Linked pairs of additive functions.

In 1990, Benz asked whether a real additive mapping satisfying $$xf(y)=yf(x)$$ for all points ( x, y) on the unit circle must be linear. In 2005, Boros and Erdei showed that it must be so. Here we generalize the problem to a pair of additive functions f, g related by the functional equation $$xf(y)=yg(x)$$ for all points ( x, y) on a specified curve. We find that for many (but not all) types of curves this forces f and g to be equal and linear. [ABSTRACT FROM AUTHOR]