This paper is concerned with studying some results by introducing a new Hadamard product $M_{\gamma ,b,v,\delta }^{C,\eta }$ operator for differential subordination and superordination for certain univalent functions in the open unit disc U. Firstly, we state some basic definitions and required theorems. Furthermore. Some sandwich theorems are derived. The differential subordination theory’s features and outcomes are symmetric to those derived using the differential superordination theory.

This paper investigates a new subclass of bi-univalent analytic functions defined on the open unit disk in the complex plane, associated with the subordination to $1+\mathrm{s}\mathrm{i}\mathrm{n}z$. Coefficient bounds are obtained for the initial Taylor–Maclaurin coefficients, with a particular focus on the second- and third-order Hankel determinants. To illustrate the non-emptiness of the proposed class, we consider the function $\sqrt{1+tanhz}$, which maps the unit disk onto a bean-shaped domain. This function satisfies the required subordination condition and hence serves as an explicit member of the class. A graphical depiction of the image domain is provided to highlight its geometric characteristics. The results obtained in this work confirm that the class under study is non-trivial and possesses rich geometric structure, making it suitable for further development in the theory of geometric function classes and coefficient estimation problems.

On The Solutions of Wave Equation in ThreeDimensions using D'alembert Formula

In this paper, we explore and introduce a novel study employing a new operator denoted as W_(ν,γ,δ,τ,n,r,d)^(η,s) within the field of geometric function theory, particularly in the context of sandwich theorems. We derive results pertaining to differential subordination and superordination for this new formula operator. Additionally, we establish specific sandwich theorems.

Deriving the General Laplace Inversion Formula using Complex Integration Results and its Applications in Solving Partial Differential Equations

In this study, we provide a class of bi-univalent functions defined via the Riemann-Liouville subordination derivative with the sine function. we establish that this class is non-empty and provide best bounds for the initial coefficients, alongside the maximal estimate for the Fekete-Szego functional.