Making use of Horadam polynomials, we propose a special family of regular functions of the type which are bi-univalent (or bi-schlicht) in the disc . We find estimates on the coefficients and and the functional of Fekete–Szegö for functions in this subfamily. Relevant connections to existing results and new observations of the main result are also presented.

The prime purpose of this article is to derive a necessary and sufficient condition for a linear operator associated with the Pascal distribution series to be in the class of analytic functions. Moreover, inclusion relation and an integral operator linked to the Pascal distribution series is considered. We have also provided some results as corollaries of our theorems.

The purpose of this paper is to establish common fixed points of six mappings in the context of bicomplex valued metric spaces. In this way, we generalize some previous well-known results from the literature. Moreover, we provide a non-trivial example to demonstrate the authenticity of established outcomes. As an application, we investigate the solution of an Urysohn integral equation by applying our results.

This work aims to explore the solution of a nonlinear fractional integro-differential equation in the complex domain through the utilization of both analytical and numerical approaches. The demonstration of the existence and uniqueness of a solution is established under certain appropriate conditions with the use of Banach fixed point theorems. To date, no research effort has been undertaken to look into the solution of this integro equation, particularly due to its fractional order specification within the complex plane. The validation of the proposed methodology was performed by utilizing a novel strategy that involves implementing the Rationalized Haar wavelet numerical method with the application of the Bernoulli polynomial technique. The primary reason for choosing the proposed technique lies in its ability to transform the solution of the given nonlinear fractional integro-differential equation into a representation that corresponds to a linear system of algebraic equations. Furthermore, we conduct a comparative analysis between the outcomes obtained from the suggested method and those derived from the rationalized Haar wavelet method without employing any shared mathematical methodologies. In order to evaluate the precision and effectiveness of the proposed method, a series of numerical examples have been developed.

In this paper, we define two classes of meromorphic multivalent functions in the punctured disc ${\mathbb{U}}^{*}=\left\{w\in \mathbb{C}:0<\left|w\right|<1\right\}$ by using the principle of subordination. We investigate a number of useful results including subordination results, some connections with a certain integral operator, sandwich properties, an inclusion relationship, and Fekete-Szegö inequalities for the functions belonging these classes. Our results are connected with those in several earlier works, which are related to this field of Geometric Function Theory (GFT) of Complex Analysis.

The Volterra-Hammerstein integral equation plays a crucial role in the control of robotic manipulators, which are widely used in industrial automation, medical robotics, and space exploration. These systems present significant control challenges due to their nonlinear behaviors and memory-dependent effects. In this research, we establish novel common fixed point theorems for generalized rational contractions within the framework of complex-valued suprametric spaces. Leveraging these theoretical advancements, we apply the derived results to solve the Volterra-Hammerstein integral equation, demonstrating its significance in robotic manipulator control. To emphasize the originality and practical applicability of our findings, a comprehensive illustrative example is provided.