Sharpening of Turán-type inequality for polynomials

For the polynomial $P(z) = \displaystyle\sum_{j=0}^{n} c_jz^j$ of degree $n$ having all its zeros in $|z|\leq k$, $k\geq 1$, V. Jain in \textquotedblleft On the derivative of a polynomial\textquotedblright, Bull. Math. Soc. Sci. Math. Roumanie Tome 59, 339–347 (2016) proved that
$$\begin{align*} \max_{|z|=1}|P^\prime(z)|\geq n\bigg(\frac{|c_0| +|c_n|k^{n+1}}{|c_0|(1+ k^{n+1}) +|c_n|(k^{n+1}+ k^{2n})}\bigg)\max_{|z|=1}|P(z)|. \end{align*}$$
In this paper we strengthen the above inequality and other related results for the polynomials of degree $n\geq 2$.