In the present paper, by making use of the new generalized operator, some results of third order differential subordination and differential superordination consequence for analytic functions are obtained. Also, some sandwich-type theorems are presented.

In this article, a new class of analytic functions which is defined by terms of a quasi-subordination is introduced. The coefficient estimates, including the classical  inequality of functions belonging to this class, are then derived. Also, several special improving results for the associated classes involving the subordination are presented.

         In the present paper, the authors introduce and investigates two new subclasses and, of the class k-fold bi-univalent functions in the open unit disk. The initial coefficients for all of the functions that belong to them were determined, as well as the coefficients for functions that belong to a field determining these coefficients requires a complicated process. The bounds for the initial coefficients and are contained among the remaining results in our analysis are obtained. In addition, some specific special improver results for the related classes are provided.

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 aim object of this paper is to study some classes of the differential subordination concept for the subclasses of univalent functions by using the convolution operator. Some interesting geometric properties like coefficient bound, growth and distortion theorems, closure theorem, radii of convexity, and starlikeness of functions are obtained. Extreme points and integral operators are also investigated.

In this paper, we conduct an investigation into a novel subclass of bi-univalent functions within the unit disk. The introduced subclass incorporates the generalized Srivastava-Attiya operator and satisfies specific quasi-subordination conditions. Through this study, the researchers determine the coefficient estimates and for functions within these subclasses and unveil new results by applying the operator to this particular subclass. The implications of these findings extend to complex analysis, number theory, and other branches of mathematics. Overall, this research significantly contributes to the theory of quasi-subordination of bi-univalent functions and enhances the understanding of the diverse applications of the Generalized Srivastava-Attiya Operator.