Jensen’s Inequality for Quasiconvex Functions

This class of functions strictly contains the class of convex functions defined on a convex set in a real linear space. See [8] and citations therein for an overview of this issue. Some recent studies have shown that quasiconvex functions have quite close resemblances to convex functions – see, for example, [4], [6], [7], [10] for quasiconvex and even more general extensions of convex functions in the context of Hadamard’s pair of inequalities. Apart from generalizations to theory, weakening the convexity condition can increase applicability. Thus in [9] use is made of quasiconvexity to obtain a global extremum with rather less effort than via convexity. In this article we pursue the concept further and derive a number of Jensen–type inequalities for quasiconvex functions. See also [5] for functions of Godunova–Levin type in the context of Jensen’s inequality.