The Pexider type generalization of the Minkowski inequality (Q432371)

Let \(\varphi :(0,\infty) \rightarrow (0,\infty)\) be a bijection. The functional \(\mathbb{P}_{\varphi} : S \rightarrow [0,\infty)\) is given byNEWLINENEWLINENEWLINE\[NEWLINE\mathbb{P}_{\varphi} ({\mathbf x})=\begin{cases} \varphi^{-1}\left( \int_{\Omega({\mathbf x})} \varphi \circ |{\mathbf x}| d\mu \right), & \mu(\Omega({\mathbf x})) >0 \cr 0, & \mu(\Omega(\mathbf x))=0\cr \end{cases} NEWLINE\]NEWLINE where \((\Omega, \Sigma, \mu)\) is a measure space, \(S\) is a linear space of all \(\mu\)-integrable simple functions \(\mathbf x : \Omega \rightarrow \mathbb{R}\), \(\Omega({\mathbf x})=\{\omega \in \Omega : {\mathbf x}(\omega)\not= 0\}\) and \(S_+=\{{\mathbf x}\in S: {\mathbf x} \geq 0 \}\).NEWLINENEWLINEInequalityNEWLINENEWLINENEWLINE\[NEWLINE \mathbb{P}_{\varphi} ({\mathbf x}+{\mathbf y}) \leq \mathbb{P}_{\psi} ({\mathbf x}) +\mathbb{P}_{\gamma} ({\mathbf y}), \;\;\;{\mathbf x}, {\mathbf y} \in S_+, \;\;\;\;\;\;(1) NEWLINE\]NEWLINE is called Pexiderization of the Minkowski inequality.NEWLINENEWLINEAuthor gives theorems of existence of the broad classes of triples \((\varphi, \psi, \gamma)\) of non-power functions satisfying inequality (1) in the case when the underlying measure space \((\Omega, \Sigma, \mu)\) satisfies some additional conditions. Main results are given in the following theorems.NEWLINENEWLINETheorem 1. Let \((\Omega, \Sigma, \mu)\) be a measure space with \(\Sigma = \{\emptyset, \Omega, A, \Omega\backslash A\}\) for some \(A\subset \Omega\) such that \(\mu(A)\), \(\mu(\Omega\backslash A)\) are positive real numbers and \(\min \{ \mu(A), \mu(\Omega\backslash A)\} \geq 1\). Suppose that \(\varphi, \psi, \gamma : (0,\infty) \rightarrow (0,\infty)\) are bijective and such that \(\varphi \) is geometrically convex, \(\varphi'_-\) and \(\gamma'_-\) (or \(\varphi'_+\) and \(\gamma'_+\)) exist and the functions \(u\mapsto \frac{\varphi(u)}{u}\), \(u\mapsto \frac{\gamma(u)}{u}\) are nondecreasing. Suppose that functions \(\frac{\varphi}{\psi}\) and \(\frac{\varphi}{\gamma}\) are nondecreasing and for each \(r\in \mu(\Sigma)\backslash \{0\}\) the functions \(f_r=\varphi^{-1}\circ (r\varphi)\), \(g_r=\psi^{-1}\circ (r\psi)\), \(h_r=\gamma^{-1}\circ (r\gamma)\) satisfy the ``Pexiderized'' subadditivity condition NEWLINE\[NEWLINEf_r(u+v) \leq g_r(u)+h_r(v), \;\;u,v>0.NEWLINE\]NEWLINE Then inequality (1) holds for \({\mathbf x}, {\mathbf y} \in S_+\).NEWLINENEWLINENEWLINENEWLINENEWLINENEWLINETheorem 2. Let \((\Omega, \Sigma, \mu)\) be a measure space with \(\mu(\Omega)=1\), \(\Sigma = \{\emptyset, \Omega, A, \Omega\backslash A\}\) for some \(A\subset \Omega\) such that \(0<\mu(A)<1\). Suppose that \(\varphi, \psi, \gamma : (0,\infty) \rightarrow (0,\infty)\) are bijective.NEWLINENEWLINE(i) If \(\varphi \) is increasing, then inequality (1) holds if and only if the function \(\Phi :(0,\infty)^2 \rightarrow (0,\infty)\) defined by \(\Phi(s,t)=\varphi(\psi^{-1}(s)+\gamma^{-1}(t)), \;\;s,t>0\) is concave.NEWLINENEWLINE(ii) If \(\varphi \) is decreasing, then inequality (1) holds if and only if the function \(\Phi\) is \(\mu(A)\)-convex.