Tricorns and Multicorns in Noor Orbit With s-Convexity

In today’s world, complex patterns of the dynamical framework have astounding highlights of fractals and become a huge field of research because of their beauty and unpredictability of their structure. The purpose of this paper is to visualize anti-Julia sets, tricorns, and multicorns by means of the Noor iteration with s-convexity. Various patterns are displayed to investigate the geometry of antifractals for antipolynomial <inline-formula> <tex-math notation="LaTeX">$\overline {z}^{k+1}+c$ </tex-math></inline-formula> of complex polynomial <inline-formula> <tex-math notation="LaTeX">$z^{k+1}+c$ </tex-math></inline-formula>, for <inline-formula> <tex-math notation="LaTeX">$k\geq 1$ </tex-math></inline-formula> in Noor orbit with s-convexity.


I. INTRODUCTION
In 1918, French mathematician Julia [1] attained a Julia set by exploring the iteration procedure of complex mapping z → z 2 +c, here c is a complex number. The object Mandelbrot set presented by Mandelbrot in 1979 by utilizing c as a complex parameter in complex mapping z → z 2 + c [2]. In 1983, Crowe et al. [3] examined in formal closeness with Mandelbrot set and called it "Mandelbar sets" and exhibited its appearance bifurcations on circular segments rather at points. Milnor instituted the word "Tricorn" for the connectedness locus for antiholomorphic polynomials z 2 +c, which plays out a transitional job between quadratic and cubic polynomials [4]. The three-cornered nature, the fundamental characteristic of a tricorn, repetition with deviation at distinct scales, follow the similar kind of self-similarity as the Mandelbrot set.
Winters deciphered that boundary of the tricorn comprise of a smooth curve [5]. Lau and Schleicher [6] investigated the symmetries of tricorn and multicorns. Nakane and Schleicher [7] considered different qualities of tricorn and multicorns and extricated that the multicorns are generalized tricorns. They also examined that the Julia set of antipolynomial A c (z) = z k+1 + c for k ≥ 1, either connected or disconnected and if the Julia set of A c is connected then the arrangement The associate editor coordinating the review of this manuscript and approving it for publication was Jun Shen. of similar parameters c is known as the multicorn. Tricorn prints, for example, tricorn coffee cups, containers and tricorn T-shirts are being utilized for business reason.
These antifractals have been generalized in a few distinctive ways. One of these speculations is the use of different fixed point iterative procedures from the fixed point hypothesis. In the fixed point hypothesis there exist many estimated techniques for discovering fixed points of a given mapping, that depend on the utilization of various feedback iteration procedures. These procedures can be utilized in the generalization of antifractals. Rani [8], [9] studied and explored the dynamics of antiholomorphic complex polynomials z k+1 + c for k ≥ 1, by using Mann iteration which is a one-step iteration process. Chauhan et al. [10] introduced relative superior tricorns and relative superior multicorns via Ishikawa iteration which is a two-step iteration process. Antifractals have been studied extensively by Rani and Chugh [11], Kang et al. [12] and Partap et al. [13] for various fixed point iteration processes. The association of s-convex combination [14] and different iteration procedures examined in a few papers. Mishra et al. [15] got fixed point results in formation of tricorns and multicorns through Ishikawa iteration technique with s-convex combination. Nazeer et al. [16] handled the Jungck-Mann and Jungck-Ishikawa iteration procedures and Kang et al. [17] presented new fixed point results for formation of fractals with s-convexity in Jungck-Noor orbit. In [18] Noor iteration and s-convexity used to generate Mandelbrot sets and Julia sets. Recently, Kwun et al. [19] presented Mandelbrot sets, Julia sets and tricorns and multicorns via Jungck-CR iteration with s-convexity.
In this article we present and exhibit another class of tricorns, multicorns and ant-Julia sets by means of Noor iteration process with s-convex combination which is a further generalization of Noor iteration. The results of this paper are the extension of results presented in [18]. This paper is organized as: In section II we present some fundamental definitions. Section III contains the escape criterion for tricorn and multicorns by means of Noor iteration process with s-convexity. In section IV we visualize images of anti-Julia sets, tricorns and multicorns by utilizing proposed threestep iterative procedure with s-convex combination. Finally, section V contains some concluding comments.

