Sandwich Subordinations Imposed by New Generalized Koebe-Type Operator on Holomorphic Function Class

. In the complex field, special functions are closely related to geometric holomorphic functions. Koebe function is a notable contribution to the study of the geometric function theory (GFT), which is a univalent function. This sequel introduces a new class that includes a more general Koebe function which is holomorphic in a complex domain. The purpose of this work is to present a new operator correlated with GFT. A new generalized Koebe operator is proposed in terms of the convolution principle. This Koebe operator refers to the generality of a prominent differential operator, namely the Ruscheweyh operator. Theoretical investigations in this effort lead to a number of implementations in the subordination function theory. The tight upper and lower bounds are discussed in the sense of subordinate structure. Consequently, the subordinate sandwich is acquired. Moreover, certain relevant specific cases are examined

Motivated by these scientific contributions, this paper investigates the new generalized Koebe operator by proposing a more generalized Koebe -type function that is holomorphic in Λ.This is employed to discuss of several interesting subordination and superordination implementations.As a consequence, the subordinate sandwich is derived.In addition, relevance's geometric outcomes are exanimated.In order to illustrate our main outcomes, we include the following concept and central lemmas.

Proposed Generalized Koebe Operator ℒ 𝜇,𝜎 𝜑(𝑧)
This section imposes the new generalized Koebe function, which is a holomorphic function on Λ. Afterward, a generalized Koebe operator is suggested according to the convolutional structure.This new operator is a generalization of the Ruscheweyh derivative operator.
Corollary 3.7.Let the hypotheses of Theorem 3.5 hold.Then, the subordination and superordination relation which implies that are consecutively the best subordinate and best dominate.
and  2 () =  in Theorem 3.5, we derive the required assertion, we attain the required assertion.

Conclusion
During this research, the remarkable contribution is to explore a new special function as an amended and generalized formula of the Koebe function based on the complex gamma function principle so that the Koebe function as a private case can be derived from it.Then, its action on operator theory over a specific complex field is attainable in view of convolution structure.This affords visions of the emergence of a new operator as one of the main consequences of this study.The proposed complex operator is a general formulation of an interesting operator that is the Ruscheweyh operator.Furthermore, by applying subordination and superordination methodology, the validated conclusions are sandwich outcomes that include this innovative operator.For future investigations, discuss by proposing generalizations and modifications to suggest Koebe-type functions and create numerous subclasses of holomorphic functions in the sense of multivalent and harmonic functions.Accordingly, the development offered in this sequel will motivate further attention and discussion in this significant area of mathematics.
Then, the relation(3.16)follows the implementation of Lemma 1.1.In view of Theorem 3.2, we yield the following interesting outcome.Corollary 3.4.Let the hypotheses of Theorem 3.2 hold.Then, the subordination relation By utilizing (3.8) and (3.25), we gain the relation(3.31),and then it is followed by an implementation of Lemma 1.2.
≺  2 (), and  1 and  2 are consecutively the best subordinate and best dominate.
and  1 and  2 are consecutively the best subordinate and best dominate.In view of Theorem 3.6, we yield the following outcome.