On 𝑛-parameter families of functions and associated convex functions

Abstract

Introduction. Let f(x) be a real-valued function continuous on the interval a < x ? b. Then f(x) is said to be strictly convex if and only if the graph of any linear function for a < x < b meets the graph of f(x) in at most two points. In this situation, one may consider the linear functions on a < x < b as a twoparameter family-for each pair of points (xi, yi) and (x2, y2), xl 5x2, there is exactly one linear graph through these points-and the strictly convex functions as with the linear functions. Beckenbach and Bing [1, 3](1) generalized this situation by replacing the linear functions by a more general 2-parameter family, that is, a family of continuous functions such that for each pair of points (xi, yi) and (x2, y2), X1i X2, there is one and only one member of the family through these points; then in a natural way they have introduced the associated convex functions. These authors have shown that many properties of the class of linear functions and convex functions hold for 2-parameter families and their associated convex functions. One surprising result was the observation that a 2-parameter family need not be topologically equivalent to the family of linear functions on the interval 0< x <1. T. Popoviciu [9] has given the definition for n-parameter families, but stated no properties. We obtain results here for such families of functions and their associated convex functions which are in part generalizations of those obtained by Beckenbach and Bing. We also obtain results related to the work of T. Popoviciu [7, 8] on convex functions associated with linear families, to that of M. M. Peixoto [6] on the derivatives of generalized convex functions, and to that on approximation discussed by S. Bernstein [4] and C. J. de la Vallee Poussin [5 ]. 1. Definitions and elementary properties. DEFINITION 1. An n-parameter family is a set of single-valued, real, continuous functions f(x) on an interval a < x < b such that for every set of points (xi, yi) (i = 1, * , n) with a ?<Xl<X2< * * * <x,<b there is exactly one f(x) with f(x) =yi. Hereafter all functions are assumed to be single-valued, real, and continuous on a < x < b. We designate an n-parameter family by F. Simple examples of n-parameter families are the set of all polynomials of