  Recently the behavior of operator monotone functions on unbounded intervals
with respect to the relation of strictly positivity has been investigated. In
this paper we deeply study such behavior not only for operator monotone
functions but also for operator convex functions on bounded intervals. More
precisely, we prove that if $f$ is a nonlinear operator convex function on a
bounded interval $(a,b)$ and $A, B$ are bounded linear operators acting on a
Hilbert space with spectra in $(a,b)$ and $A-B$ is invertible, then
$sf(A)+(1-s)f(B)>f(sA+(1-s)B)$. A short proof for a similar known result
concerning a nonconstant operator monotone function on $[0,\infty)$ is
presented. Another purpose is to find a lower bound for $f(A)-f(B)$, where $f$
is a nonconstant operator monotone function, by using a key lemma. We also give
an estimation of the Furuta inequality, which is an excellent extension of the
L\"owner--Heinz inequality.
