Malaysian Journal of Mathematical Sciences, September 2023, Vol. 17, No. 3
$L^{\gamma}$ Inequalities for the Polar Derivative of Polynomials
Singh, M. S., Reingachan, N., Devi, M. T., and Chanam, B.
Corresponding Email: barchand_2004@yahoo.co.in
Received date: 5 December 2022
Accepted date: 13 July 2023
Abstract:
In this paper, firstly, we obtain an inequality in $L^{\gamma}$ analogue concerning the polar derivative for a polynomial $p(\xi)=\displaystyle{\sum_{\nu=0}^{m}}c_{\nu}\xi^{\nu}$ of degree $m$ having no zero in $|\xi| < r$, $r\geq 1$ proved by Govil et al. \cite{GoRaSm}. Secondly, we also prove $L^{\gamma}$ version for the polar derivative of an ordinary inequality for a polynomial having all its zeros in $|\xi|\leq r$, $r\leq 1$ proved in that same paper. Our results generalize and improve some known inequalities.
Keywords: polynomials; polar derivative; $L^{\gamma}$ inequality; Bernstein's inequality
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