This paper presents a newly defined subclass of analytic functions and explores several significant properties within the class, which use for their definitions the q-analogues of the derivative and the subordinations. Thus, we tried to connect different notions of the q-calculus with those of the Geometric Function Theory of one variable function. We identify the bounds of the initial coefficients and found upper bounds of the Fekete–Szegő functional for these classes. We investigate the relationship between the coefficients of an univalent function and those of its inverse by examining the difference between their second Hankel determinants. Furthermore, we analyze the behavior of the quantity module of the difference between the second Hankel determinant of a function and the same determinant for its inverse. To improve the obtained results by finding sharp estimations remains an interesting open question.

This paper introduces a novel subclass of analytic functions defined by the product of a modified sigmoid function and the lemniscate Bernoulli function. We initiate the study by deriving initial coefficient bounds for functions within this subclass, followed by an investigation into several key analytic properties. Specifically, we establish the Fekete–Szegö inequality and analyze Hankel determinants of various orders. Furthermore, the study provides estimates for the logarithmic coefficients and establishes bounds for both the inverse coefficients and the logarithmic inverse coefficients of functions in the subclass. This comprehensive analysis offers a significant contribution to the theory of analytic function subclasses associated with the products of special functions.