On a functional equation related to a generalization of Flett's mean value theorem
The authors offer a proof, without regularity assumptions, that the general solutions \(f: \mathbb{R} \to\mathbb{R}\) of the functional equation (\(q\in ]0,1[\) constant) \[ f((1-q)x+qy)-f(x)=q(y-x)h((1-q)x+qy)-(h(y)-h(x))q^2(y-x)/2 \] (\((x,y)\in\mathbb{R}^2, x\neq y\)) are quadratic polynomials for arbitrary \(q\in]0,1[\) and, only if \(q=3/4,\) also cubic polynomials. The second unknown function, \(h\), turns out to be the derivative of \(f\), thus establishing a connection to \textit{T. M. Flett}'s mean value theorem [Math. Gaz. 42, 38-39 (1958)]. [Remark: The trivial case \(a=0\) (\(x=0\) in our notation), neglected in the lines after eqs. (2.18) and (2.22), can easily be taken care of by substituting \(a=0\) into eqs. (2.17) and (2.21), respectively].
functional equation with two unknown functions
scientific article; zbMATH DE number 1441870
scientific article