Around Strongly Operator Convex Functions

In this paper, we obtain the subadditivity inequality of strongly operator convex functions on (0, ∞) and (-∞,0). Applying the properties of operator convex functions, we deduce the subadditivity property of operator monotone functions on (0, ∞). 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, ∞). Moreover, we study the strongly operator convex functions on (a, ∞) with a>-∞ and also on left half line (-∞,b) with b< ∞. We show that any non-constant strongly operator convex function on (a, ∞) is strictly operator decreasing, and any non-constant strongly operator convex function on (-∞,b) is strictly operator monotone. Consequently, if g is a strongly operator convex function on (a, ∞) or (-∞,b), we estimate lower bounds of $|g(A)-g(B)|$ whenever $A-B>0$.