II. PRELIMINARIES
The multicorn A * for A c is defined as the collection of all c ∈ C for which the orbit of 0 under the action of A c is bounded, i.e., It is noticed that at m = 2, multicorns reduce to tricorn. Definition 2: (Julia set [21]) Let f : C −→ C symbolize a polynomial of degree ≥ 2. Let F f be the set of points in C whose orbits do not converge to the point at infinity. That is, F f = {x ∈ C : {|f n (x)| , n varies from 0 to ∞} is bounded}. F f is called as filled Julia set of the polynomial f . The boundary points of F f is called as the points of Julia set of the polynomial f or simply the Julia set.
Definition 3: (Mandelbrot set [20]) The Mandelbrot set M consists of all parameters c for which the filled Julia set of Q c is connected, that is + c is defined as the collection of all c ∈ C for which the orbit of the point 0 is bounded, that is We determine the initial point 0 that is the only critical point of Q c . Definition 4: Let T : C → C is a mapping. Then Picard iteration process is defined by the following sequence {x n }: Definition 5: Let T : C → C be a mapping. The Mann iteration process [22] is defined by the following sequence {x n }: where η 1 n ∈ (0, 1]. Definition 6: Let T : C → C is a mapping. Then Ishikawa iteration process [23] is defined by the following sequence {x n }: where η 1 n ∈ (0, 1] and η 2 n ∈ [0, 1]. where η 1 n ∈ (0, 1] and η 2 n , η 3 n ∈ [0, 1]. The above sequence is called Noor orbit, that is a function of five tuples (T , z 0 , η 1 n , η 2 n , η 3 n ). It is noticed that Noor iteration diminishes to the: In the literature convex combination has been generalized in different manners. The s-convex combination is one of them.
Definition 8: (s-convex combination [14]) Let z 1 , z 2 , ..., z n ∈ C and s ∈ (0, 1]. The s-convex combination is defined in the following way: where λ k ≥ 0 for 1 ≤ k ≤ n and n k=1 λ k = 1. It is seen that for s = 1 the s-convex combination changed to the standard convex combination. We take z o = z ∈ C, η 1 n = η 1 , η 2 n = η 2 and η 3 n = η 3 then the Noor iteration scheme with s-convex combination can be write in the following way, where Q c (z n ) be a quadratic, cubic or (k + 1)th degree function.

III. ESCAPE CRITERION
There exists two distinct kinds of points in functional dynamics. First type of points exists in a stable set of infinity which escape the interval after a limited number of iterations the set of these points is called the escape set and second kind of points never escape the interval after any number of iterations the set of such points is known a prisoner set. These sets perform important role in choosing the escape criterion of polynomials under different fixed point iterative procedures. These escape criteria are important to create the antifractals which are at the core of different applications in PC illustrations. We establish a generalized escape criterion for antipolynomials Q c (z) = z k+1 + c where k ≥ 1, in modified Noor orbit.

IV. GRAPHICAL EXAMPLES
In this section tricorns and multicorns are presented for functions of the form z → z k+1 + c for k ≥ 1 via modified Noor iteration scheme. Also, anti-Julia sets are introduced for quadratic and cubic antipolynomials. To produce the images we applied the escape time algorithm with the general escape criterion implemented in the software Mathematica 9.0. Pseudocode of the tricorns and multicorns generation algorithm    is exhibited in Algorithm 1, while Algorithm 2 presents the pseudocode for the anti-Julia set generation algorithm.

V. CONCLUSIONS
In this paper escape criterion for antifractals has presented with respect to Noor orbit with s-convexity and visualized the pattern of symmetry among them. We attained quite different antifractals from those presnted in Noor orbit by Rani and Chugh [11]. In dynamics of antipolynomials z → z k+1 + c for k ≥ 1, we created a few examples of tricorns and multicorns for a similar estimation of k and various estimations of η 1 , η 2 , η 3 and s in Noor orbit with s-convexity. We observed that the quantity of branches joined to the main body of the tricorns and multicorns are k + 2, where k + 1 is the power of z for k ≥ 1. We likewise seen that when k + 1 is odd the symmetry of multicorn is around x-axis and y-axis and for k + 1 is even the symmetry is preserved just along x-axis. Many connected anti-Julia sets presented for quadratic and cubic functions. Attractive changes can be seen in antifractals generated in Noor orbit with s-convexity for different values of η 1 , η 2 , η 3 and s. We believe that consequences of this paper will be impress those who are interesting in generating aesthetic graphics automatically.