New versions of weighted multidimensional functional and Stolarsky means
In this article, the authors provide an elegant introduction of various multidimensional means, and study their properties. Let us cite a couple of definitions. Definition (doubly weighted multidimensional functional means). Let \(I\subset{\mathbb R}\) be an interval, \({\mathbf x}:=\left(x_1,\dots,x_n\right)\in I^n\) (\(n\geq 2\)), \({\mathbf p}:=\left(p_1,\dots,p_n\right)\) a discrete distribution, \(h,g:\,I\to{\mathbb R}\) be continuous and strictly monotone functions, and let \(\mu\) be a probability measure on the Borel \(\sigma\)-algebra of the simplex \[ E_{n-1}=\left\{\left(t_1,\dots,t_{n-1}\right)\in{\mathbb R}^{n-1}\mid t_1,\dots,t_{n-1}\geq 0,\,\sum_{j=1}^{n-1}t_j\leq 1\right\}. \] Define \[ M\left({\mathbf x},{\mathbf p},h,g,\mu\right):=h^{-1}\left(\int_{E_{n-1}}h\circ g^{-1}\left(\frac{\sum_{j=1}^n t_j p_j g\left(x_j\right)}{\sum_{j=1}^n t_j p_j}\right)\,d\mu\left({\mathbf t}\right)\right), \] where \(t_n:=1-\sum_{j=1}^{n-1}t_j\), and \(d\mu\left({\mathbf t}\right):=d\mu\left(t_1,\dots,t_{n-1}\right)\). Definition (doubly weighted multidimensional Stolarsky means). Let \({\mathbf x}:=\left(x_1,\dots,x_n\right)\in\left(0,\infty\right)^n\) (\(n\geq 2\)), let \({\mathbf p}:=\left(p_1,\dots,p_n\right)\) be a discrete distribution, and let \(\mu\) be a probability measure on the Borel \(\sigma\)-algebra of the simplex \(E_{n-1}\). Define \[ S_w\left({\mathbf x},{\mathbf p},s,r,\mu\right):= \begin{cases} \left(\int_{E_{n-1}}\left(M_r\left({\mathbf x},{\mathbf t}{\mathbf p}\right)\right)^{s-r} \,d\mu\left({\mathbf t}\right)\right)^{\frac{1}{s-r}}, & s\neq r,\\ \exp\left(\int_{E_{n-1}}\ln\left(M_s\left({\mathbf x},{\mathbf t}{\mathbf p}\right)\right)\,d\mu\left({\mathbf t}\right)\right), & s=r, \end{cases} \] where \[ M_r\left({\mathbf x},{\mathbf p}\right):= \begin{cases} \left(\sum_{j=1}^n p_j x_j^r\right)^{1/r}, & r\neq 0,\\ \prod_{j=1}^n x_j^{p_j}, & r=0 \end{cases} \] are weighted multidimensional \(r\)-th power means, and \[ {\mathbf t}{\mathbf p}:=\left(\frac{t_1 p_1}{\sum_{k=1}^n t_k p_k},\dots,\frac{t_n p_n}{\sum_{k=1}^n t_k p_k}\right). \] The first result on these means says that the function \(S_w\left({\mathbf x},{\mathbf p},\cdot,\cdot,\mu\right)\) is continuous on \({\mathbb R}^2\). Furthermore, the multidimensional Stolarsky means with discrete weights are introduced, and interesting relations are obtained. After that, the authors introduce the doubly weighted multidimensional \(r\)-th power means, various logarithmic, and identric means, and investigate their properties. In particular, monotonicity properties are well-investigated. Moreover, inequalities between various means are established. Let us cite one of such results on relations between the doubly weighted multidimensional Stolarsky means \(S_w\left({\mathbf x},{\mathbf p},s,r,\mu\right)\) and the doubly weighted multidimensional \(r\)-th power means \(M_w\left({\mathbf x},{\mathbf p},r,\mu\right)=S_w\left({\mathbf x},{\mathbf p},2r,r,\mu\right)\). Proposition. Let \({\mathbf x}:=\left(x_1,\dots,x_n\right)\in\left(0,\infty\right)^n\) (\(n\geq 2\)), \({\mathbf p}:=\left(p_1,\dots,p_n\right)\) a discrete distribution, and let \(\mu\) be a probability measure on the Borel \(\sigma\)-algebra of \(E_{n-1}\). (a) If \(s\geq 2r\), then \[ M_w\left({\mathbf x},{\mathbf p},r,\mu\right)\leq S_w\left({\mathbf x},{\mathbf p},s,r,\mu\right) \leq M_w\left({\mathbf x},{\mathbf p},s-r,\mu\right). \] (b) If \(s<2r\), then reversed inequalities hold. The article should be interesting for specialist in wide area of analysis, probability, and other areas of mathematics.
scientific article; zbMATH DE number 6699216