Results for Analytic Function Associated with Briot–Bouquet Differential Subordinations and Linear Fractional Integral Operators

: In this paper, we present a new class of linear fractional differential operators that are based on classical Gaussian hypergeometric functions. Then, we utilize the new operators and the concept of differential subordination to construct a convex set of analytic functions. Moreover, through an examination of a certain operator, we establish several notable results related to differential subor-dination. In addition, we derive inclusion relation results by employing Briot–Bouquet differential subordinations. We also introduce a perspective study for developing subordination results using Gaussian hypergeometric functions and provide certain properties for further research in complex dynamical systems.


Introduction
The geometric function theory is one of the mathematical science domains where the theory of differential and integral operators has been used.In [1], Salagean covered differential operators and devised a few enlarged classes in the open unit disk of normalized analytic functions.Since then, a number of scholars have used differential and integral operators to justify the geometric characteristics of the analytic functions [2].In the development of several classes of univalent functions, several fractional integral operators have been used recently [3].
Let D be the unite disc D = {v : v ∈ C, |v| < 1} normalized by f ′ (0) = 1 and f (0) = 0 and A = { f (v) : v ∈ D}.Then, every function f ∈ A has a series expansion as In A, the univalent function class is represented by S, the starlike function class is represented by S * , and the convex function class is represented by K [4].The authors of [5][6][7] explore the class C(γ) of convex functions of complex orders γ (γ ∈ C − {0}) and analyze many extensions of the class of univalent functions.
In [8], Mocanu and Miller introduce the concept of differential subordination, which was later applied to many classes of analytic functions.If f 1 , f 2 ∈ A and there is a Schwartz function w, with w(0) = 0 and |w| < 1, such that f 1 (v) = f 2 (w(v)), for v ∈ D, then the function f 1 is subordinate to the function f 2 , expressed as The differential and integral operators have made major contributions as a result of such development in the differential subordination principle; see [9][10][11] for further details.
Assume h ∈ S, ψ : C 3 × D → C and p is an analytic function on D that satisfies the subordination ψ(p, vp ′ , z 2 p ′′ ; v) ≺ h(v), v ∈ D. ( If p ≺ q for every p such that the connection (2) holds, then the univalent function q is said to be a dominant to the solution of the differential solution (2).If q ≺ p is satisfied by any dominant p of (2), then that dominant p is the best dominant.It is evident that on the open union disk D, the best dominant is unique [8].By employing differential subordination, the authors in [12] introduce a subclass of univalent functions and deduce numerous features of these functions.Oros uses differential subordination theory in [13] to study the Gaussian hypergeometric function.
Specific problems are connected to the differential subordination (2) and also to the differential equations associated with the subordination presented in (2).It was used in a number of physics and mathematics branches.For instance, R. W. Ibrahim uses the Brin-Bouquet differential subordination in [14] to study the class of analytic functions.By using the appropriate Schwarz function, we arrive at the Briot-Bouquet differential equation.The Briot-Bouquet differential equation is used by the authors in [15] to solve the holomorphic dynamical system Rong in [16] studied holomorphic dynamical systems by using Briot-Bouquet differential equation.The authors in [17] used the Briot-Bouquet differential equation to obtain the solution of the equation of nano-shells.In this situation, the fields of transposition of the nano-shell possess a dynamic system as follows where t is in any interval, and θ is the angle between z and w and their conjugates.
A special type of such problems is known as the Briot-Bouquet differential subordination.The Briot-Bouquet differential subordination is very useful in discussing various original results; see the following lemma.
and h(v) is the best dominant of (3) Lemma 2 ([8,18]).Assume γ, β ∈ C and h is a function which is convex on D, where Re{βh(v) + γ} > 0. Suppose that p is an analytic mapping on D, with h(0) = p(0), and If the differential equation has a univalent solution g(v), then The function g(v) is the best dominant.
Remark 1 ([8,18]).In view of Lemma 2, we can easily show that if In [19], Miller and colleagues examine analytical functions through the lens of differential subordination theory, yielding novel insights into the hypergeometric function.The Gaussian hypergeometric function and its use in differential subordination-based analytic function theory have been highlighted in previous studies [20 -22].More recently, refs.[23][24][25] have derived a number of conclusions about the differential subordination for the classical Gaussian hypergeometric function.In [26], authors explore many applications and describe innovative fractional integral operators using the Gaussian hypergeometric function.
Let a, b and c be complex numbers with c / ∈ {−1, −2, −3, ...}.Then, the classical Gaussian hypergeometric function where (α) k represents a Pochhammer symbol that is defined in terms of the Gamma function as The classical Gaussian hypergeometric function 2 F 1 (a, b, c; z) is an analytic function on D. If a or b is a negative integer, then the classical Gaussian hypergeometric function reduces to a polynomial.
In this study, we use classical Gaussian hypergeometric functions to construct the integral operator and introduce the linear differential operator D µ,η λ .As a study of convex function subclasses, we examine this operator.Next, we establish analytic functions by following the Briot-Bouquet differential subordination.
Proof.Let f 1 and f 2 be arbitrary functions in S δ (λ, µ, η) such that Then, it suffices to establish that the function This implies that Therefore, we obtain Thus, the result is obtained.
If f ∈ S δ (λ, µ, η) and then, the subordination vL and this result is sharp.
Proof.By following Equation ( 7) and employing simple computations, we derive .
Computing (twice) the logarithmic derivative of the above relation reveals Now, define the following notation Therefore, the subordination (8) can be given as From Lemma 2, we get Thus, the differential subordination ( 8) is established and h is the best dominant.This ends the proof of our result. where Then the differential equation is a formal solution to (6).
Proof.Consider the function g(z) in ( 13), we get By computing the logarithmic derivative, we derive The relations ( 11) and ( 13) becomes Hence, from ( 14) and ( 15), we obtain Now, from the Equation ( 12), we have Computing (twice) the logarithmic derivative of the above equation reveals Thus, by aid of ( 16) and ( 17), the relation ( 6) is established.The proof of Proposition 1 is completed.
Consequently, utilizing Theorem 2, we derive the following result.
If f ∈ S δ (λ, µ, η) and implies that and this result is sharp. where Then, the differential equation is a formal solution of (18).
Proof.By considering the function g(z) in ( 21), we get By computing the logarithmic derivative, we derive Therefore, the relations (20) and ( 21) become Hence, by utilizing ( 22) and ( 23), we obtain Now, by using the equation included in (20), we have Once again, computing twice the logarithmic derivative of the above equation reveals Thus, taking into account ( 24) and ( 25), the relation ( 18) follows.The proof of Proposition 2 is completed.
In the next result, we derive the fascinating conclusion for the class S δ (λ, µ, η). where Proof.By virtue of Theorem 2, we reach the following differential subordination where p is given by (10).Also, by applying Theorem 2 and Remark 1, we derive

