Applications of First-Order Differential Subordination for Subfamilies of Analytic Functions Related to Symmetric Image Domains

: This paper presents a geometric approach to the problems in differential subordination theory. The necessary conditions for a function to be in various subfamilies of the class of starlike functions and the class of Carathéodory functions are studied, respectively. Further, several consequences of the ﬁndings are derived.


Introduction
Let C be the complex plane and E = {z : z ∈ C and |z| < 1} be the open unit disk.Let A represent the collection of all analytic functions, u, defined on E and fulfill the criteria u(0) = 0 and u (0) − 1 = 0. Thus, each function, u, in class A has the following Taylor series expansion: The various subclasses of A have been studied intensively, for instance, Ref. [1].Further, let S denote a subfamily of A, whose members are univalent in unit disk E. Let S * (α) and C(α) be the subfamilies of A for 0 ≤ α < 1, where S * (α) represents starlike functions of order α, and C(α) represents convex functions of order α.Analytically, these families are represented by S * (α) = u ∈ A : Re zu (z) u(z) > α , and In particular, if α = 0, then we can observe that S * (0) = S * and C(0) = C are well-known families of starlike functions and convex functions, respectively.Moreover, for two functions, u 1 , u 2 ∈ A, the expression u 1 ≺ u 2 denotes that the function u 1 is subordinate to the function u 2 if there exists an analytic function, µ, with the following properties: |µ(z)| ≤ |z| and µ(0) = 0 such that u 1 (z) = u 2 (µ(z)) ∀z ∈ E.
In addition, if u 2 ∈ S, then the aforementioned conditions can be expressed as follows: u 1 ≺ u 2 if and only if u 1 (0) = u 2 (0) and u 1 (E) ⊂ u 2 (E).

Considering the function φ(
we get the family S * nep = S * 1 + z − 1 3 z 3 , which was introduced and investigated recently by Wani and Swaminathan [7].The image of E under the function φ(z) = 1 + z − 1 3 z 3 is bounded by a nephroid-shaped region.5.For φ(z) = e z , the class S * e = S * (e z ) has been defined and studied by Mendiratta [8]. 6. Taking φ(z) = z + √ 1 + z 2 , we then get the family S * cres = S * z + √ 1 + z 2 , which maps E to a crescent-shaped region and was given by Raina et al. [9].
7. The function φ(z) = 1 + sinh −1 z gives the following class introduced by Kumar and Arora [10]: S * ρ = S * 1 + sinh −1 z .The natural extensions of differential inequalities on the real line into the complex plane are known as differential subordinations.Derivatives are an essential tool for understanding the properties of real-valued functions.Differential implications can be found in the complex plane when a function is described using differential subordinations.For example, Noshiro and Warschawski provided the univalency criteria for the analytical function theorem, which showed such differential implications.The range of the combination of the function's derivatives is frequently used to determine the properties of a function.
Let h be an analytic function defined on E, with h(0) = 1.Recently, Ali et al. [11] have investigated some differential subordination results.More specifically, they studied the following differential subordinations for some particular ranges of α.
In this paper, we consider the following two subfamilies of analytic functions.
S * car = u ∈ A : and where the family defined in (3) was introduced by Kumar and Kamaljeet [17], and the family defined in (4) was introduced by Gandhi [18].The lemma below underlies our considerations in the following sections.
Lemma 1. [19] For the univalent function q : E → C and the analytic functions λ and v in q(E) ⊆ E with λ(z) = 0 for z ∈ q(E), define then h ≺ q, and q is the best dominant.

Subordination Results for the Class S *
car Theorem 1.Let h be an analytic function with h(0) = 1 in the unit disc E and satisfy Then, we have the following. .

Proof. Consider the analytic function
which is a solution of the differential subordination equation Let us take z ∈ E, q(z) = a β (z), ν(z) = 1, and Since the function φ car (z) maps E into a starlike region (with respect to 1), the function h is starlike.Further, h satisfies Re zh (z) Θ(z) > 0. As an application to Lemma 1, we possess the following property: Each subordination of Theorem 1, is similar to 1). ( This yields the necessary condition for which h(z) ≺ ω(z), z ∈ E. Looking at the geometry of each of these functions ω(z), it is noticed that the condition is also sufficient.
and these inequalities can be reduced to = β 2 .We note that β 1 − β 2 < 0, and hence the following subordination holds.

