Zygmund’s Type Inequality to the Polar Derivative of a Polynomial

International Journal of Advanced Research in Mathematics ◽

10.18052/www.scipress.com/ijarm.5.52 ◽

2016 ◽

Vol 5 ◽

pp. 52-59

Author(s):

M.S. Pukhta

Keyword(s):

Type Inequality ◽

Polar Derivative ◽

Zygmund’S Inequality ◽

Derivative Of A Polynomial

In this paper we improve a result recently proved by Irshad et al. [On the Inequalities Concerning to the Polar Derivative of a Polynomial with Restricted Zeroes, Thai Journal of Mathematics, 2014 (Article in Press)] and also extend Zygmund's inequality to the polar derivative of a polynomial.

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ON ZYGMUND–TYPE INEQUALITIES CONCERNING POLAR DERIVATIVE OF POLYNOMIALS

Ural mathematical journal ◽

10.15826/umj.2021.1.007 ◽

2021 ◽

Vol 7 (1) ◽

pp. 87

Author(s):

Nisar Ahmad Rather ◽

Suhail Gulzar ◽

Aijaz Bhat

Keyword(s):

Unit Circle ◽

Type Inequality ◽

Polar Derivative ◽

Minimum Modulus ◽

Derivative Of A Polynomial ◽

Restricted Zeros ◽

Ordinary Derivative

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.

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