  In this paper, we obtain the subadditivity inequality of strongly operator
convex functions on $(0, \infty)$ and $(-\infty,0)$. Applying the properties of
operator convex functions, we deduce the subadditivity property of operator
monotone functions on $(0, \infty)$. We give new operator inequalities
involving strongly operator convex functions on an interval and the weighted
operator means. We also investigate relations between strongly operator convex
functions and Kwong functions on $(0, \infty)$. Moreover, we study the strongly
operator convex functions on $(a, \infty)$ with $a>-\infty$ and also on left
half line $(-\infty,b)$ with $b< \infty$. We show that any non-constant
strongly operator convex function on $(a, \infty)$ is strictly operator
decreasing, and any non-constant strongly operator convex function on
$(-\infty,b)$ is strictly operator monotone. Consequently, if $g$ is a strongly
operator convex function on $(a, \infty)$ or $(-\infty,b)$, we estimate lower
bounds of $|g(A)-g(B)|$ whenever $A-B>0$.
