A Diophantine equation with the harmonic mean

Let f is an element of Q[x] be a polynomial without multiple roots and degf >= 2. We give conditions for f=x2+bx+cunder which the Diophantine equation 2f(x)f(y)=f(z)(f(x)+f(y))\ has infinitely many nontrivial integer solutions and prove that this equation has infinitely many rational parametric solutions for f=x2+bx with nonzero integer b. Moreover, we show that it has a rational parametric solution for infinitely many cubic polynomials.