Difference formula defined by a new differential symmetric operator for a class of meromorphically multivalent functions

Symmetric operators have benefited in different fields not only in mathematics but also in other sciences. They appeared in the studies of boundary value problems and spectral theory. In this note, we present a new symmetric differential operator associated with a special class of meromorphically multivalent functions in the punctured unit disk. This study explores some of its geometric properties. We consider a new class of analytic functions employing the suggested symmetric differential operator.


Introduction
The study of the operator is narrowly connected with problems in the theory of functions. Various operators that were studied are operators on the space of holomorphic functions. For instance, Beurling's theorem defines the invariant subspaces of bounded holomorphic functions on the open unit disk. Beurling deduced the idea as multiplication of the independent variable on the Hardy space. The realization in studying multiplication operators is seemed in Toeplitz operators, specifically in the Bergman space of holomorphic functions. The geometric function theory is likewise ironic covering a long list of operators, counting differential, integral, and convolution operators. Limited symmetric operators are studied in this field. Newly, Ibrahim and Darus (see [1] and for applications see [2][3][4][5]) offered new symmetric differential, integral, and linear symmetric operators for a class of normalized functions in the open unit disk.
In this note, we proceed to consider a differential symmetric operator (DSO) associated with a class of meromorphically multivalent functions in the punctured unit disk. Consequently, we suggest a new class of analytic functions based on DSO to study it in view of the geometric function theory. Moreover, we investigate the real case of a formula con-taining the DSO. We show that this operator is a solution of a type of Sturm-Liouville equation. Some examples are illustrated in the sequel.

Construction
In this paper, we construct a new DSO connected with the following class of multivalent meromorphic functions k (℘) consisting of functions ϕ with the power series expansion where k ∈ N = {1, 2, 3, . . .} and n -℘ ∈ N. Recall that the functions ϕ of the form (2.1) are called meromorphic with a pole at z = 0 so that ϕ(z)z -℘ is analytic in ∪ (see Komatu [6] or Hayman [7]). We then concentrate on a subclass of k (℘) formulated by a subordination and explore inclusion properties and sufficient inclusion conditions for this class and check its closure property under convolution or Hadamard product.

Differential symmetric operator (DSO)
In this place, we state a few definitions and a lemma that we shall need in the next section. First, we define a conformable differential operator for the class of meromorphic functions k (℘) defined by (2.1).
Clearly, mα ϕ(z) ∈ k (℘) as well as, for two functions ϕ and ψ ∈ k (℘), we have So, in general, we have the following proposition.

Proposition 2.2 (Semigroup property)
The class of DSO constructed by mα has the semigroup property since, for ϕ and ψ in k (℘), we obtain We will need the following subordination definition for our class of meromorphic functions. For functions ϕ and ψ in k (℘), we call that ϕ is subordinate to ψ, denoted by ϕ ≺ ψ, if there is a Schwarz function with (0) = 0 and | (z)|≤|z| < 1 so that ϕ(z) = ψ( (z)) in ∪ (see [8] or [9]).
To prove our outcomes in the next section, we need the following lemmas which are due to Miller and Mocanu [9].
Note that when λ 1 = λ 2 = 1, then we have the following result.
We have the following geometric results.

Theorem 3.7
For the function ϕ ∈ k (℘), define a functional where dυ is a probability measure. Moreover, where C is the class of analytic convex in ∪.
Proof For the first part of the theorem, we suppose that then by the majority concept, we have (z) ∈ S * . The second and third assertions are verified by [9]-Corollary 3.6a.1.
Similarly, we have the next result.

Conclusion
From what has been presented above, it is apparent that we formulated a new differential symmetric operator (DSO) associated with a class of meromorphically multivalent functions. We presented some outcomes covering the geometric studies of the suggested operator joining the Janowski function in the open unit disk. Our consequences indicated, under some conditions, that the proposed operator converges to the Janowski function. Moreover, we discussed the functional (z) + z (z) and the solution for real cases when ℘ = 1 and ℘ = 2 (z) + z (z) = 0.
We discovered that the real cases are converging to the Sturm-Liouville equation, and the solutions are found to be a combination of special functions. We presented the condition that gives (Theorem 3.3) for λ 2 > 0.