A criterion for<i>n</i>-convexity

Abstract

The development of the P n -integral of R. D. James and W. H. Gage is based on certain properties of -convex functions. In order to develop this integral systematically a more detailed study of w-convex functions is needed. In the second section of this paper various derivatives are defined and some of their properties given; in the third and last sections properties of %-convex functions are developed. 2* Definitions and some simple properties of generalized derivatives* Suppose F is a real-valued function defined on the bounded closed interval [, b] then if it is true that for x 0 e ]a, b[ (1)