On a functional equation related to a generalization of Flett′s mean value theorem

Abstract

Abstract. In this paper, we characterize all the functions that attain their Flett mean value at a particular point between the endpoints of the interval under consideration. These functions turn out to be cubic polynomials and thus, we also characterize these.

Polynomials and divided differences

“…We note that results related to those in this paper, in the sense of connecting polynomial and similar functions with divided differences, appear inter alia in papers by Aczél and Kuczma [3], Andersen [4], Davies and Rousseau [5], Deeba and Simeonov [6], Haruki [7], Sablik [12,13] and Schwaiger [16]. In [10] and [11], Riedel and Sablik characterize polynomial functions by functional equations derived from Flett's mean value theorem. Further examples and references may be found in the book by Sahoo and Riedel [15].…”

Characterizing polynomial functions by a mean value property

“…It turns out that specifying c to be equal to a+3b 4 implies that (2) characterizes cubic polynomials. More exactly, we obtained in [4] the following.…”

“…In [4], following the approach of Aczél [1] and Haruki [3] who solved functional equations related to the Lagrange Mean Value Theorem, we replaced in (1) the derivative of f by an unknown function h and so we obtained the following functional equation:…”

On Flett’s mean value theorem

“…Dini's derivatives [19], symmetric derivatives [21], v-derivatives [12], etc.). Also, a characterization of all the functions that attain their Flett's mean value at a particular point between the endpoints of the interval [20], other functional equations and means related to Flett's theorem should be mentioned in the future.…”