In this investigation, by using the Komatu integral operator, we introduce the new class of bi-univalent functions based on the rule of subordination. Moreover, we use the Faber polynomial expansions and Fibonacci numbers to derive bounds for the general coefficient of the bi-univalent function class.

The primary motivation of the paper is to investigate the power series distribution (Pascal model) for the analytic function classes TG, TGS and TGC. Furthermore, we give necessary and sufficient conditions for the Pascal distribution series belonging to these classes.

In this current work, by using a relation of subordination, we define a new subclass of starlike harmonic functions. We get coefficient bounds, distortion theorems, extreme points, convolution and convex combinations for this class of functions. Moreover, some relevant connections of the results presented here with diverse known results are briefly denoted.

In this paper, we introduce a new subclass of harmonic functions that significantly improves our understanding of these functions in geometric function theory. We provide a comprehensive analysis of this subclass by deriving several important properties, including coefficient bounds and decay bounds, which are necessary to evaluate the behavior and limitations of functions in this class. Additionally, we establish sufficient coefficient conditions for harmonic functions to belong to this class. Moreover, we rigorously show that this subclass is closed under both convex combinations and convolutions, meaning that any convex combination or convolution of functions in this class will also belong to the class. These results provide valuable insights into the stability and applicability of the subclass and provide a solid framework for further theoretical explorations and practical applications in complex analysis.

In this paper, we establish some results concerning the convolutions of harmonic mappings convex in the horizontal direction with harmonic vertical strip mappings. Furthermore, we provide examples illustrated graphically with the help of Maple to illuminate the results.