An estimate of quasi-arithmetic mean for convex functions (Q2839854)

For a selfadjoint operator \(A\) on a Hilbert space \(H\), a normalized positive linear map \(\Phi\) and a strictly monotone function \(\varphi\), a quasi-arithmetic mean is defined by \(\varphi^{-1}(\Phi(\varphi(A)))\). \textit{A. Matsumoto} and \textit{M. Tominaga} [Sci. Math. Jpn. 61, No. 2, 243--247 (2005; Zbl 1092.47018)] investigated the relation between \(\varphi^{-1}(\Phi(\varphi(A)))\) and \(\Phi(A)\) for a convex function \(\varphi\). In this paper, the authors provide some conditions under which an order among quasi-arithmetic mean of the form \(\varphi^{-1}(\Phi(\varphi(A)))\leq \psi^{-1}(\Phi(\psi(A)))\) holds. Some difference and ratio type complementary inequalities among quasi-arithmetic means are given by using the Mond-Pečarić method, see [\textit{J. Pečarić} et al., Mond-Pečarić method in operator inequalities. Inequalities for bounded selfadjoint operators on a Hilbert space. Monographs in Inequalities 1. Zagreb: ELEMENT (2005; Zbl 1135.47012)].