Generation of Julia and Mandelbrot sets using an s-convex extension of the Noor iteration for a generalized rational complex function

Abstract In this paper, we propose a method for generating Julia sets and Mandelbrot sets by a three-step iteration process in which we use s -convex combination and a complex function ζ c ( z ) = <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mi>a</mml:mi> <mml:msup> <mml:mrow> <mml:mi>z</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>p</mml:mi> </mml:mrow> </mml:msup> <mml:mo>+</mml:mo> <mml:mfrac> <mml:mrow> <mml:mi>b</mml:mi> </mml:mrow> <mml:mrow> <mml:msup> <mml:mrow> <mml:mi>z</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>q</mml:mi> </mml:mrow> </mml:msup> </mml:mrow> </mml:mfrac> <mml:mo>+</mml:mo> <mml:mi>log</mml:mi> <mml:msup> <mml:mrow> <mml:mi>c</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>t</mml:mi> </mml:mrow> </mml:msup> </mml:math> . The Julia sets are obtained as a collection of points determined through the escape criterion of the sequence established herein. It is the central concept in the study undertaken in this work. Mandelbrot sets are correspondingly determined in the parameter spaces. Visual patterns are generated for both the Julia and Mandelbrot sets for certain choices of parameters involved in the iteration. The changes in these patterns with the variations of the iteration parameters in certain ranges are qualitatively studied. Numerical experiments are performed using the MATLAB software. The average number of iterations (ANI) and the execution time in the course of the numerical experiments for the above two generation processes are noted and analyzed. In both cases, these two measures of the generation processes are observed to be in qualitative agreement.