On Functions with Nonnegative Divided Differences

Abstract

Let V_n (F) = V_n (F;x₀,x₁,...,x_n) be the nth divided difference of F with respect to the n + 1 points X₀,X₁, ...,x_n on an interval [a,b]. If the inequality V_n(F) ≥ 0 holds for all choices of points x₀,x₁, ...,x_n in [a,b], then F is said to be n-convex on [a,b]. It is here shown that if F(x) is n-convex on [a,b], if F^(r)(x) exists and is continuous on [a,b], 0 ≤ r ≤ n − 2, and if F_(n−1),+(a) is finite, then

$$F\left(x\right)\,=\,G\left(x\right)+\sum\limits_{k=0}^{n-1}{F_{(k),+}}\left(a\right)\frac{{(x-a)^k}}{{kl}},\,x\,\varepsilon\,[a,b]$$

, where G(x) is monotonic increasing on [a,b], and F_(k),+(a) is the one-sided Peano derivative of F at a. The result has applications in approximation theory.