Now, by using Proposition 1, we have
.
By using Equation ( 11), we obtain Equation ( 13) hence implies that Therefore, we derive In view of Equation ( 13) and the fact that g is convex, g(D) is symmetric with respect to the real axis.Therefore, we derive This ends the proof.
Following the proof of Theorem 5, we then prove an inclusion for the class S δ (λ, µ, η) by utilizing Proposition 2 and Theorem 3.
Then, we have the following inclusion where If the mapping f ∈ A satisfies the following differential subordination The result is sharp.
. By computing the logarithmic derivative, it can be read as The relation (27) becomes By applying Lemma 2 and Remark 1, we establish that p(v) ≺ g(v) ≺ h(v).Thus, the differential subordination in (30) is established.The proof is finished.
and this result is sharp.
, v ∈ D. By computing the logarithmic derivative, we get The relation (29), for z ∈ D, becomes By employing Lemma 2 and Remark 1, we infer that p ≺ g ≺ h.Thus, the differential subordination in (28) is established.The proof of Theorem 7 is completed.
Similarly, from Theorems 6 and 7, we establish the following results.
then we have The result is sharp.
The result is sharp.

Application
As an application of this theory, we introduce the following results.
then the following differential subordination implies that The result is sharp.
Proof.First, we define where p ∈ P. We compute the logarithmic derivative of relation (33) to have Therefore, we can rewrite the differential subordination (31) in the form Now, by taking β = 1 − ζ and γ = ζ, we apply Lemma 2 to obtain the differential subordination (32) and announce h as the best dominant.This ends the proof.
Then, the following differential subordination Proof.In view of Theorem 3, there can be formed a convex function g such that Now, by applying Equation (20) and the fact that β = 1 − ζ, we write Thus, by invoking Equation ( 19) and the fact that γ = ζ, we establish that Equation ( 21), therefore, reveals Thus, the subordination (37) is derived.This finishes the proof of Corollary 3.
Proof.First, we define For this function, we find that ′ Thus, by using Lemma (1), we derive where w(v) is a Schwarz function.Since we obtain that ≥ (Re(v)) 1 ζ for Re(v) > 0 and ζ ≥ 1, from inequality (42), the inequality (39) is derived.This end the proof.

Conclusions
This article uses the inspirational classical Gaussian hypergeometric functions to define a linear fractional differential operator D µ,η λ .The new differential operator is then used to introduce a number of analytic function subclasses.Moreover, the convex function class S δ (λ, µ, η) is provided, and its characteristics are described.Furthermore, a number of conclusions involving starlike and convex functions of complex order are achieved by using the Briot-Bouquet differential subordination.Briot-Bouquet differential subordination was used to construct a variety of inclusion relationships.Following the establishment of this idea, scholars have considered a number of applications for this theory.As a result, the findings of this research might be expanded to examine Gaussian hypergeometric functions and significant properties of complex dynamical systems.