Intensive studies aiming to extend the Hermite-Hadamard inequalities and to explore some properties and applications of these inequalities have been recently carried out. The contribution of this paper falls within this framework. We investigate some refinements and reverses for the Hermite-Hadamard inequalities when the integrand map is pointwise convex in its functional arguments. Our theoretical results obtained here immediately imply those of operator versions without referring to the techniques of the operator theory.