On The Third-Order Complex Differential Inequalities of ξ-Generalized-Hurwitz–Lerch Zeta Functions

In the zdomain, differential subordination is a complex technique of geometric function theory based on the idea of differential inequality. It has formulas in terms of the first, second and third derivatives. In this study, we introduce some applications of the third-order differential subordination for a newly defined linear operator that includes ξ-Generalized-Hurwitz–Lerch Zeta functions (GHLZF). These outcomes are derived by investigating the appropriate classes of admissible functions.


Introduction and Terminology
Complex Function Theory (CFT) is a mathematical branch dating back to the 18th century. It investigates the functions of complex numbers. This branch has attracted the concern of several researchers. Among the remarkable names are Euler, Gauss, Riemann, Cauchy and others. It has numerous implementations in diverse fields of mathematics and science. These functions have many interesting properties that are not owned by real-valued functions. For instance, infinitely differentiable functions, holomorphic functions, every holomorphic function in the open unit disk can be represented as a Taylor series, conformal functions (that is, they preserve angles when f (z) = 0), line integrals, and all types of handy formulas. The considerable area in CFT is the Geometric Function Theory (GFT). The study of GFT includes investigating the interaction between the analytical properties of the complex holomorphic function and the geometrical properties of the image domain. Riemann [1] in 1851 introduced the first major result in GFT named the Riemann Mapping Theorem. Later, in 1907, Koebe [2] was a prominent scientist who studied the univalent functions in the open unit disk. Thereafter, in 1912, Koebe [3] presented a modified version of the Riemann's mapping theorem by utilized univalent functions. The theory tends towards the principle of "univalent" and "holomorphic", Riemann's mapping theorem plays a significant role in the collection of both principles. This synthesis interprets the formula of a domain where the complex functions being defined, for details see [1,4].
On the other hand, differential inequality theory (inequalities including derivatives of functions) impacted the development of GFT due to it giving much information regarding the behavior of the holomorphic function. Further, there are many differential implications in which characterization of a and suppose that H 0 ≡ H[0, 1] and H 1 ≡ H [1,1]. Let A denote the class of all holomorphic functions ϑ in D, normalized by the conditions ϑ(0) = ϑ (0) − 1 = 0, and of the formula The subclass of A involving holomorphic univalent function is denoted by S, [1]. In [4] the concept of subordination between holomorphic functions given as: for two functions ϑ 1 and ϑ 2 , holomorphic in D, the function ϑ 1 is said to be subordinate to ϑ 2 , or ϑ 2 superordinate to ϑ 1 in D, written ϑ 1 ≺ ϑ 2 , if there is a holomorphic functionh in D withh(0) = 0 and |h(z)| < 1 for all z ∈ D, such that ϑ 1 (z) = ϑ 2 (h(z)). In particular, if the function ϑ 2 is univalent in D, then the following characterization for subordination is gained as: The natural generalization of holomorphic univalent function is a p-valent (multivalent) function, that is, if for each ω, the equation ϑ(z) = ω has at most p roots in a domain D ⊂ C, and if there is ω 0 such that the equation ϑ(z) = ω 0 has exactly p roots in a Domain D. Let A p (p ∈ N = {1, 2, 3, ...}) denote the class involves all p-valent functions in D of the form If ϑ is the p-valent function with p = 1, then ϑ is the holomorphic univalent function, [4]. As one of the most remarkable tools, namely Hadamard (convolution) product, utilizes to formulate assorted operators: differential, integral and convolution operators. The term "Hadamard product" is attributed to Hadamard in 1899 [1] and defined as: for two functions ϑ ∈ A of the form ϑ (z) = z + ∑ ∞ =2 α , z  , = 1, 2, their convolution, ϑ 1 * ϑ 2 , is given by More generally, the convolution product of two functions ϑ ∈ A p of the formula ϑ (z) = z p + ∑ ∞ =p+1 α , z  , = 1, 2, p ∈ N, is the function ϑ 1 * ϑ 2 given by In 1915, Alexander [32] was the first to introduce a linear integral operator which drafted in terms of the convolution, namely "Alexander operator" as follows: let ϑ ∈ A and I A : A → A be defined as Later on, in 1965, Libera [33] given another linear integral operator so-called "Libera operator" I L : A → A by the formula In 1969, Bernardi [34] imposed a more general linear integral operator I ε : A → A, for ϑ ∈ A and ε > −1, as The operator I ε is called the generalized Bernardi-Libera-Livingston integral operator. For ε = 0, the operator I ε reduces to the Alexander operator I A given by Equation (6) and for ε = 1, it reduces to the Libera operator I L defined by Equation (7).
Corresponding to the Ruscheweyh operator D , ∈ N 0 given by Equation (10), in 1999, Noor [36] considered the following linear operator: let ϑ ∈ A, ∈ N 0 and I : A → A be defined as such that Evidently, I 0 ϑ(z) = zϑ (z), I 1 ϑ(z) = ϑ(z), z ∈ D. This reverse relationship between the operators I and D gives a a cause for naming the Noor operator an integral operator. The operator I is called as the Noor integral operator of th order of ϑ.
Analogous to D ℘ , ℘ > −1 written by Equation (9), in 2002, Choi, Saigo and Srivastava [37] defined the linear operator I ℘,F : A → A, for ϑ ∈ A, ℘ > −1 and F > 0 by such that The operator I ℘,F is called the Choi-Saigo-Srivastava operator. For ℘ = and F = 2 reduces to the Noor integral operator I of Equation (11).
In 2002, Liu and Noor [38] provided a linear operator as: for ϑ ∈ A p , ℘ > −p and I ℘+p : A p → A p defined by such that Obviously, I 0+p ϑ(z) = zϑ (z)/p and I 1+p ϑ(z) = ϑ(z). The operator I ℘+p is an extended Noor integral operator I of Equation (11). In addition, the operator I ℘+p is closely related to the Choi-Saigo-Srivastava operator I ℘,F of Equation (12).
The Theory of Hypergeometric Functions (HFT) has been incorporated in GFT. Employing hypergeometric functions in the proof of the famed problem "Bieberbach conjecture" by de Branges in 1984 [39] has given complex analysts a renewed attention to study the role of special functions. In this regard a lot of implementations and generalizations are found. The study of this theory gained an independent status. The Gauss Hypergeometric Function (GHF), denoted by F (µ, ν; τ; ω), was first introduced by Gauss in 1812 [39]. It is given as follows: for µ, ν and τ be complex numbers with τ other than 0, −1, −2, ..., and where ( )  is the Pochhammer symbol given by Another important special function related to GHF is the incomplete beta function Other generalized Noor-type linear integral operators between classes of holomorphic functions associated with hypergeometric functions and its generalizations have been posed by authors. For instance, Al-Janaby et al. ( [40,41]).
In terms of the Hurwitz-Lerch Zeta function Φ(z, γ, η) defined by (see, for example, [58][59][60]) The following new family of the (GHLZF) was considered systematically by Srivastava [61]: and the equality in the convergence condition holds true for suitably bounded values of |z| given by

