SOME INEQUALITIES FOR STARSHAPED AND CONVEX FUNCTIONS

Necessary and sufficient conditions are obtained on a function G of bounded variation such that for all increasing x for which φ(0) = 0 for some specified U, and all convex φ for which φ(0)= 0; the conditions are otherwise independent ofφ and x. Similar results are obtained when the inequality is reversed. Necessary and sufficient conditions for both directions of inequality are also obtained when φ isstarshaped and φ(0) = 0. The relationship to previous results is sketched. Applications to statistical tolerance limits are indicated. © 1969 by Pacific Journal of Mathematics.