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

In this article, we explore and analyze the different variants of Julia set patterns for the complex exponential function W ( z ) = αe z n + βz 2 + log γ t and complex sine function T ( z ) = sin( z n ) + βz 2 + log γ t , where n ≥ 2 , α, β ∈ C , γ ∈ C \ 0 , and t ∈ R , t ≥ 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.


Introduction
The fascinating field of fractal mathematics has attracted the interest of scientists, mathematicians, and artists, offering profound insights into the intricate balance of complexity and order within the natural world.Among the vast array of mathematical shapes and patterns, Julia sets have emerged as a focal point of study, displaying mesmerizing visual representations and intriguing mathematical relationships.Named after the French mathematician Gaston Julia, these sets lie at the core of the broader field of complex dynamics, constructed through iterations of complex functions that concentrate on specific values within the complex plane.The work of another prominent mathematician, P. Fatou [7], has extended the study of Julia sets by introducing the Fatou set as their complement within the domain, see e.g., [1,4,5,6,13,14,15,16,17].
One of the most fascinating aspects of Julia sets is their complex nature.The term "fractal" derived from the Latin language, meaning "split" or "break", accurately describes these self-similar patterns in complex graphics.Fractals, with their infinitely complex and similar patterns, find numerous real-life applications and are prevalent in nature, effectively describing phenomena such as leaf patterns, tree branches, lightning, clouds, rivers, and crystals.The importance of fractals extends to various fields, playing a vital role in surveying or investigating various natural or living structures, including microbial culture.Additionally, fractal theory is widely employed in cryptography, image compression, encryption, radar systems, computational architectural design, and engineering models, which underscores its wide and influential applications in [13].
In the existing literature, numerous studies have explored the application of explicit and implicit iteration schemes in constructing fractal sets.In [2], Ashish et al. utilized Noor iteration to construct Julia sets.Subsequently, Cho et al. [3] expanded upon the results of Ashish et al. by employing s-convex combination.Kumari et al. demonstrated the fractal patterns obtained through the iterative method introduced by Abbas and Nazir in [9].Recent studies include the utilization of Picard-Mann iteration with s-convexity [18] and Picard-Mann iteration [21] for constructing Julia sets.
In the majority of studies focusing on the Julia set, researchers commonly employ the n th degree polynomial in the form of z n + c.However, in a distinctive approach, Tanveer et al. [19,21] introduced a modification to the constant term in this function and a new fixed point iteration.Instead of c, they proposed using log(c t ), where t ∈ R and t ≥ 1.Moreover, they chose to utilize the Mann and Picard-Mann iterations and furnished a proof for the escape criterion applied in the escape-time algorithm that generated the Mandelbrot sets images in their study.
Motivated by the incorporation of the logarithmic function for the constant term c, our paper adopts a similar approach by replacing the constant c with log(c t ), where t ∈ R and t ≥ 1.Furthermore, instead of the Mann and Picard-Mann iterations, we employ the viscosity approximation-type iterative method with s-convexity, rigorously establishing escape criteria for the considered iterations.Our study encompasses the presentation and analysis of graphical examples.Inspired by Tanveer et al. [21], the present work investigates Julia sets of complex exponential function W (z) = αe z n + βz 2 + log γ t and complex sine function T (z) = sin(z n ) + βz 2 + log γ t , where n ≥ 2, α, β ∈ C, γ ∈ C\0, and t ∈ R, t ≥ 1, utilizing a viscosity approximation-type iterative method with s-convexity to develop the escape criterion.
The remainder of the paper is structured as follows: In Section 2, we provide fundamental definitions and results essential for accomplishing the objectives of this paper.Section 3 is dedicated to the investigation of escape criteria for the viscosity approximation-type iterative method with s-convexity, focusing on both the complex exponential function W (z) = αe z n + βz 2 + log γ t and complex sine function T (z) = sin(z n ) + βz 2 + log γ t .In Section 4, we present some graphical examples of Julia sets obtained using the proposed approach.These examples demonstrate the correlation between the size, color, and appearance of the fractal patterns of the generated Julia sets and the values of the parameters.Finally, in Section 5, we conclude our work.

