Greatest prime factor of a polynomial

It is established that for the greatest prime factor $P(x)$ of the value of an integral irreducible polynomial $f(x)$ of degree $n\ge2$ for integral $x>0$ the estimate $P(x)>c_f\ln\ln x$, $x>x_0(f)$ holds, where $c_f$ is a positive value effectively defined by the coefficients of the polynomial.