Using convolution (or Hadamard product), we define the El-Ashwah and Drbuk linear operator, which is a multivalent function in the unit disk $U=\left[w:\left|w\right|<1 \mathrm{and} w\in ₵\right]$, and satisfy its specific relationship to derive the subordination, superordination, and sandwich results for this operator by using properties of subordination and superordination concepts.

In this article, the authors introduce two new subclasses of a class m-fold symmetric biunivalent functions in open unit disk. Coefficient bounds for the Taylor-Maclaurin coefficients |am+1| and |a2m+1| are are obtain . Furthermore, we solve ”Fekete-Szeg” ”o” functional problems for functions in FP,m(γ, µ, ϑ) and MP,m(κ, η, ϑ) . Also, several certain special improver results for the associated classes are presented .

This article aims to significantly advance geometric function theory by providing a valuable contribution to analytic and multivalent functions. It focuses on differential subordination and superordination, which characterize the interactions between analytic functions. To achieve our goal, we employ a method that relies on the characteristics of differential subordination and superordination. As one of the latest advancements in this field, this technique is able to derive several results about differential subordination and superordination for multivalent functions defined by the new operator ${\mathcal{M}}_{\lambda ,p}^m\left(v,\rho ;\eta \right)\mathcal{F}\left(\xi \right)$ within the open unit disk $\mathfrak{A}$. Additionally, by employing the technique, the differential sandwich outcome is achieved. Therefore, this work presents crucial exceptional instances that follow the results. The findings of this paper can be applied to a wide range of mathematical and engineering problems, including system identification, orthogonal polynomials, fluid dynamics, signal processing, antenna technology, and approximation theory. Furthermore, this work significantly advances the knowledge and understanding of the analytical functions of the unit and its interactive higher relations. The characteristics and consequences of differential subordination theory are symmetric to those of differential superordination theory. By combining them, sandwich-type theorems can be derived.

The main aim of the present work is to introduce the class of multivalent harmonic functions defined by the general integral operator. Thus, We get some geometric properties, like coefficients estimate, extreme point and distortion theorem, convolution property, radii of starlikeness, and convexity.

In the present study, a subclass of harmonic univalent functions defined by a differential operator acting on complex harmonic functions is tackled. A sufficient condition and a representation theorem for the subclass are derived. Some geometric properties associated with it are also investigated, including coefficient bound, extreme points, distortion and convex combinations in connection to the subclass

In this work, the authors introduce and study a new subclass ofMh,uΣr(γ,η,κ,λ), which is r-fold bi-univalent functions in open-unit disk 𝔄. Using the Rushwaya derivative operator, the study investigates the properties and behavior of an analytically generalized r-fold symmetric two-monovalent function in the open-unit disk 𝔄. The analysis calculates the Taylor-Maclarin coefficients |ar+1| and |a2r+1|, which represent one of its key terms these estimates Mh,uΣr(γ,η,κ,λ). Subclass of functions important for understanding analytical and geometric behavior they provide valuable insights into the development of these concepts and their relationship to the geometric properties of the functions. Furthermore, research goes beyond coefficient estimates to examine the significance of results. By analyzing this expression, the study highlights the circumstantial importance of the Mh,uΣr(γ,η,κ,λ) subclass plant. The importance of this new subclass is essentially its ability to add special information to specific operator users, as well as the inclusion of multiple values to increase its flexibility and robustness. Furthermore, by linking us to prior research in the field, the study strengthens the theoretical framework and deepens our understanding of these activities, thus contributing to extensive knowledge in this area.