Inequalities between operator means based on the Mond-Pečarić method and its applications

In this paper, we shall consider the estimates of $alpha$-operator divergence by terms of the spectra of positive operators. For this purpose, we shall investigate the estimates of the difference of two operator means in general setting. We prove that for positive invertible operators $A$, $B$ and a given $alpha >0$, there exists the most suitable real number $beta $ such that $$Phi (A sigma_1 B)geq alpha Phi (A) sigma _2 Phi (B) +beta Phi (A)$$ where $Phi $ is a unital positive linear map and $sigma _1$, $sigma _2$ are operator means. In particular, if we put $alpha =1$ and $Phi $ is the identity map, then we have the lower bound of the difference of $A sigma _1 B$ and $A sigma _2 B$. Consequently we obtain the estimates of $alpha$-operator divergence by terms of the spectra of positive operators.