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We revisit and improve joint concavity/convexity and monotonicity propertiesof quasi-entropies due to Petz in a new fashion. Then we characterize equalitycases in the monotonicity inequalities (the data-processing inequalities) ofquasi-entropies in several ways as follows: Let$\Phi:\mathcal{B}(\mathcal{H})\to\mathcal{B}(\mathcal{K})$ be atrace-preserving map such that $\Phi^*$ is a Schwarz map. When $f$ is anoperator monotone or operator convex function on $[0,\infty)$, we presentseveral equivalent conditions for the equality$S_f^K(\Phi(\rho)\|\Phi(\sigma))=S_f^{\Phi^*(K)}(\rho\|\sigma)$ to hold forgiven positive operators $\rho,\sigma$ on $\mathcal{H}$ and$K\in\mathcal{B}(\mathcal{K})$. The conditions include equality cases in themonotonicity versions of Lieb's concavity and Ando's convexity theorems.Specializing the map $\Phi$ we have equivalent conditions for equality cases inLieb's concavity and Ando's convexity. Similar equality conditions arediscussed also for monotone metrics and $\chi^2$-divergences. We furtherconsider some types of linear preserver problems for those quantum informationquantities.

Equality cases in monotonicity of quasi-entropies, Lieb's concavity and Ando's convexity