Third-Order Differential Subordination for Analytic Functions Involving the Lommel Function of the First Kind

This study investigates third-order differential subordination and its influence on classes of analytic functions associated with the Lommel function of the first kind. By employing a newly defined operator Lwjf(z), we identify and characterize the admissible function classes that satisfy the corresponding third-order differential subordinations. These admissibility conditions enable the derivation of several key results, including a sandwich-type theorem obtained as a direct consequence of the established framework. The findings contribute to a broader understanding of analytic functions governed by higher-order differential constraints and highlight the significant role played by the Lommel function in shaping these geometric properties.