On the stability of the equation stemming from Lagrange MVT

The stability behaviour of the functional equation F(y)−F(x)=(y−x)f((x+y)/2) is studied. It is proved that this equation is superstable i.e. if f,F satisfy |F(y)−F(x)(y−x)f((x+y)/2)|≤ε then f satisfies this equation (with some F). In order to obtain this result the equation hΔh2f(x)=0 is considered and it is proved that also this equation is superstable.