General escape criteria for the generation of fractals in extended Jungck–Noor orbit

Abstract

The aesthetic patterns play a key role in the field of fractals. Due to self similarities in the nature of fractals, researchers used the fractals in many fields of sciences (i.e. in Mathematics, Computer Science, Physics, Image Encryption, Biology and Chemistry). The most studied fractals types are the Mandelbrot sets (MSs) and Julia sets (JSs). To generate fractals, escape criteria is required. In this work, a general escape criteria is proved via extended Jungck-Noor iteration with s-convexity. These results are used in algorithms to present the generation of fractals in extended Jungck-Noor orbit for general complex polynomial f(x)=∑i=0paixi with p≥2, where ai∈ℂ for i=0,1,2,…,p. The graphics of MSs and JSs are demonstrated in the examples. The variations in MSs and JSs for different values of involved parameters are also shown.