Julia sets of transcendental functions via a viscosity approximation-type iterative method with s-convexity

Abstract

In this article, we explore and analyze the different variants of Julia set patterns for the complex exponential function $W(z) =\alpha e^{z^n}+\beta z^2 + \log{\gamma^t}$ and complex sine function $T(z) =\sin({z^n})+\beta z^2 + \log{\gamma^t}$, where $n\geq 2, \alpha, \beta\in\mathbb{C}, \gamma\in\mathbb{C}\backslash {0}$, and $t\in\mathbb{R}, ~t\geq 1$ by employing a viscosity approximation-type iterative method with $s$-convexity. We utilize a viscosity approximation-type iterative method with s-convexity to derive an escape criterion for visualizing Julia sets. This is achieved by generalizing the existing algorithms, which led to visualization of beautiful fractals as Julia sets. Additionally, we present graphical illustrations of Julia sets to demonstrate their dependence on the iteration parameters. Our study concludes with an analysis of variations in the images and the influence of parameters on the color and appearance of the fractal patterns. Finally, we observe intriguing behaviors of Julia sets with fixed input parameters and varying values of $n.$