Bounds for the Second Hankel Determinant and Its Inverse in Specific Function Classes

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.