Properties of Functions Formed Using the Sakaguchi and Gao-Zhou Concept

: This paper introduces a new class related to close-to-convex functions denoted by K k , N s . This class is based on combining the concepts of starlike functions with respect to N -ply symmetry points of the order α , introduced by Chand and Singh; and K ( k ) s , introduced by Wang, Gao, and Yuan, which are generalizations of the classes of functions introduced by Sakaguchi and Gao and Zhou, respectively. We investigate the class for several properties including coefﬁcient estimates, distortion and growth theorems, and the radius of convexity.


Introduction
The study of geometric functions is the study of the geometric properties of analytic functions. They have been studied extensively by many authors throughout the decades, pioneered by mathematicians Cauchy and Riemann. Example results of their work are the Cauchy Integral Formula ( [1], p. 2), which states that f (n) (z) = n! 2πi C ζ (ζ−z) n+1 dζ, where C is a rectifiable Jordan curve, f is analytic inside and on C, and z is inside C; and the Riemann Mapping Theorem ( [2], p. 10), which allows the mapping of any simply connected domain in C conformally onto the unit disc U = {z ∈ C : |z| < 1}.
Let A be the class of analytic functions in the unit disc U of the form f (z) = z + ∞ ∑ n=2 a n z n , The class P consists of functions with a positive real part of the form p(z) = 1 + ∞ ∑ n=1 c n z n , which satisfies the conditions {p(z)} > 0 and p(0) = 1. Duren [1] and Goodman [2] provide a more in depth study on the class S and its subclasses, as well as class P.
Sakaguchi [3] introduced a class of functions that are starlike with respect to symmetric points, denoted by S * s , with the following definition, Definition 4. Let f ∈ S. Then f ∈ S * s if, and only if, Chand and Singh [4] generalized S * s by introducing the class of starlike functions with respect to N-ply symmetry points of order α (0 ≤ α < 1) for N ∈ N, which they denoted using S * ,N s (α), with the following definition.
Definition 5. Let f ∈ S. Then f ∈ S * ,N s (α) for 0 ≤ α < 1 if, and only if, As can be seen, S * ,2 s (0) = S * ,2 s = S * s . They also introduced the class of close-to-convex functions with respect to N-ply symmetry points of order α (0 ≤ α < 1) for N ∈ N, denoted by K N s (α), with the following definition.
In another direction, Gao and Zhou [5] introduced a subclass of close-to-convex functions, denoted by K s , where the authors considered the products of functions in the denominators as opposed to the sums used by Chand and Singh [4]. The definition of K s is shown as follows.
Motivated by these papers, the authors considered combining these two approaches to generalize the classes further and developed a unified class of functions. The following definition introduces the class K k,N s , which unites K where k, N ∈ N, such that where for ε = e i 2π and for γ = e i 2π As it can be observed, K k,1 s = K (k) s and K 1,N s = K N s (0). This paper looks into properties for the class K k,N s . Part of this study is to observe if these properties are in conjunction with the previously obtained results.

Preliminary Results
In establishing the properties, some preliminary lemmas are needed to prove these results are stated and established as follows. Lemmas 1 and 2 are known results for functions in P [2]. Lemma 1. Let p ∈ P. Then |c n | ≤ 2 for each n ∈ N. The result is sharp for p(z) = 1+z 1−z .

Lemma 2.
Let p ∈ P and z = re iθ . Then Lemma 3 is cited from a work by Silverman in [14].
Moreover, suppose that |z| = r and a = 1+r 2 1−r 2 . Then where r γ is the unique root of the equation This result is sharp with equality in (7) attained at the point z = −r for and cos θ 0 is defined by the equation In Lemma 4, we modify Lemma 1 obtained in [6] by Wang et al.
. This makes it simple to deduce that Next, let Differentiating (10) logarithmically gives Thus, This completes the proof.

Results
This section presents the properties found for the class K k,N s . In the following, we present the distortion and growth theorems, coefficient estimates, and the radius of convexity. Before proceeding to the properties, the following theorem proves that G k,N given by (4) is starlike.

Remark 2.
As G k,N ∈ S * , the above theorem shows that the class K k,N s is a subclass of the close-to-convex functions K, K k,N s ⊂ K. Theorem 2. The distortion and growth bounds for f ∈ K k,N s are given as follows where |z| = r < 1 and m is the highest common factor of N and k. Equality is attained at the right-hand side for the function Proof. Suppose f ∈ K k,N s for k, N ∈ N, then there exists a function g ∈ S * ,N s ( k−1 k ), such that (3) holds.
where m is the highest common factor of N and k. Then, Thus, 2m Nk , and for |z| < r = 1, Combining (13) and (14) results in As there exists a p ∈ P such that z f (z) the following is obtained using (6), (15), and (16). From (17), the upper bound for | f (z)| is To prove the lower bound, z 0 ∈ U with |z 0 | = r (0 < r < 1) such that | f (z 0 )| = min{| f (z) : |z| = r}. It is sufficient to prove that that the left-hand side inequality holds for the point z 0 . Morever, | f (z)| ≥ | f (z 0 )| with |z| = r (0 ≤ r < 1). As f is univalent in the unit disc U, as f is a close-to-convex function, the original image of the line segment 0, f (z 0 ) is a piece of arc R in |z| ≤ r , then, in accordance to (17), This proves that f 1 ∈ K k,N s with respect to g 1 ∈ S * ,N s k−1 k and completes the proof.
Theorem 3. Let f (z) = z + ∞ ∑ n=2 a n z n ∈ K k,N s for k, N ∈ N. Then The inequalities are sharp with the extremal function Proof. Let f ∈ K k,N s . Then there exists g ∈ S * ,N s k−1 k , such that where k, N ∈ N. G k,N of the form (12) is a starlike function by Theorem 2 and there exists p ∈ P such that z f (z) G k,N (z) = p(z).
Comparing the coefficients on both sides yieds Using Lemma 1 and the triangle inequality, the following inequalities are obtained.
Using (19), (20) and Lemma 3 when δ = 0, . The next step is to find r such that which can be rearranged to for all 0 ≤ r < 1. This shows that F is a monotonically decreasing function within r ∈ [0, 1). As F(0) = 1 and F(1) = −4, this implies that there exists a root R c within (0, 1) such that F(R c ) = 0. Therefore, > 0 for 0 ≤ |z| = r ≤ R c , i.e., f is convex. This completes the proof.

Conclusions
All results obtained are consistent with prior results. In particular, when N = k = 1, the results obtained for the growth and distortion theorems (Theorem 2), coefficient estimates (Theorem 3), and radius of convexity (Theorem 4) are equal to those obtained by Libera [15] in 1964 for the class of close-to-convex functions. This suggests that our obtained results are accurate and sharp. Similarly, for the case N = k ∈ N, the results concur with those obtained by Chand and Singh [4] in 1979, which again illustrate that the results are accurate and that equality occurs for certain extremal functions. However, new results for Theorems 1 and 3 can be found by considering a different function From a different perspective, the idea used in introducing the new class demonstrates that it is feasible to merge multiple approahces, generalizing the classes and results. Thus, it is foreseeable to utilise subordination principles in merging multiple approaches to expand other classes (such as class K s p (φ) by Kant [10] and class K