In this paper, we introduce a new class of complex functions incorporating an exponential factor and generate its Mandelbrot- and Julia-like sets using the \(K^*\) iteration with s-convexity. An escape criterion is established for the combination of the function and iteration scheme and is then implemented in the escape-time algorithm for the visualization of the fractal sets. The dynamical behavior of the resulting fractals is analyzed through graphical and numerical experiments using the average escape time and the non-escaping area index. The results demonstrate that the proposed iteration framework significantly influences the geometry and escape dynamics of the generated sets, leading to distinct structural variations governed by the iteration parameters.
