  We consider a quantum quasi-relative entropy $S_f^K$ for an operator $K$ and
an operator convex function $f$. We show how to obtain the error bounds for the
monotonicity and joint convexity inequalities from the recent results for the
$f$-divergences (i.e. $K=I$). We also provide an error term for a class of
operator inequalities, that generalize operator strong subadditivity
inequality. We apply those results to demonstrate explicit bounds for the
logarithmic function, that leads to the quantum relative entropy, and the power
function, which gives, in particular, a Wigner-Yanase-Dyson skew information.
In particular, we provide the remainder terms for the strong subadditivity
inequality, operator strong subadditivity inequality, WYD-type inequalities,
and the Cauchy-Schwartz inequality.
