P-functions, Quasi-convex Functions, and Hadamard-type Inequalities1

We establish some results concerning P-functions from the standpoint of abstract convexity. In particular, we show that the set of all P-functions on a segment is the least set closed under pointwise sum, supremum, and convergence and containing the class of all nonnegative quasi-convex functions on that segment. Further, generalizations are derived of a recent inequality of Hadamard type for P-functions