets \(S=\{x_1,\cdots,x_k\}\) of cardinality \(k\) by \[ \mathcal{A}^{(k)} (S) = \mathcal{A}^{(k)} (x_1,\dots,x_k) = \frac{1}{k} \sum_{i=1}^k x_i. \] Here, a data set stands for a set with possible repetitions. For every continuous strictly monotone function \(f :\mathbb R \longrightarrow \mathbb R\) and every natural number \(k\), one associates the \(k\)-dimensional quasi-arithmetic mean \(\mathcal{A}_f^{(k)}\) defined by \[ \mathcal{A}_{f}^{(k)}(S) = \mathcal{A}_{f}^{(k)}(x_1,\dots,x_k) = f^{-1} \left(\mathcal{A}^{(k)}(f(x_1),\dots,f(x_k))\right). \] If \(g :\mathbb R \longrightarrow\mathbb R\) is another continuous strictly monotone function, and if \(k < n\) are any natural numbers, then one associates with \(f\), \(g\), \(k\), and \(n\) the \(n\)-dimensional mixed quasi-arithmetic mean \(\mathcal{A}_{f,g;k}^{(n)}\) defined for sets \(S\) of cardinality \(n\) by \[ \mathcal{A}_{f,g;k}^{(n)} (S) = \mathcal{A}_g^{(r)}\left(\mathcal{A}_f^{(k)} (T_1),\dots,\mathcal{A}_f^{(k)} (T_r)\right), \] where \(T_1,\dots,T_r\) are the data subsets of \(S\) of cardinality \(k\) and where \(r = \left(\begin{matrix} n \\ k \end{matrix} \right)\) is the number of such subsets. Let \(f, g, F, G : \mathbb R \longrightarrow \mathbb R\) be continuous and strictly monotone and assume that \(\mathcal{A}_f^{(k)} \leq \mathcal{A}_F^{(k)}\) and \(\mathcal{A}_g^{(k)} \leq \mathcal{A}_G^{(k)}\), where \(\mathcal{A}_f^{(k)} \leq \mathcal{A}_F^{(k)}\) means that \(\mathcal{A}_f^{(k)}(T) \leq \mathcal{A}_F^{(k)}(T)\) for all data sets \(T\) of cardinality \(k\). The paper under review proves that (i) if \(1 < k < n\), then \(\mathcal{A}_{f,g;k}^{(n)} \leq \mathcal{A}_{F,G;k}^{(n)}\), (ii) if \(G \circ F^{-1}\) is convex, if \(G\) is increasing, and if \(2 < k < n\), then \(\mathcal{A}_{f,g;k}^{(n)} \leq \mathcal{A}_{F,G;k-1}^{(n)}\), (iii) if \(g \circ f^{-1}\) is concave, if \(g\) is increasing, and if \(2 < k < n\), then \(\mathcal{A}_{F,G;k}^{(n)} \leq \mathcal{A}_{f,g;k-1}^{(n)}\). These results generalize results that were obtained by \textit{W.~Ness} [Prax. Math. 8, 293--296 (1964)], by \textit{B.~C. Carlson, R.~K. Meany}, and \textit{S.~A. Nelson} [Pac. J. Math. 38, 343--349 (1971; Zbl 0195.34603)], by \textit{E.~K.~Godunova} [Am. Math. Soc., Transl., II. Ser. 88, 57--66 (1970); translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1965, No. 4(47), 45--53 (1965; Zbl 0158.05204)], and by \textit{M.~Marcus} and \textit{H.~Minc} [``A survey of matrix theory and matrix inequalities'' (1964; Zbl 0126.02404)].
scientific article; zbMATH DE number 5155522
scientific article