Role of $s$-convexity in the generation of fractals as Julia and Mandelbrot sets via three-step fixed point iteration

In this manuscript, we investigate the dynamics of higher-order polynomials with complex coefficients by employing the Jungck–Noor iteration scheme (one of the iterative methods) in conjunction with $s$-convexity, which controls the weighting of previous iterates versus current polynomial evaluations, tuning convergence speed, escape dynamics, and fractal density from compact, high-brightness patterns to intricate, detailed structures. This framework enables us to establish new escape criteria and to visualize nonclassical deviations of the celebrated Mandelbrot and Julia sets. The resulting fractal structures display intricate geometries that not only enrich the theoretical study of complex dynamics but also resemble patterns observed in natural systems. To highlight the novelty of our work, we provide both graphical and numerical illustrations that demonstrate how variations in polynomial parameters and iteration settings influence shape transformations, symmetry, color distributions, and computational complexity. A key observation is that each point in the Mandelbrot set encodes detailed information about the corresponding Julia fractal, reinforcing the deep interplay between the two families. Moreover, when real-valued parameters are considered in the polynomial map and iteration process, some fractals exhibit striking motifs suggestive of potential applications, for example, in pattern design within the textile industry. Our study also opens pathways for extending this framework to noise-perturbed systems and physical models, which will be explored in future work.