In this research, we define a few subclasses of analytic functions which are connected to sine functions and the cardioid domain in the unit disk. We investigate initial coefficient bounds, the Fekete–Szego problem and second and third Hankel determinants for the functions f belonging to these newly defined classes. We also define the class of m-fold symmetric functions related with the sine function and then investigate the bounds of the third Hankel determinant for twofold symmetric and threefold symmetric functions.

In this research article, we introduce new family ${\mathcal{R}}_G$ of holomorphic functions, which is related to the generalized bounded turning and generating functions of Gregory coefficients. Leveraging the concept of functions with positive real parts, we acquire the first five coefficients for the functions belonging to this newly defined family, demonstrating their sharpness. Furthermore, we find the third Hankel determinant for functions in the class ${\mathcal{R}}_G$. Moreover, the sharp bounds for logarithmic and inverse coefficients of functions belonging to the under-considered class ${\mathcal{R}}_G$ are estimated.

This study investigates quasi bound states in the continuum (qBICs) in profile and resonance signals arising from polarization phase quantization in Sm ${}^{\mathbf {3+}}$ :BiPO ${}_{\mathbf {4}}$ . Two types of profile qBICs (third-order non-resonance profile qBIC and fourth-order resonance profile qBIC), result from destructive and constructive quantization phase, respectively. In addition, two types of third-order resonance qBICs, namely two-dark/one-bright states (2D1B) and one-dark/two-bright states (1D2B), are observed at constructive and destructive channel phase, respectively. The third-order 2D1B states resonance qBIC together with two optical jump, comes from eigenvector amplitude splitting near the exceptional point (EP) due to quasi-constructive quantization phase. These findings suggest potential applications in the interference-based quantum memory devices.