On splitting the area under curves

AbstractTwo area-splitting problems involving real-valued functions of a real variable are investigated. The second of these is essentially equivalent to finding all functions a ϵ C1((0, r)) with 0 < a(x) < x which satisfy the functional differential equation a′ (a(x)) = a(x)x for x ϵ (0, r). All solutions analytic at x = 0 (and many which are not) are exhibited in closed form