A relationship between the Mahler measure and the discriminant of algebraic numbers

In this note we show that in the well-known Dobrowolski estimate $\ln M(\alpha)\gg(\ln\ln d/\ln d)^3$, $d\to\infty$, where $\alpha$ is a nonzero algebraic number of degree $d$ that is not a root of unity and $M(\alpha)$ is its Mahler measure, the parameter $d$ can be replaced by the quantity $\delta=d/\Delta(\alpha)^{1/d}$, where $\Delta(\alpha)$ is the modulus of the discriminant of $\alpha$. To this end, $\alpha$ must satisfy the condition $\deg\alpha^p=\deg\alpha$ for any prime $p$.