Escape criteria for one parameter family of complex functions \(\mathbf{f}_{\mathbf{k}}\left( \mathbf{z}\right)\mathbf{=}\mathbf{\text{kcsc}}\left( \mathbf{z} \right)\) via non-standard iterations

Document Type : Research Paper


1 Department of Mathematics and Computer Applications College of Science, Al-Nahrain University, Iraq

2 Department of Mathematics, College of Science, Baghdad University, Iraq

3 Al Ain University, Abu Dhabi, UAE


In this research we stated and proved the some escape criteria theorems of the one parameter family of the transcendental meromorphic-functions \(F\mathbf{=}\left\{ f_{k}\left( z \right) = k\ csc\left( z \right):k\mathbb{\in C\ }\text{and}\text{\ z}\mathbb{\in C} \right\}\). Furthermore, we used non-standard iterations: Mann, Ishikawa and Noor iterations in the complex plane. This research can be considered as an extension of [1].


[1] C. Beck, Physical meaning for Mandelbrot and Julia sets, Phys. D 125 (3–4) (1999) 171–182.
[2] P.J. Bentley, Methods for improving simulations of biological systems: Systemic computation and fractal proteins, J. Royal Soc. Interface 6 (2009) 451–466.
[3] W. D. Crowe, R. Hasson, P.J. Rippon and P.E.D. Strain-Clark, On the structure of the Mandelbar set, Nonlinearity 2(4) (1989) 541–553.
[4] S.V. Dhurandhar, V.C. Bhavsar, and U.G. Gujar, Analysis of z−plane fractal images from z → zα+c for α < 0., Comput. Graph. 17(1) (1993) 89–94.
[5] P. Dom´ınguez and N. Fagella, Residual Julia sets of rational and transcendental functions, In P.J. Rippon and G.M. Stallard, editors, Transcendental Dynamics and Complex Analysis, Cambridge University Press, 2010. pages 138–164.
[6] P. Fatou, Sur les ´equations fonctionnelles, Bull. Soc. Math. France 47 (1919) 161–271.
[7] G. Julia, M´emoiresurl’it´eration des fonctionsrationnelles, Journal de Math´ematiquesPuresetAppliqu´ees, 8 (1) 1918 47–246.
[8] L. Koss, Elliptic functions with disconnected Julia sets, Int. J. Bifurc. Chaos 26(6) (2016) 1650095.
[9] A. Lakhtakia, V.V. Varadan, R. Messier, and V.K. Varadan, On the symmetries of the Julia sets for the process z → zp + c, J.ournal of Phys. A Math. Gen. 20(11) (1987) 3533–3535.
[10] F. Peherstorfer and C. Stroh, Connectedness of Julia sets of rational functions, Comput. Meth. Function Theory 1(1) (2001) 61–79.
[11] H. Qi, M. Tanveer, W. Nazeer and Y. Chu, Fixed point results for fractel generation of complex polynomials involving sine function via non-standard iterations, IEEE 8 (2020) 154301–154317.
[12] M. Rani and V. Kumar, Superior Mandelbrot set, J. Korea Soc. Math. Education Ser. D 8(4) (2004) 279–291.
[13] K. Shirrif, Fractals from simple polynomial composite functions, Comput. Graph. 17(6) (1993) 701–703.
[14] P. Zahadat, D.J. Christensen, S. Katebi and K. Stoy, Sensor-coupled fractal gene regulatory networks for locomotion control of a modular snake robot, In A. Martinoli, F. Mondada, N. Correll, G. Mermoud, M. Egerstedt, M.A. Hsieh, L.E. Parker, and K. Støy, editors, Distributed Autonomous Robotic Systems, Springer Tracts in Advanced Robotics, pages 517–530. Springer Berlin Heidelberg, 2013.
[15] C. Zou, A.A. Shahid, A. Tassaddiq, A. Khan and M. Ahmd, Mandelbrot sets and Julia sets in Picard-Mann orbit, IEEE Access, 31 (2020) 64411–6442.
Volume 13, Issue 1
March 2022
Pages 1537-1543
  • Receive Date: 09 August 2021
  • Accept Date: 28 November 2021