Region of variablity for Bazilevic functions

: Let H be the family of analytic functions deﬁned in an open unit disk U = { z : | z | < 1 } and


Introduction and definitions
For a fixed point ξ in U, the set of values of log f (ξ) ξ as f ranging on the family of injective or univalent mappings is always a closed disk. This useful and important fact was proved by Grunsky [6]. The region of variability for various subfamilies of univalent functions became a fascinating area of the current research. Several authors have studied about such regions for certain subfamilies of analytic functions. These regions for the functions of bounded derivative were discussed by Yanagihara in 2005 as seen in [21]. Functions in such a subfamily satisfy the conditions | f (z)| ≤ 1 and Re( f (z)) > 0. Later on, Ponnusamy discussed the region of variability for the Kaplan family of functions K as described in [9]. Moreover, Yanagihara found the range of values for a subfamily of convex function, as seen in [23]. Ponnusamy discussed these problems for subfamilies of S * and K, as found in [10]. In 2008, see [15], Ponnusamy again considered these aspects for the spirallike functions. Vasudevarao discussed similar results when f is ranging over the functions with positive real parts, for detail see [11].
Most of the authors have introduced new subfamilies and studied the region of variability for these subfamilies of analytic and univalent functions. In 2010, Chen and Aiwu discussed these problems for functions with bounded Mocanu variations as seen in [2]. In 2011, Ponnusamy et al. [12] investigated these regions for the families of exponentially convex functions.
More useful and interesting results on these regions are also discussed by many authors. In 2014, Sunil Varma et al. [18], also worked out these issues for the related subfamilies along with Bappaditya who utilized the idea of subordination in his work as found in [1]. Also some related findings are seen in [10,14,16,19,20].
Let H be the family of analytic functions defined in U = {z : |z| < 1} the open unit disk with the center at origin O included in the z-plane and A ⊂ H. Then any function f ∈ A takes the form: f (z) = z + ∞ n=2 a n z n , z ∈ U. (1.1) From (1.1), we note that f (0) = f (0) − 1 = 0. The subfamily of univalent or injective mappings is represented by S and obviously S ⊂ A. A holomorphic or analytic mapping is called univalent as it never takes the same value twice. For example, in the unit disk U, f (z) = 1 + 2z + z 2 ∈ S. Formally, we have: On the other hand, for a mapping f ∈ S in U, from z 1 z 2 we have f (z 1 ) f (z 2 ). The Koebe function k (z) as defined by and the mapping M (z) = ηz + β νz + µ , η, ν, β, µ ∈ C : νβ − ηµ 0, are univalent. The family S is preserved under basic transformations, for reference, see [5]. Furthermore, from the condition f (z 0 ) 0, we have local univalence of f at z 0 . Let P be the family of holomorphic mappings p : p (U) is the right half ω-plane having series form Obviously p (0) = 1 and Re p(z) > 0. This and other related families have a significant role in the recent development of the subject. Many subfamilies of S are connected with the class P. For p ∈ P, it may not be necessary that p ∈ S. For example p (z) = 1 + z j ∈ P, but p(z) S for j ≥ 2. Obviously, the mapping L 0 (z) so that We take B as a family of analytic function ω : U −→ U with |ω(z)| < 1 and ω(0) = 0. A function f is subordinate to a function F and we write f ≺ F, if there exists ω ∈ B such that f (z) = F(ω(z)). Particularly, if F ∈ S, then f ≺ F can be equivalently reformulated as f (0) = F(0) and f (U) ⊂ F(U).
For A ∈ C, B ∈ [−1, 0), the function p ∈ P γ [A; B] can be written as: where ω ∈ B. Moreover, we can defined the family P γ [ξ, A; B] of analytic functions as: The family S * contains starlike functions f such that f (U) is starlike about O in the w-plane. This family has been extensively studied in the literature, as seen in [5].
Spacek [17] extended the family S * by using the logarithmic spirals besides the line segments. Let γ ∈ − π 2 , π 2 . The curve ϑ γ : R → C : ϑ γ (t) = te iγ , t ∈ R and its rotation e iθ ϑ γ (t), θ ∈ R are called γ-spirals. A domain D ⊂ C is known as γ-spirallike about the origin, if the spiral has initial point at the origin and terminal of the spiral is any other point of D. A function f ∈ A is spirallike, if f (U) is spirallike about the origin. The family of spirallike functions represented by S γ is defined by For details, we refer [5]. A normalized analytic or holomorphic function f ∈ C γ iff z f ∈ S * γ . For details see [16]. For β with Re β > 0, let B(ψ, β) be the family of functions f so that These Bazilevic functions are obviously univalent in U. Let B γ (ψ, β) be the family of holomorphic functions f in U such that These functions are called spirallike Bazilevic functions. Let A ∈ C, B ∈ [−1, 0), β > 0 and γ ∈ − π 2 , π 2 . Then f ∈ B γ ψ, ξ, β, A; B if e iγ z f (z) where ψ is a starlike function. In view of Herglotz form of Janowski functions, we write f ∈ B γ ψ, ξ, β, A; B as which can further take the form Integrating on both sides to have then for ω ∈ B, we see that We can also write or we see that which on simplifications yields . (1.8) In view of (1.6), on differentiating (1.7), we note that which on simplifications proves that Then by using classical Schwarz lemma [5] and a related result as seen in [4], we note that |ω (0)| ≤ 1. So we obtain Again on differentiating (1.9), we can write Also we see that (1.10) Now, we let This shows that Using Schwarz lemma for |ξ| < 1, we write |g(z)| < |z| and |g (0)| < 1. For equality, we see that g(z) = e iε z, ε ∈ R. Now |g (0)| < 1 shows that there is l ∈ U : g (0) = l. Thus by using (1.10) and (1.11), we find that It follows from (1.5) that for each fixed z 0 ∈ U, the region of variability V γ [ψ, z 0 , ξ, β, A; B] is a set defined as: when f ranges over the family B γ ψ, ξ, β, A; B .
The region of variability problems provide accurate information about the family S, than theorems about bounds on functions, their derivatives and rotation theorems. It typically refers to a range or interval within which a certain variable can vary or fluctuate. It is commonly used in research to describe the extent variation of region in a particular set. It is important to note that the region of variability can vary depending on the context and the specific variable being analyzed. Different methods and techniques can be used to determine and characterize the region of variability for different types of sets.
Here, we find the region of variability , where f ranges over the family B γ ψ, ξ, β, A; B is defined above by (1.7) and where q is described as a part of (1.5). As special cases, region of variability V γ [ψ, z 0 , ξ, β, A; B] can also described for different choices of parameters. As described above, Mohsan et al. [16] found V γ [ψ, z 0 , ξ, β, A; B] for a certain related subfamily of holomorphic functions. Ponnusamy et al. [8,9] studied these regions for f ∈ S. Yanagihara [22] determined such region for the family of convex functions. For work on such regions, see [10, 11, 13-16, 19, 21] and others. For any p ∈ N, we let S * In the case that p = 1, f is starlike univalent. We refer to [21,22] for the Lemma 1.2 given subsequently.

Main results
We initiate our investigations by studying certain properties of the family V γ [ψ, z 0 , ξ, β, A; B] such as compactness and convexity.

Concluding remarks
The region of variability problems are more useful for the family S, than the related classical theorems about this family S. In this study, we discussed the region of variability V γ [ψ, z 0 , ξ, β, A; B] for f (z 0 ), where f ranges over the family B γ ψ, ξ, β, A; B .

Use of AI tools declaration
The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.