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/