A geometric characterization ofnth order convex functions

Abstract

A new geometric characterization is presented for a function convex of order n on an open interval, distinct from the whole of R. We shall prove that if /: (a, b)~^R with b < + oo, if a is an arbitrarily fixed number, a ^ 6, and if F(x) denotes the of the point of intersection in the x, y-plsme between the vertical line x = a and the osculating parabola of order n to the graph of / at the point (x, /(#)), then / is convex of order n on {a, b) iff F is increasing thereon. l Introduction* Let / be a real-valued convex function <yi the interval (α, 6) where a ^ o: it is stated in [1; p. I. 51, Exercise 7], and is indeed elementary to prove, that the function F(x) = f(x) — xfuix) is decreasing on (α, 6), fή denoting the right derivative of /. F(x) is none other than the ordinate at the origin of the right tangent line to the graph of / at the point (x, f(x)). Besides proving the converse of this proposition in this paper we shall extend the result to nth order convex functions, the role of the tangent line being played by the osculating parabola of order n. The result we present provides a meaningful geometric characterization of such a class of functions (Theorem 2.1 below). The notion of higher-order convexity is classical: its systematic study essentially began with a paper by Popoviciu [4] and was continued in many other works by the same author. The entire theory is surveyed in his monograph [5]. Other properties can be found in books [2; Chp. 4 §3 and Chp. 3 §2] and [3; Chp. XI] in the context of generalized convex functions. Many characterizations of nth order convex functions can be obtained from these references; in order to establish our main result we shall only need a few of such characterizations and shall state them here for the sake of convenience.