Summary: In this paper we consider some generalizations of the Ando inequality \[ |||f(A)-f(B)|||\leqslant |||f(|A-B|)||| \] with the ``weight'' \((A-B)^p\). More precisely, for \(p\geqslant 1\) such that \((-1)^p=-1\) and for a non-negative function \(f\) on \([0,\infty)\) such that \(f(0)=0\), we study the following inequality: \[ \mathrm{Tr}((A-B)^p(f(A)-f(B)))\geqslant\mathrm{Tr}(|A-B|^pf(|A-B|)), \] whenever \(A\) and \(B\) are positive semidefinite matrices. We show that the inequality is true for any operator convex function \(f\) and it is reversed whenever \(f\) is operator monotone.