Converse theorem for the Minkowski inequality (Q947567)

Let \((\Omega,\Sigma,\mu)\) be a measure space such that \(0<\mu(A)< 1<\mu(B)<\infty\) for some \(A,B\in\Sigma\). Under some natural conditions on the bijective functions \(\varphi,\varphi_1,\varphi_2, \psi,\psi_1,\psi_2: (0,\infty)\to (0,\infty)\), the author proves that if \[ \psi\Biggl(\int_{\Omega(f+ g)} \varphi\circ(f+ g)\,d\mu\Biggr)\leq \psi_1\Biggl(\int_{\Omega(f)} \varphi_1\circ f\,d\mu\Biggr)+ \psi_2\Biggl(\int{\Omega(g)} \varphi_2\circ g\,d\mu\Biggr) \] for all nonnegative \(\mu\)-integrable simple functions \(f,g:\Omega\to R\) where \[ \Omega(f):= \{\omega\in\Omega: f(\omega)\neq 0\}, \] then there exists a real \(p\geq 1\) such that \[ {\varphi(t)\over\varphi(1)}= {\varphi_1(t)\over\varphi_2(1)}= t^p,\quad {\psi(t)\over \psi(1)}= {\psi_1(t)\over \psi_1(1)}= t^{{1\over p}},\quad i= 1,2. \] Some generalizations and relevant results for the reversed inequality are also presented.