Generalized escape criteria for fractals via convex viscosity approximation iterations

In this manuscript, we present the generation of Mandelbrot sets, Julia set fractals, and Biomorphs using a two-step viscosity approximation method applied to the complex function W ( z ) = z n + u z + r , where n ≥ 2 and u , r ∈ C . The two-step viscosity approximation method is employed to establish new escape criteria for Julia sets, Mandelbrot sets, and Biomorphs. Subsequently, the viscosity approximation process is extended by incorporating m -convexity, and the corresponding escape criteria are generalized for these fractals. Further, we introduce the viscosity approximation process with s -convexity and develop the associated escape criteria for the same fractal structures. Additionally, we provide a comparative visualization of Mandelbrot sets, Julia set fractals, and Biomorphs using the standard viscosity approximation process, as well as its m -convex and s -convex variants, applied to the same complex polynomial function. High-resolution fractal images are produced using MATLAB R2024a with 50 iterations and a figure resolution of 800, highlighting the differences in the resulting structures for identical parameter values.