As a generalization of the quasi-arithmetic mean, 
we consider a mean-like transformation of operator functions.
Let $\Phi$ be a unital positive linear map of $B(H)$, 
the algebra of all bounded linear operators on a Hilbert space $H$, 
and $f(t)$ (resp. $g(t)$) a continuous function 
on an interval $[m,M]$ (resp. $f([m,M])$).
Then it is defined by $(g \circ \Phi \circ f)(A)$ 
for a selfadjoint operator $A$ with $m \le A \le M$.
We give a lower bound of the difference 
between $(g \circ \Phi \circ f)(A)$ and $\Phi(A)$.
Precisely we prove that if $f(t)$ is concave on $[m,M]$ and $g(t)$ 
is increasing and convex on $f([m,M])$, then for each $\l \in \R$,
$(g \circ \Phi \circ f)(A) - \l \Phi(A) \ge \min_{t \in [m,M]} 
\left\{ g(\a_ft+\b_f) - \l t \right\}$ 
where $\a_f:=\frac{f(M)-f(m)}{M-m}$ and $\b_f:=\frac{Mf(m)-mf(M)}{M-m}$.
It is an extension of our previous estimation 
for $\Phi=\omega_x$, the vector state for a unit vector $x \in H$.