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Recently the behavior of operator monotone functions on unbounded intervalswith respect to the relation of strictly positivity has been investigated. Inthis paper we deeply study such behavior not only for operator monotonefunctions but also for operator convex functions on bounded intervals. Moreprecisely, we prove that if $f$ is a nonlinear operator convex function on abounded interval $(a,b)$ and $A, B$ are bounded linear operators acting on aHilbert 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 resultconcerning a nonconstant operator monotone function on $[0,\infty)$ ispresented. 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 givean estimation of the Furuta inequality, which is an excellent extension of theL\"owner--Heinz inequality.

Estimates of operator convex and operator monotone functions on bounded intervals