The Julia and Mandelbrot sets for the function [formula omitted] exhibit Mann and Picard–Mann orbits along with [formula omitted]-convexity

Abstract

This research paper introduces a novel approach to visualize Julia and Mandelbrot sets by employing iterative techniques, which play a crucial role in creating fractals. The primary focus is on complex functions of the form F(z)=zp−qz2+rz+sincw for all z∈ℂ, where p∈N∖{1}, q∈ℂ, r,c∈ℂ∖{0} and w∈[1,∞). The Mann and Picard–Mann iteration schemes with s-convexity are utilized throughout the study. Innovative escape criteria are developed to generate Julia and Mandelbrot sets using these iterative methods. These criteria serve as guidelines for determining when the iterative process should terminate, leading to the creation of captivating fractal patterns. The research investigates the impact of parameter variations within the iteration schemes on the resulting fractal’s shape, size, and color. By manipulating these parameters, a wide range of captivating fractal patterns can be generated and visualized, encompassing various aesthetic possibilities. Additionally, we discuss the numerical examples related to Julia and Mandelbrot sets generated through the proposed iteration. We also delve into discussions concerning execution time and the average number of iterations.