On Sahoo-Riedel equations on a real interval
The author determines the general solutions \(f, h, k : I \to \mathbb{R}\) of the functional equations \[ f(x) - f(y) = (x-y) \left [ h\left ( {{x+y} \over 2} \right ) + k(x) + k(y) \right ] \tag{1} \] and \[ xf(y) -y f(x) = (x-y) \left [ h\left ( {{x+y} \over 2} \right ) + k(x) + k(y) \right ] \tag{2} \] for all \(x, y \in I\), where \(I \subset \mathbb{R}\) is an arbitrary interval on the real line \(\mathbb{R}\) with positive length. The motivation for studying these equations on the restricted domain came from the fact that they arise on some problems connected with the convexity of a derivative. Using ideas from \textit{P. K. Sahoo} and \textit{T. Riedel} [ Mean value theorems and functional equations, World Scientific, Singapore (1998; Zbl 0980.39015)] and some clever substitutions, the author determines solutions of these equations. The following results are obtained. Lemma: Assume that \(I \subset \mathbb{R}\) is an interval of positive length and \(0 \in \operatorname {int}I\). The functions \(f, h, k : I \to \mathbb{R}\) satisfy equation (2) if and only if \(f(x) = 3 ax^3 + 2b x^2 + cx + (2\beta -\alpha )\), \(h(x) = -8a x^3 - 4b x^2 - 2 A(x) + \alpha \), and \( k(x) = ax^3 + bx^2 + A(x) + \beta - \alpha \), where \(A : \mathbb{R} \to \mathbb{R}\) is an additive function and \(a, b, c, \alpha , \beta\) are arbitrary real constants. Theorem: Assume that \(I \subset \mathbb{R}\) is an interval of positive length. The functions \(f, h, k : I \to \mathbb{R}\) satisfy equation (1) if and only if \(f(x) = 3 ax^4 + 2b x^3 + cx^2 + (\beta -\alpha )x + d\), \(h(x) = 8a x^3 + 4b x^2 + 2 A(x) -(\beta + \alpha ) \), and \( k(x) = 2ax^3 + bx^2 - A(x) + cx + \beta \) for all \(x \in\operatorname {int} I\), where \(A:\mathbb{R}\to \mathbb{R}\) is an additive function and \(a, b, c, d, \alpha , \beta\) are arbitrary real constants. If the interval \(I\) is closed, then \(h\) may be arbitrary on the boundary of \(I\).
scientific article; zbMATH DE number 1742895