We treat the behavior of operator monotone and operator convex functions on bounded and unbounded intervals with respect to the relation of strict positivity. 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 $A>B$, then $sf(A)+(1-s)f(B)>f(sA+(1-s)B)$. Our main purpose is to find a lower bound for $f(A)-f(B)$, where $f$ is a nonconstant operator monotone function.

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