General inequalities for quasideviation means (Q1106968)

Let \(I_ j\subseteq {\mathbb{R}}\) \((j=0,1,...,m)\) be open intervals. The functions \(E_ j:\) \(I\) \(2_ j\to {\mathbb{R}}\) are quasideviations if \(t\mapsto E_ j(x,t)\) are continuous, \(sign E_ j(x,t)=sign(x-t(x) (x,t\in I_ j)\) and \(t\mapsto E_ j(y,t)/E_ j(x,t)\) are strictly increasing \((x,y\in I_ j,\quad x<t<y).\) Let \({\mathbf{x}}\quad j=(x_ 1^{(j)},...,x_ n^{(j)})\in I\quad n_ j,\) \(\mathbf{\lambda}=(\lambda_ 1,...,\lambda_ n)\in {\bar {\mathbb{R}}}\quad n_+\setminus \{0\};\) the unique solutions \(t_ j=M({\mathbf{x}}\quad j,\mathbf{\lambda};E_ j)\) of \(\sum^{n}_{k=1}\lambda_ kE_ j(x_ k^{(j)},t_ j)=0\quad (j=0,1,...,m)\) are quasideviation means. The author's main result is that, for arbitrary \(f:\quad I_ 1\times...\times I_ m\to I_ 0\) and arbitrary continuous \(F:\quad I_ 1\times...\times I_ m\to {\mathbb{R}},\) \[ M({\mathbf{f}}({\mathbf{x}}\quad 1,...,{\mathbf{x}}\quad m),\mathbf{\lambda};E_ 0)\quad \leq \quad F(M({\mathbf{x}}\quad 1,\mathbf{\lambda};E_ 1),...,M({\mathbf{x}}\quad m,\mathbf{\lambda};E_ m)), \] where \textbf{f}(\textbf{x} 1,...,\textbf{x} \(m)=(f(x_ 1^{(1)},...,x_ n^{(1)}),...,f(\) \(x_ 1^{(m)},...,x_ n^{(m)}))\), is equivalent to its special case \textbf{\(\lambda\) }\(=(1,...,1)\) and both are equivalent to the existence of \(G_ j:\quad I_ 1\times...\times I_ m\to {\mathbb{R}}\quad (j=1,...,m)\) such that \[ E_ 0(f(x_ 1,...,x_ m),F(t_ 1,...,t_ m))\quad \leq \quad \sum^{m}_{j=1}A_ j(t_ 1,...,t_ m)E_ j(x_ j,t_ j) \] for all \(x_ j,t_ j\in I_ j\) \((j=1,2,...,m)\). This is applied to several older (Hölder, Minkowski type) and newer inequalities (comparison of quasideviation means, in particular of homogeneous and/or differentiable ones).