Preliminaries
Definition 2.1 (Julia set [7]) Let p : C → C. The filled Julia set of p is denote by J p and is defined as where p k denotes the k times composition of the function p. Noticeably, it is a set of complex numbers for which the orbits do not converge to a point at infinity.The Julia set of p is the boundary of J p , that is, J p = ∂J p .
In the manuscript, let η = log(γ t ) γ . Consequently, we can express log(γ t ) = ηγ.Let p, W, T : C → C be complex-valued mappings such that p is a contraction mapping.In the complex plane, consider the sequence {z k } ∞ k=0 of iterates for any starting point z 0 ∈ C, with parameters µ, ν ∈ (0, 1), and k ≥ 0, is referred to as the viscosity approximation-type iterative method with s-convexity, and it is expressed as: and The viscosity approximation-type iterative method with s-convexity reduces to: • The viscosity approximation-type iterative method [21] when s = 1.
To generate fractals and escape limitations are the basic key to run the algorithms.Since it is well known that | sin(z n )| ≤ 1 for some z ∈ C, and the Maclaurin expansion for the sine function is where 0 < |ω| ≤ 1 except the values of z ∈ C for which |ω| = 0 and satisfying the bound

Escape criteria for the considered complex functions
In this section, we introduce the escape time algorithms via the viscosity approximation-type iterative method with s-convexity combination for novel complex function of the type As a result, we establish a novel threshold escape radii and leverage them to visualize some non-classical variants of classical fractals, as illustrated in the subsequent outcomes.

Escape criterion for
In this subsection, we prove the escape criteria for transcendental function via viscosity approximation-type iterative method with s-convexity.

