Fractal Generation via CR Iteration Scheme With S-Convexity

Abstract

The visual beauty, self-similarity, and complexity of Mandelbrot sets and Julia sets have made an attractive field of research. One can find many generalizations of these sets in the literature. One such generalization is the use of results from fixed-point theory. The aim of this paper is to provide escape criterion and generate fractals (Julia sets and Mandelbrot sets) via CR iteration scheme with s-convexity. Many graphics of Mandelbrot sets and Julia sets of the proposed three-step iterative process with s-convexity are presented. We think that the results of this paper can inspire those who are interested in generating automatically aesthetic patterns.

Some new local fractional inequalities associated with generalized $(s,m)$-convex functions and applications

“…In the future the above theory and analysis can be extended to more complicated and applicable problems of convexity involving fractal domains. Finally, our consequences have a potential connection in fractal theory and machine learning [19,20]. This new concept will be opening new doors of investigation toward fractal differentiations and integrations in convexity, preinvexity, fractal image processing, and camouflage in the garments industry.…”

“…Individuals accept that the items in nature can be made or can be depicted by images, for example, lines, circles, conic areas, polygons, circles, and quadratic surfaces. The utilization of new scientific tools and concepts in this field of research will have an inordinate impression on enlightening image compression, where fractals and fractal-concerned techniques have demonstrated applications [18][19][20]. It is interesting that the authors [21,22] investigated the local fractional functions on fractal space deliberately, which comprises of local fractional calculus and the monotonicity of functions.…”

Exploration of Filled-In Julia Sets Arising from Centered Polygonal Lacunary Functions

“…Some trajectories can be found that orbit them many times before moving either to a stable orbit around the zero fixed point or diverging from the unit disk. For a very recent analysis of orbits and escape time algorithms for fractals associated with finite polynomials, the reader is directed to the work of Nazeer and Kang's group [28][29][30][31]. The points originating close to the center are the primary fixed points, whereas the points very close to the unit circle are the secondary fixed points.…”

“…Costin and Huang have recently published an interesting investigation of lacunary functions near the naural boundary and have shown self-similar and fractal-like character in these systems. In the last couple of years, Nazeer and Kang have collaborated on developing fractal generating and escape time algorithms for finite polynomials based on the concept of S-convexity [28][29][30][31].…”

Some new Simpson-type inequalities for generalized p-convex function on fractal sets with applications

“…The most significant utilization of fractals in software engineering is fractal picture compression. This sort of compression utilizes the way that this present reality is very well portrayed by fractal geometry [52,55,56]. Interestingly, the author of [53] investigated the local fractional functions on fractal space deliberately, which comprises of local fractional calculus and the monotonicity of functions.…”