Applications of Fractional Calculus and Quantum Calculus in Subordination and q-Derivative Operators

The theory of analytic functions remains a fundamental area of geometric function theory, with particular emphasis on coefficient problems, differential subordinations, and determinant estimates. Motivated by recent developments in fractional calculus and quantum calculus, this paper introduces two new subclasses of normalized analytic functions by employing the subordination principle in combination with the q-derivative operator and the q-Sălăgeăn differential operator within the framework of quantum calculus. The inclusion of fractional and q-calculus techniques provides a more flexible and generalized approach to classical problems in complex analysis, enabling deeper structural insights into analytic function classes. Using the subordination framework, we derive coefficient relations for the proposed subclasses. Furthermore, we establish sharp upper bounds for the Fekete–Szegö functional |a3−δa22| and for the second Hankel determinant H2,2(f)=a2a4−a32. The obtained results extend and unify several known works in the literature and demonstrate how the interaction between fractional calculus, quantum operators, and subordination theory can be effectively used in geometric function theory. Finally, the presented approach opens the door for further investigations involving higher-order Hankel determinants, other subclasses of analytic functions, and potential extensions involving special functions and fractional operators.