Exploring the Complex Dynamics of Julia and Mandelbrot Sets for the Complex-valued Mapping sin(z<sup>k</sup>) + az + c Using Four-Step Iterative Scheme with s-Convexity

In this paper, we investigate unique variations of the Julia and Mandelbrot sets, establishing criteria for the escape of a function Tcz = sin(zk) + az + c for all z ∈ C, where k ∈ N \ {1} and a, c ∈ C with a̸ = 0 by utilizing a four-step iterative scheme extended with s-convexity, we analyze the dynamics of these fractals by formulating and implementing the algorithms. Our exploration focuses on understanding how distinct parameters impact these fractals’ color, dynamics, and overall visual characteristics. Our findings reveal that specific fractal patterns resemble exquisite natural objects like flowers and spiders. These fractal designs are also employed as visually striking artistic motifs on garments and textile materials. The captivating and complex properties of fractal patterns within dynamic systems make them an enticing and visually appealing area of research.