For a polynomial $P(z) = z^s(a_0 + a_1 z + \cdots + a_{n-s} z^{n-s}) \in P_n$, $0 \le s \le n$, it was shown recently by Rather et al. [12] that if $P(z) \neq 0$ in $|z| > k$, $k \le 1$, then for every $\alpha \in \mathbb{C}$ with $|\alpha| \ge k$, $$\max_{|z|=1} |D_\alpha P(z)| \ge n \frac{|\alpha| - k}{1 + k} \left\{ 1 + \frac{k}{n} \left( s + \frac{k^{n-s}|a_{n-s}| - |a_0|}{k^{n-s}|a_{n-s}| + |a_0|} \right) \right\} \max_{|z|=1} |P(z)|.$$ In this paper, we obtain certain refinements of this inequality and related results.