Application of Fractals
In this section, we tailor two algorithms: one for the Julia set of W (z) = αe z n + βz 2 + log γ t and the other for T (z) = sin(z n ) + βz 2 + log γ t employing the viscosity approximation-type iterative method with s-convexity, where n ≥ 2, β ∈ C, γ ∈ C\0, and t ∈ R, t ≥ 1.We illustrate graphs of Julia sets at various input parameters, generating Julia sets through the viscosity approximation-type iterative method with s-convexity using Algorithm 1 and 2, and comparing the resultant images.Finally, we visualize Julia sets for various input parameters and different values of n.Throughout the paper, a maximum number of iterations k = 100 is consistently applied.In this example, we present two cases.In the first case, we take an integer value of t, i.e., t = 3, 6, 9 and in the second one a non-integer value, i.e., t = 0.5, 2.5, 6.5.Note that the last column in all the tables displays the image execution time (in short, IET) in seconds.Julia sets for W (z) = αe z n + βz 2 + logγ t via viscosity approximationtype iterative method with s-convexity are generated here with the following inputs: In Figure 2, we fixed the value of γ to 9 (purely real) and varying the value of t, in the first case, we take an integer value of t, i.e., t = 3, 6, 9 (Figure 2 (i)-(iii)), and in the second one a non-integer value, i.e., t = 0.5, 2.5, 6.5 (Figure 2 (iv)-(vi)).From the images, we see that the Julia set becomes larger with the increase of the value of t.Thus, the value of t has a great impact on the shape, size and color to the fractals, and the generated Julia sets looks like flowers and Rangoli or may be compared to glass painting.We can see some spiral structures but the spiral pattern arms and the size of lashes of bunches slightly increase with the increase the value of t. (i (iv) t = 0.5 (v) t = 2.5 (vi) t = 6.5In Figure 3, we fixed the value of γ to 6i (purely imaginary) and varying the value of t.From the images, we see that the Julia set becomes larger with the increase of the value of t.Thus, the value of t has a significant impact on the shape, size and color to the fractals, and the generated Julia sets looks like flowers and Rangoli or may be compared to glass painting.We can see some spiral structures but the spiral pattern arms and the size of lashes of bunches slightly increase with the increase the value of t.
(iv) t = 0.5 (v) t = 2.5 (vi) t = 6.5 (iv) t = 0.5 (v) t = 2.5 (vi) t = 6.5In Figure 4, we fixed the value of γ to 3 + 2i and varying the value of t.From the images, the value of t has a significant impact on the shape, size and color to the fractals, and the generated Julia sets looks like spiral galaxy, leaf of Rex Begonia and Rangoli or may be to glass painting.However, when we look closely, then we notice that, we can see some beautiful spiral structures and the spiral pattern arms slightly different with the increase the value of t.At first sight, the spiral arm in the lower in Figure 4(i, v), and the spiral arm in the upper in Figure 4(ii, vi), or the two spiral arms upper as well as lower in Figure 4(iii, iv) are slightly different from the other arms.
Table 4 : Fixed all the parameters values complex and varying t.
Stat., Optim.Inf.Comput.Vol. 12, September 2024  In Figure 6, we fixed all the parameters values and changed the values of µ and ν.From the images, the values of µ and ν have the significant changes on the shape, size and color to the fractals, and the generated Julia sets looks like leaf of Rex Begonia, spiral galaxy or may be compared to glass painting.We notice that, the generated Julia set in Figure 6(i, ii) are sightly same, the sets in Figure 6(v, vi) are sightly same, the set in Figure 6(iii) and Figure 6(iv) are different from the others for different vales of µ and ν.The image execution time is strictly increase with the increase the value of t which is given in the last column in Table 6.In Figure 7, we fixed all the parameters values and varying the value of s.From the images, value of s has a significant change on the shape, size and color to the fractals, and the generated Julia sets looks like spiral galaxy, leaf of Rex Begonia and Rangoli or may be compared to glass painting.Julia sets are slightly different with the increase the value of s.However, when we look closely, then we notice that, we can see some beautiful spiral structures and the spiral pattern arms slightly different in Figure 7(iv, v), t or the two spiral arms upper as well as lower in Figure 7(vi), the sets in Figure 7(i, ii) and Figure 7(ii) are different from the other.The image execution time consistently increases as the value of t rises, as shown in the last column of Table 7.In Figure 8, we fixed all the parameters values and varying the value of n.From the images, the value of n has a significant change on the shape, size and color to the fractals.The generated Julia sets have different patterns with the increase the value of n.The image execution time consistently increases as the value of n rises, as shown in the last column of Table 8.
In this example, we present two cases.In the first case, we take an integer value of t, i.e., t = 3, 6, 9 and in the second one a non-integer value, i.e., t = 0.5, 2.5, 6.5.Note that the last column in all the tables displays (iv) t = 0.5 (v) t = 2.5 (vi) t = 6.5 Table 11 : Fixed all the parameters values complex and varying t.
(iv) t = 0.5 (v) t = 2.5 (vi) t = 11.5 Figure 12.Julia sets for n = 2 with all the parameters values complex and varying t.
Stat., Optim.Inf.Comput.Vol. 12, September 2024    of the lashes gradually decreases from the center of the bunch, and the angle between every two bunches is π 3 .In Figures 9-15, all the figures look similar to each other but differ in their Julia points.Different parameter values enhance the symmetrical pattern's beauty.It is observed that • the parameters t, γ, µ, ν and s play a very important role in giving shape, size and color to the fractals.
• the convergence criteria derived for the fractals play a crucial role in enhancing their resolution and pixel richness.• all the fractals developed in this paper are very novel, aesthetic, and pleasing as the function T (z) and W (z) incorporate a special type of sine function combined with a logarithmic component.

Conclusion
We derived an escape criterion for generating fractals using the proposed iterative method for the complex exponential functions W (z) = αe z n + βz 2 + log γ t and T (z) = sin(z n ) + βz 2 + log γ t , where n ≥ 2, β ∈ C, γ ∈ C\0, and t ∈ R, t ≥ 1.The visualization of Julia sets is facilitated by implementing these results in Algorithms 1 and 2. Using MATLAB software, we generated compelling non-classical variants of the Julia fractals, which were subsequently discussed and evaluated for various parameter values.We observed that these parameters significantly impact not only the shape but also the symmetry of the generated sets.We believe that the results of this research will be valuable for those interested in creating aesthetically pleasing graphics and designer printing patterns.Additionally, the textile sector can benefit from these findings for designing and printing purposes.

Figure 1 .
Figure 1.Color map used in sketching the fractals.

Figure 5 .
Figure 5. Julia sets for n = 2 with all the parameters values complex and varying t.

Table 1 :
Fixed value of γ as purely real and varying t.

Table 2 :
Fixed value of γ as purely complex and varying t.

Table 3 :
Fixed value of γ as complex and varying t.

Table 5 :
Fixed value of t and varying the parameters µ and ν.

Table 6 :
Effect of change in the value of s.

Table 7 :
Effect of the value of n.

Table 12 :
Effect of change in the values of t, µ and ν.

Table 13 :
Effect of change in the values of t and s.