Remarks on some problems of Th. M. Rassias (Q1888668)

During the 40th ISFE meeting, Th. M. Rassias posed the following problems: (1) Find all functions \(f:(0,\infty)\to\mathbb{R}\) such that \[ f(\sqrt {xy})+f \left(\frac {x^2+y^2}{x+y}\right) =f(x)+f(y).\tag{a} \] (2) Find all functions \(f:(0,\infty) \to\mathbb{R}\) such that \[ f\left(\frac{x+y+xy} {2} \right)+f \left(\frac{2xy}{x+y+xy} \right)=f(x)+f(y)+f(xy).\tag{b} \] The author shows that a continuous function \(f:(0,\infty)\to\mathbb{R}\) satisfies (a) if and only if \(f\) is constant, and that a function \(f:(0,\infty)\to\mathbb{R}\) satisfies (b) if and only if \(f\) is identically zero.