Some lower bound estimates for the generalized derivative of a polynomial
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Abstract
\begin{align*}
Re\left(z\frac{P^\prime(z)}{P(z)}\right) \geq \frac{n}{1+k}\left\lbrace 1 + \frac{k}{n}\left(\frac{k^{n}\left|a_n\right| - \left|a_0\right|}{k^{n}\left|a_n\right| + \left|a_0\right|}\right)\right\rbrace.
\end{align*}
In this paper, we extend this inequality to the generalised derivative by taking $s$-folded zeros at origin. As an application, we obtain some lower bound estimates for the generalized derivative and generalized polar derivative of a polynomial with restricted zeros, which include various results due to Tur\'{a}n, Malik, Dubinin, Aziz, Rather and Govil as special cases.
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