  Motivated by some recently established operator Jensen-type inequalities
related to a usual convexity, in the present paper we derive several more
accurate operator Jensen-type inequalities for certain subclasses of convex
functions. More precisely, we obtain interpolating series of Jensen-type
inequalities for log-convex and non-negative superquadratic functions. In
particular, we obtain the corresponding refinements of the Jensen-Mercer
operator inequality for such classes of functions.
