Abstract
In this paper, we investigate some properties of quite fast escaping set Q(f) and \(Q_{\epsilon} (f)\), a subset of Q(f), where f is a transcendental entire function. We first define level of \(Q_{\epsilon} (f)\), based on iterations of \(\mu _\epsilon (r,f)={M(r,f)^\epsilon }\) over r. Then, we show some relations between levels of \(Q_{\epsilon} (f)\) and observe that every component of \(Q_{\epsilon} (f)\) is unbounded. Finally, we give condition under which \(Q_{\epsilon ,R}(f)\) (0th level of \(Q_{\epsilon} (f)\)), \(Q_{\epsilon} (f)\), Q(f) and I(f) form a spider’s web.
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Acknowledgements
Authors would like to thank Prof. G. M. Stallard for her comments regarding example of function with I(f) as spider’s web, but Q(f) is not. The third author sincerely acknowledges the financial support rendered by the National Board of Higher Mathematics, Department of Atomic Energy, Government of India, sponsored project with Grant No.02011/17/2022/NBHM(R.P)/R &D II/9661 dated: 22.07.2022. Also, the authors are very much thankful to the referees for their valuable comments towards the improvement of the paper.
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Majee, S., Chatterjee, S. & Chakraborty, G. On the level of \(Q_{\epsilon} (f)\) in quite fast escaping set and spider’s web. Annali di Matematica 202, 1335–1342 (2023). https://doi.org/10.1007/s10231-022-01282-y
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DOI: https://doi.org/10.1007/s10231-022-01282-y