Abstract
If is a polynomial of degree n having all its zeros in then Aziz [Inequalities for the derivative of a polynomial. Proc Am Math Soc. 1983;89(2):259–266] proved that Recently, Kumar [On the inequalities concerning polynomials. Complex Anal Oper Theory. 2020;14(6):1–11 (Article ID 65)] established a generalization of this inequality and proved under the same hypothesis for a polynomial , that In this paper, we sharpen the above inequalities and further extend the obtained results to the polar derivative of a polynomial. As a consequence, our results also sharpens considerably some results of Dewan and Upadhye [Inequalities for the polar derivative of a polynomial. J Ineq Pure Appl Math. 2008;9:1–9 (Article ID 119)].
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Acknowledgments
The authors would like to thank the anonymous referee for many useful comments. The first author was supported by the National Board for Higher Mathematics (R.P), Department of Atomic Energy, Government of India (No. 02011/19/2022/R&D-II/10212).
Disclosure statement
No potential conflict of interest was reported by the author(s).
Data availability statement
Data sharing not applicable to this article as no datasets were generated or analysed during the current study.