A note on an irreducible class of polynomials over integers

In this note, we prove an irreducibility criterion for the polynomial of the form $f(x)=a_n{x}^n+a_{n-1}{x}^{n-1}+\cdots +a_m{x}^m+p^u\in \mathbb{Z}[x]$, where $p$ is a prime number, $u⩾1$, $\gcd (u,m)=1$, $p\nmid a_m$ and $p^u>\mathrm{\mid }{a}_n\mathrm{\mid }+\mathrm{\mid }{a}_{n-1}\mathrm{\mid }+\cdots +\mathrm{\mid }{a}_m\mathrm{\mid }$. In particular, we show that the conjecture of Koley and Reddy is true.