Turán type inequalities for the generalized derivative of a polynomial

Let \(P(z)=\prod _{j=1}^{n}(z-z_j)\) $P(z)={\prod }_{j=1}^n(z-z_j)$ be a polynomial of degree n with \(|z_j|\le k_j\le 1,1\le j\le n\) $|z_j|\le k_j\le 1,1\le j\le n$ , it is known that \(\begin{aligned} \max _{|z|=1}|P^\prime (z)|\ge \frac{n}{2}\left\{ 1+\frac{1}{1+\left( \frac{2}{n}\right) \sum _{j=1}^{n}\frac{k_j }{1-k_j}}\right\} \max _{|z|=1}|P(z)|. \end{aligned}\) $$\begin{array}{c}\hfill \underset{|z|=1}{max}|P^{\prime }(z)|\ge \frac{n}{2}\left\{1,+,\frac{1}{1+\left(\frac{2}{n}\right){\sum }_{j=1}^n\frac{k_j}{1-k_j}}\right\}\underset{|z|=1}{max}|P(z)|.\end{array}$$ In this paper, we shall extend this inequality to the generalized derivative of a polynomial without perturbing the condition on the zeros of underlying polynomial. We shall also present the polar derivative analogue of the result obtained for the generalized derivative of the polynomial. Our results include different relevant inequalities as special cases.