## On Zygmund-type inequalities concerning polar derivative of polynomials.(English)Zbl 1476.30026

Summary: 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.

### MSC:

 30C10 Polynomials and rational functions of one complex variable 30A10 Inequalities in the complex plane

### Keywords:

polynomials; polar derivative; $$L^p$$-inequalities
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### References:

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