Growth estimate for rational functions with prescribed poles and restricted zeros

	Growth estimate for rational functions with prescribed poles and restricted zeros
International Journal of Nonlinear Analysis and Applications
مقاله 19، دوره 13، شماره 1، خرداد 2022، صفحه 247-252 اصل مقاله (338.89 K)
نوع مقاله: Research Paper
شناسه دیجیتال (DOI): 10.22075/ijnaa.2021.23465.2544
نویسندگان
Ishfaq Ahmad Dar^* ؛ Nisar Ahmad Rather؛ Mohd Shafi Wani
Department of Mathematics, University of Kashmir, Srinagar-190006, India
چکیده
Let $r(z)= f(z)/w(z)$ where $f(z)$ be a polynomial of degree at most $n$ and $w(z)= \prod_{j=1}^{n}(z-a_{j})$, $\|a_j\|> 1$ for $1\leq j \leq n.$ If the rational function $r(z)\neq 0$ in $\|z\|< k$, then for $k =1$, it is known that $$\left\|r(Rz)\right\|\leq \left(\frac{\left\|B(Rz)\right\|+1}{2}\right) \underset{\|z\|=1}\sup\|r(z)\|\,\,\, for \,\,\,\|z\|=1$$ where $ B(z)= \prod_{j=1}^{n}\left\{(1-\bar{a_{j}}z)/(z-a_{j})\right\}$. In this paper, we consider the case $k \geq 1$ and obtain certain results concerning the growth of the maximum modulus of the rational functions with prescribed poles and restricted zeros in the Chebyshev norm on the unit circle in the complex plane.
کلیدواژه‌ها
Rational functions؛ Polynomial Inequalities؛ Zeros
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