6.
u ∈ S * ρ , for β ≥ Theorem 2. Let h be analytic with h(0) = 1 in unit disc E and assume that Then, we have the following. .

Proof. Consider the analytic function b
Then, b β is a solution of the differential equation If we take z ∈ E, q(z) = b β (z), ν(z) = 1, and λ(z) = β z in Lemma 1, then the function Since the function φ car (z) maps E into a starlike region (w.r.to 1), the function h is starlike.Further, h satisfies Re > 0. Applying this to Lemma 1, we possess that Each subordination of Theorem 1 is similar to for each subordinate function in the theorem, which is valid if b β (z) ≺ ω(z), z ∈ E. Here, we use the same technique as in Theorem 1, omitting the rest of the proof.

Theorem 3.
Let h be an analytic function with h(0) = 1 in unit disc E and satisfy that Then, the following results. .
Proof.Consider the function c β : E → C, defined by , which is the solution of the differential equation: In Lemma 1, let z ∈ E, q(z) = c β (z), ν(z) = 1, and λ(z) = β z 2 .Then, the function Since the function φ car (z) maps E into a starlike region (w.r.to 1), the function h is starlike.Further, h satisfies Re(zh (z)/Θ(z)) > 0. Therefore, from Lemma 1, we possess that Each subordination of Theorem 2 is similar to for each subordinate function in the theorem, which is valid if s β (z) ≺ ω(z), z ∈ E. Here, we use the same technique as we used in Theorem 1, so we omit the rest of the proof.
At the end of Section 2, as a geometric approach to the problems in differential subordination theory, the following figures in Figure 1 graphically represent the results in the section.

3L
Theorem 4. Let h be an analytic function with h(0) = 1 in unit disc E and satisfy that Then, we have the following.
It is easy to verify that the analytic function t β : E → C, defined by is the solution of Equation (6).
Proof.Consider the analytic function s β : E → C, defined by Then, s β is a solution of the differential equation: Let z ∈ E, q(z) = s β (z), ν(z) = 1, and λ(z) = β z .Applying for Lemma 1, the function Θ : E → C is given by Θ(z) = zs β (z)λ s β (z) = φ car (z) − 1, and so h(z) = 1 + Θ(z) = φ car (z).Since the function φ car (z) maps E into a starlike region (w.r.to 1), the function h is starlike.Further, h satisfies Re(zh (z)/Θ(z)) > 0. Applying Lemma 1, we possess that Each subordination of Theorem 1 is similar to for each subordinate function in the theorem, which is valid if s β (z) ≺ ω(z), z ∈ E. Here, we use the same technique as in Theorem 1, omitting the rest of the proof.
Corollary 5. Let u ∈ A that satisfies the following subordination.
Proof.Consider the function d β : E → C, defined by , which is the solution of the following differential equation.

Conclusions
In this article, we have studied the first-order differential subordination for two symmetric image domains, namely the cardioid domain and the domain bounded by three leaf functions.Further, we examined some graphical interpretations of these results.Moreover, this concept can be extended to meromorphic, multivalent, and quantum calculus functions.

Figure 1 .
Figure 1.Graphical Representation of Results in Section 2.

Figure 2 .
Figure 2. Graphical Representation of Results in Section 3.
Since the function φ car (z) maps E into a starlike region (w.r.to 1), the function h is starlike.Further, h satisfies Re(zh (z)/Θ(z)) > 0. It follows from Lemma 1 that 1 Let h be an analytic function with h(0) = 1 in open unit disc E and satisfy that Each subordination of Theorem 2 is similar toh(z) ≺ ω(z),for each subordinate function in the theorem, which is valid if d β (z) ≺ ω(z), z ∈ E. Here, we use the same technique as in Theorem 1, omitting the rest of the proof.Let u ∈ A that satisfies the following subordination.