Geodesic
Some Inequalities for the Polar Derivative of a Polynomial
Kragujevac Journal of Mathematics, Tome 47 (2023) no. 4, p. 567

Voir la notice de l'article provenant de la source eLibrary of Mathematical Institute of the Serbian Academy of Sciences and Arts

Let $P(z)$ be a polynomial of degree $n$ which has no zeros in $|z|1$, then it was proved by Liman, Mohapatra and Shah \cite{moh} that \begin{align*} |zD_lpha P(z) + n\beta eft( \frac{|lpha|-1}{2}\right) P(z)\right| eq {}\frac{n}{2}eftbrace eft|lpha +\beta eft( \frac{|lpha|-1}{2}\right) \right|+eft|z+\betaeft(\frac{|lpha|-1}{2}\right)\right|\right\rbrace \maximits_{|z|=1}|P(z)| -\frac{n}{2}eftbrace eft|lpha + \betaeft(\frac{|lpha|-1}{2} \right) \right|-eft|z +\betaeft( \frac{|lpha|-1}{2}\right) \right|\right\rbrace \minimits_{|z|=1}|P(z)|, \end{align*} for any $\beta$ with $|\beta|\leq 1$ and $|z|=1$. In this paper we generalize the above inequality and our result also generalizes certain well known polynomial inequalities.
Classification : 30A10, 30C15, 30D15
Keywords: polynomial, Bernstein inequality, polar derivative 
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M. H. Gulzar; B. A. Zargar; Rubia Akhter. Some Inequalities for the Polar Derivative of a Polynomial. Kragujevac Journal of Mathematics, Tome 47 (2023) no. 4, p. 567 . http://geodesic.mathdoc.fr/item/KJM_2023_47_4_a6/