Translativity of beta-type functions (Q2697667)

Let \(I\subset \mathbb{R}\) be an interval closed under addition, and let \(f\colon I\to ]0, +\infty[\) be a function. The function \(B_{f}\colon I^{2}\to ]0, +\infty[\) defined by \[ B_{f}(x, y)= \dfrac{f(x)f(y)}{f(x+y)} \qquad \left(x, y\in I\right) \] is called a beta-type function in the interval \(I\). We call \(f\) the generator of \(B_{f}\). If \(I\subset \mathbb{R}\) is an interval for which \(t+I\subset I\) for all \(t>0\), then a function \(F\colon I^{2}\to \mathbb{R}\) is positively translative if there exists a function \(\alpha \colon ]0, +\infty[\to \mathbb{R}\) such that \[ F(x+t, y+t)= F(x, y)+\alpha(t) \] holds for all \(x, y\in I\) and \(t>0\). Similarly, if \(I\subset \mathbb{R}\) is an interval for which \(t+I\subset I\) for all \(t<0\), then a function \(F\colon I^{2}\to \mathbb{R}\) is negatively translative, if there exists a function \(\alpha \colon ]-\infty, 0[\to \mathbb{R}\) such that \[ F(x+t, y+t)= F(x, y)+\alpha(t) \] holds for all \(x, y\in I\) and \(t<0\). The aim of this paper is to study positively (resp., negatively) translative beta-type functions on different types of intervals. In Theorem 1, the authors show that if \(f\colon \mathbb{R}\to ]0, +\infty[\) and \(\alpha\colon ]0, +\infty[\to \mathbb{R}\) are functions, then the following conditions are pairwise equivalent: (i) The beta-type function \(B_{f}\colon \mathbb{R}^{2}\to ]0, +\infty[\) is \(\alpha\)-translative; (ii) \(\alpha\equiv 0\) and \[ f(0)f(x+y)= f(x)f(y) \qquad \left(x, y\in \mathbb{R}\right); \] (iii) The beta-type function \(B_{f}\) is constant. In the fourth section of the paper the authors consider the same problem in the interval \([0, +\infty[\). Finally, in the fifth section, the traslativity of beta-type functions in the interval \(]0, +\infty[\) is investigated.