Around strongly operator convex functions

We establish the subadditivity of strongly operator convex functions on $(0,\infty )$ and $(-\infty ,0)$. By utilizing the properties of strongly operator convex functions, we derive the subadditivity property of operator monotone functions on $(-\infty ,0)$. We introduce new operator inequalities involving strongly operator convex functions and weighted operator means. In addition, we explore the relationship between strongly operator convex and Kwong functions on $(0,\infty )$. Moreover, we study strongly operator convex functions on $(a,\infty )$ with $-\infty <a$ and on the left half-line $(-\infty ,b)$ with $b<\infty$. We demonstrate that any nonconstant strongly operator convex function on $(a,\infty )$ is strictly operator decreasing, and any nonconstant strongly operator convex function on $(-\infty ,b)$ is strictly operator monotone. Consequently, for a strongly operator convex function g on $(a,\infty )$ or $(-\infty ,b)$, we provide lower bounds for $\left|g\right(A)-g(B\left)\right|$ whenever $A-B>0$.