21). A dominant ω(z) that achieves ω(z) ≺ ω(z) for all dominants ω(z) of Equation
The class of admissible functions related to differential subordination is presented next.
The following theorem is a key outcome in third-order differential subordination.
This operator achieves the differential recurrence relation . Throughout this paper, the generalized Noor-type linear integral operator will be denoted by M p ϑ(z).

Differential Subordination with M p ϑ(z)
This section introduces certain appropriate class of admissible functions and studies some third-order differential subordination outcomes for the operator M p ϑ(z) defined by Equation (27).

Definition 6.
Let A be a set in C, ω ∈ J 0 and  ∈ N\{1}. The class of admissible functions Σ M [A, ω] consists of those functions : C 4 × D −→ C that satisfy the following admissibility condition: where z ∈ D, χ ∈ ∂D\G(ω), and κ ≥ .
If ϑ ∈ A p and ω ∈ J 0 achieve the following conditions: and Proof. Define the following holomorphic function υ(z) in D by From Equations (29) and (33), we have Further computations show that and Define the parameters u 1 , u 2 , u 3 and u 4 as: and then M p ϑ(z) < Q.
Proof. Let (u 1 , u 2 , u 3 , u 4 ; z) = u 2 − u 1 . Utilizing Corollary 3 with A =h(D) and We have to find the condition so that ∈ Σ M [A, Q], that is, the admissibility condition of Equation (48) is achieved. This follows since The required outcome is obtained.
Proof. Let (u 1 , u 2 , u 3 , u 4 ; z) = u 3 − u 2 . Utilizing Corollary 3 with A =h(D) and It is enough to show that ∈ Σ M [A, Q], that is, the admissibility condition of Equation (48) is achieved. This follows since This completes the proof. It is adequate to show that ∈ Σ M [A, Q], that is, the admissibility condition of Equation (48) The required outcome is derived.

Conclusions and Future Directions
In the terms of the ξ-Generalized Hurwitz-Lerch Zeta functions (GHLZF) in the z-domain, a new generalized Noor-type linear integral operator is introduced. This operator was utilized to study new classes of holomorphic functions in D. In addition, new applications of the third-order differential subordination outcome that involves this new operator were investigated. The third-order differential inequalities were imposed in this work to show the uppercase of this new generalized Noor-type linear integral operator in D.

Conflicts of Interest:
The authors declare no conflict of interest.