A Diophantine equation with the harmonic mean

Abstract

Let $$f\in \mathbb {Q}[x]$$ be a polynomial without multiple roots and $$\deg {f}\ge 2$$. We give conditions for $$f=x^2+bx+c$$ under 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 $$f=x^2+bx$$ with nonzero integer b. Moreover, we show that it has a rational parametric solution for infinitely many cubic polynomials.