Another irreducibility criterion

Let $f=a_0+a_{1}x+\cdots +a_m{x}^m\in \mathbb{Z}[x]$ be a primitive polynomial. Suppose that there exists a positive real number $\alpha$ such that $\mathrm{\mid }{a}_m\mathrm{\mid }{\alpha }^m>\mathrm{\mid }{a}_0\mathrm{\mid }+\mathrm{\mid }{a}_1\mathrm{\mid }\alpha +\cdots +\mathrm{\mid }{a}_{m-1}\mathrm{\mid }{\alpha }^{m-1}$. We prove that if there exist natural numbers $n$ and $d$ satisfying $n\ge \alpha +d$ for which either $\mathrm{\mid }f(n)\mathrm{\mid }\mathrm{/}d$ is a prime, or $\mathrm{\mid }f(n)\mathrm{\mid }\mathrm{/}d$ is a prime-power coprime to $\mathrm{\mid }{f}^{\mathrm{\prime }}(n)\mathrm{\mid }$, then $f$ is irreducible in $\mathbb{Z}[x]$.