Nisar Ahmad Rather     (University of Kashmir, Hazratbal, Srinagar, Jammu and Kashmir 190006, India)
Suhail Gulzar     (Government College of Engineering and Technology, Safapora, Ganderbal, Jammu and Kashmir 193504, India)
Aijaz Bhat     (University of Kashmir, Hazratbal, Srinagar, Jammu and Kashmir 190006, India)


Let \(P(z)\) be a polynomial of degree \(n\), then concerning the estimate for maximum of \(|P'(z)|\) on the unit circle, it was proved by S. Bernstein that \(\| P'\|_{\infty}\leq n\| P\|_{\infty}\). Later, Zygmund obtained an \(L_p\)-norm extension of this inequality. The polar derivative \(D_{\alpha}[P](z)\) of \(P(z)\), with respect to a point \(\alpha \in \mathbb{C}\), generalizes the ordinary derivative in the sense that \(\lim_{\alpha\to\infty} D_{\alpha}[P](z)/{\alpha} = P'(z).\) Recently, for polynomials of the form \(P(z) = a_0 + \sum_{j=\mu}^n a_jz^j,\) \(1\leq\mu\leq n\) and having no zero in \(|z| < k\) where \(k > 1\), the following Zygmund-type inequality for polar derivative of \(P(z)\) was obtained: 
$$\|D_{\alpha}[P]\|_p\leq n \Big(\dfrac{|\alpha|+k^{\mu}}{\|k^{\mu}+z\|_p}\Big)\|P\|_p, \quad \text{where}\quad |\alpha|\geq1,\quad p>0.$$
In this paper, we obtained a refinement of this inequality by involving minimum modulus of \(|P(z)|\) on \(|z| = k\), which also includes improvements of some inequalities, for the derivative of a polynomial with restricted zeros as well.


\(L^{p}\)-inequalities, Polar derivative, Polynomials

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DOI: http://dx.doi.org/10.15826/umj.2021.1.007

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