Ma-Minda-type analytic functions have been a central topic in geometric function theory due to their importance in studying properties such as univalence, convexity, and starlikeness. However, the class of Ma-Minda functions associated with kidney-shaped domains has not been systematically investigated before, leaving a clear research gap. In this paper, we introduced and studied the geometric properties of class ${\mathcal{S}}_k^{\ast }$. Our approach combined rigorous geometric analysis with Python programming (version 3.13.2) and graphical illustrations, providing both theoretical depth and computational support. Several inclusion relations and radius problems for ${\mathcal{S}}_k^{\ast }$ were established in comparison with other known classes. Furthermore, we examined various problems related to the initial coefficient bounds, with particular emphasis on their sharpness.

This paper discusses an elaborate investigation of the dimensionless time-dependent paraxial equation based on its varied soliton solutions obtained through an improved modified extended tanh function method. The approach is capable of producing kink, bell-shaped, singular wave, periodic, singular periodic wave, bright and dark solitary wave, breather soliton, singular bell, M-shaped, W-shaped, and V-pattern solitons expressed as hyperbolic and trigonometric functions. Visualization via three dimensional (3D) surface plots, two dimensional (2D) cross-sections and contour plots allows a thorough examination of the wave morphology. Nonlinear dynamics are investigated using bifurcation, chaotic, sensitivity, and stability analyses that uncover complex solution behaviors and stability regimes. Modulation instability analysis verifies the stability of soliton structures under perturbations. The work demonstrates the effectiveness of the improved modified extended tanh function method for solving nonlinear partial differential equations and enhances theoretical knowledge of paraxial wave phenomena.

In this investigation, a generalized form of the Gregory function, called the $\beta$-Gregory function, is derived. This function includes several well-known analytic functions as special cases. We then consider a new symmetric class of analytic and bi-univalent functions related to the $\beta$-Gregory function. The geometric and analytic properties of this class are investigated, and inclusion relationships with other known subclasses are established under suitable sufficient conditions. Furthermore, we derive accurate estimates of the coefficients of this class and discuss the associated Fekete-Szegö inequality, showing how the results obtained contribute to generalizing and improving several previous works in bi-univalent functions.