Get 20M+ Full-Text Papers For Less Than $1.50/day. Start a 14-Day Trial for You or Your Team.

Learn More →

A note on the avoidance criterion for normal functions

A note on the avoidance criterion for normal functions Abstract Let \(\varphi _{1},\varphi _{2},\varphi _{3}\) be three functions meromorphic in the unit disc \(\Delta \) and continuous on the closure of \(\Delta \) such that \(\varphi _{i} (z)\ne \varphi _{j}(z)\) on the unit circle \(|z|=1.\) Let \(\mathcal {F}\) be a family of meromorphic functions such that \(f \ne \varphi _{j}\) on \(\Delta \) for \(j=1,2,3 ~\text{ and }~f\in \mathcal {F}.\) Then there exists a constant M such that \((1-|z|^{2})f^{\#}(z)\le M \) for each \(z\in \Delta ~\text{ and }~f\in \mathcal {F}.\) In addition, this constant M depends only on the three functions \(\varphi _{1},\varphi _{2}~\text{ and }~\varphi _{3}.\) This generalizes the related theorems due to Lappan (In: Progress in analysis: proceedings of the 3rd international ISAAC congress in progress in analysis, vol 1, pp 221–228. World Scientific, 2003), and Xu and Qiu (C R Math 349:1159–1160, 2011), respectively. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Analysis and Mathematical Physics Springer Journals

A note on the avoidance criterion for normal functions

Yang, Liu
Analysis and Mathematical Physics , Volume 10 (3): 6 – Sep 1, 2020
Download PDF
6 pages

Loading next page...
 
/lp/springer-journals/a-note-on-the-avoidance-criterion-for-normal-functions-0tsJ2F3NSL

References (9)

Publisher
Springer Journals
Copyright
2020 Springer Nature Switzerland AG
ISSN
1664-2368
eISSN
1664-235X
DOI
10.1007/s13324-020-00379-y
Publisher site
See Article on Publisher Site

Abstract

Abstract Let \(\varphi _{1},\varphi _{2},\varphi _{3}\) be three functions meromorphic in the unit disc \(\Delta \) and continuous on the closure of \(\Delta \) such that \(\varphi _{i} (z)\ne \varphi _{j}(z)\) on the unit circle \(|z|=1.\) Let \(\mathcal {F}\) be a family of meromorphic functions such that \(f \ne \varphi _{j}\) on \(\Delta \) for \(j=1,2,3 ~\text{ and }~f\in \mathcal {F}.\) Then there exists a constant M such that \((1-|z|^{2})f^{\#}(z)\le M \) for each \(z\in \Delta ~\text{ and }~f\in \mathcal {F}.\) In addition, this constant M depends only on the three functions \(\varphi _{1},\varphi _{2}~\text{ and }~\varphi _{3}.\) This generalizes the related theorems due to Lappan (In: Progress in analysis: proceedings of the 3rd international ISAAC congress in progress in analysis, vol 1, pp 221–228. World Scientific, 2003), and Xu and Qiu (C R Math 349:1159–1160, 2011), respectively.

Journal

Analysis and Mathematical Physics – Springer Journals

Published: Sep 1, 2020

There are no references for this article.