Third-Order Differential Subordination for Generalized Struve Function Associated with Meromorphic Functions

the


Introduction
Furthermore, if the function  ̌ is univalent in , the following equivalence hold.
Let ,  ∈ ∑(, ) where  is given by Eq 2 and  is defined by 3 the convolution ( or Hadamard product) of two functions  and  is defined by In this study, it was considered one of these functions, the series solution of an inhomogeneous second-order Bessel differential equation, which was presented and explored by Struve. 4Struve functions and their generalization are used in a variety of areas of applied mathematics and physics.
After a series of mathematical operations, the Struve function of order has the following form 6 : and where , ,  ∈ ℂ,  =  + ( + 2) 2 ≠ 0, −1, −2, … ⁄ and ()  is the shifted factorial (or Pochhammer symbol) 6   Which is defined as follows From Eq 4 it can be easily concluded that To derive the results of this paper the following lemmas and definitions are needed.Definition 2: 5 If Ὼ ⊆ ℂ, Ҩ ∈ Ꝙ and  ≥ 2. Let   [Ὼ, Ҩ] be the family of admissible functions that include functions ϓ: ℂ 4 x  → ℂ that satisfy the requirement of acceptability as: where  ∈ , ᶉ ∈ Ә\(Ҩ) and  ≥ .
Several writers have derived many important conclusions involving numerous operators connected by differential superordination and differential subordination for example 2,3,10 .
] be the family of admissible functions, which comprises the functions : ℂ 4 x  → ℂ that meet the admissibility requirement: where  ∈ , ᶉ ∈ Ә\(Ҩ) and  ≥  ≥ 2. Proof: In unit disk , define To differentiate equation Eq 8 with respect to  and using Eq 5 and its recurrence, Once again to differentiate equation Eq 9 through applying the recurrence relation Eq 5 and with regard to , Additional calculations show that Now, the transformation is defined ϓ(, , , ; ): ℂ 4 x  → ℂ by And by making use of the equations Eq 8 -Eq 11 ϓ(ℳ(), ℳ ′ (),  2 ℳ " (),  3   When the function Ҩ() has an unknown behavior on Ә , the following results will be an extension of Theorem 1.
The next corollary explains the connection between the solution of the related 3  − order differential equation and the best dominant of a 3  −order differential subordination.has a solution Ҩ with Ҩ ∈ Ꝙ 1 that meets the requirements Eq 6, then subordination Eq 16 implies that and Ҩ is the best dominant of Eq 16.

Conclusion
The purpose of using the operator Š , , , is to provide some results for the 3  − order differential subordination for analytic function.Investigating pertinent classes of admissible function leads to the results.The findings presented in this paper offer fresh perspectives for further research and opportunities are provided for scholars to generalize the results to produce new results in the fields of meromorphic univalent and mermorphic multivalent function theory.

Theorem 1
is proved by relying on Lemma 1 because the requirements of admissibility for function  ∈ ð 1 [Ὼ, Ҩ] of Definition 3 is equivalent to the requirements of admissibility for function ϓ ∈   [Ὼ, Ҩ] in Definition 2.