Jensen's inequality for operators without operator convexity (Q624524)

It was proved by \textit{F.\,Hansen}, \textit{J.\,Pečarić} and \textit{I.\,Perić} [Math.\ Scand.\ 100, No.\,1, 61--73 (2007; Zbl 1151.47025)] that, if \(f\) is an operator convex function on an interval \(J\), then \(f\left(\sum_{i=1}^n\Phi_i(A_i)\right) \leq \sum_{i=1}^n\Phi_i(f(A_i))\) holds for every \(n\)-tuple \((A_1, \dots, A_n)\) of selfadjoint operators in \(B(H)\) with spectra in \(J\) and every \(n\)-tuple \((\Phi_1, \dots, \Phi_n)\) of positive linear mappings \(\Phi_i: B(H) \to B(K)\) \((1 \leq i\leq n)\) such that \(\sum_{i=1}^n\Phi_i(1_H)=1_K\). In the present paper, the authors show that, if \(f\) is a continuous convex function, \(J\) contains all bounds \(m_i=\inf_{\|\xi\|=1}\langle A\xi,\xi\rangle\), and \(M_i=\sup_{\|\xi\|=1}\langle A\xi,\xi\rangle\) and \((m_A,M_A)\cap[m_i, M_i]=\emptyset\) for all \(1\leq i\leq n\), where \(m_A\) and \(M_A\) are the corresponding bounds of the selfadjoint operator \(A =\sum_{i=1}^n\Phi_i(A_i)\), then the Jensen inequality above holds. They also study operator quasi-arithmetic means under the same conditions.