Inequalities for the rational functions with no Poles on the unit circle | ||
| International Journal of Nonlinear Analysis and Applications | ||
| مقاله 133، دوره 14، شماره 1، فروردین 2023، صفحه 1727-1735 اصل مقاله (355.95 K) | ||
| نوع مقاله: Research Paper | ||
| شناسه دیجیتال (DOI): 10.22075/ijnaa.2022.25770.3124 | ||
| نویسندگان | ||
| Uzma Mubeen Ahanger* ؛ Wali Mohammad Shah؛ Lubna Wali Shah | ||
| Department of Mathematics, Central University of Kashmir, Ganderbal-191201, Jammu and Kashmir, India | ||
| چکیده | ||
| Let $\mathcal{R}_{n}$ be the set of rational functions with prescribed poles. It is known that if $r \in \mathcal{R}_{n},$ such that $r(z)\neq 0$ in $ |z|<1,$ then \begin{align*} \sup_{|z|=1}|r^{'}(z)|\leq \frac{|\mathcal{B}^{'}(z)|}{2}\sup_{|z|=1}|r(z)| \end{align*} and in case $r(z)=0$ in $|z|\leq 1,$ then \begin{align*} \sup_{|z|=1}|r^{'}(z)|\geq \frac{|\mathcal{B}^{'}(z)|}{2}\sup_{|z|=1}|r(z)|, \end{align*} where $\mathcal{B}(z)$ is the Blashke product. The main aim of this paper is to relax the condition that all poles of $r(z)$ lie outside the unit circle and instead assume their location anywhere off the unit circle in the complex plane $\mathbb{C}. $ The results so obtained besides the above inequalities generalize some other well-known estimates for the derivative of rational functions $r \in \mathcal{R}_{n}$ with prescribed poles and restricted zeros. | ||
| کلیدواژهها | ||
| Inequalities؛ Polynomials؛ Rational functions؛ Off the unit circle؛ Poles؛ Zeros | ||
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