Sharp inequalities for $q$-starlike functions associated with differential subordination and $q$-calculus

This paper employs differential subordination and quantum calculus to investigate a new class of $q$-starlike functions associated with an eight-like image domain. Our study laid a foundational understanding of the behavior of these $q$-starlike functions. We derived the results in first-order differential subordination. We established sharp inequalities for the initial Taylor coefficients and provided optimal estimates for solving the Fekete-Szegö problem and a second-order Hankel determinant applicable to all $q$-starlike functions in this class. Furthermore, we presented a series of corollaries that demonstrate the broader implications of our findings in geometric function theory.