Certain Subordination and Superordination Properties of Analytic Functions

In this paper, we extract some subordination and Superordination properties using the characteristics of the generalized byproduct operator. The article aims to demonstrate some applications of the differential subordination concept to univalent function subclasses that contain specific convolutions as operators. During this time, several highly complex mathematical detectives have emerged, including Riemann, Cauchy, Gauss, Euler, and several others. Geometric function theory combines or involves geometry and analysis. The main objectives of the paper above are to investigate the dependence principle and to introduce an extra subset over polyvalent functions through a further operator related to higher-order derivative products. The results were important when taking into account the numerous geometric characteristics, including radii over stiffness, close-to-convexity, and convexity; value estimation; deformation and expansion bounds; and so on.