Quadratic functions fulfilling an additional condition along hyperbolas or the unit circle

Abstract

Let $S_1$ denote the set of all pairs (x, y) of real numbers that fulfill the condition $x^2 - y^2 = 1$, and $S_2$ denote the set of all pairs (x, y) of real numbers that fulfill the condition $x^2 + y^2 = 1 \,$. In this paper we consider quadratic real functions f that satisfy the additional equation $y^2 f(x) = x^2 f(y)$ under the condition $(x,y) \in S_i$$(i=1,2)$. We prove that each of these conditions implies $f(x) = f(1) x^2$ for all $x \in \mathbb {R}$.