On generalized convex functions
E. F. Beckenbach, R. H. Bing · Transactions of the American Mathematical Society · 1945
Introduction.Let {£(*)} be a family of real functions defined in the interval a<x<b and satisfying the following conditions: (1) Each F(x) is a continuous function of x in a<x<b.(2) There is a unique member of the family which, at arbitrary xlt x2 satisfying a<Xx<x2<b, takes on arbitrary values yi, y2 respectively.By F(x; x%, yi; x2, y2) we shall denote the member of {F(x)} which takes on values yi, y2 at Xi, x2, respectively.For a given function f(x) we shall for brevity denote F[x; Xi,f(xi); x¡, f(x¡)] by £,-,,-(#).We shall further denote the interval a<x<b by (a, b).