Third order F-differential subordination and related applications

Abstract

This paper develops a third-order framework for \(F\)-differential subordination in the setting of analytic and univalent functions defined in the open unit disk. The study extends earlier first- and second-order subordination principles by incorporating differential expressions involving derivatives up to the third order. To establish the main results, suitable classes of admissible functions are introduced in relation to an \(F\)-function, its associated image domain, and the corresponding pre-image region. These admissibility conditions connect the geometric behavior of analytic functions with algebraic inequalities involving the function and its first three derivatives. By employing third-order differential inequality techniques and appropriate transformations between the relevant domains, sufficient conditions are obtained under which an analytic function is subordinate to a prescribed univalent comparison function. The results also clarify the relationship between the newly defined admissibility classes and previously known third-order differential subordination principles. Several special cases are discussed to demonstrate that the proposed formulation contains classical results as limiting or reduced forms. As an application, a generalized version of the Schwarz lemma is presented for functions satisfying the proposed \(F\)-subordination relation. The corresponding equality case is examined, and consequences concerning disk automorphisms and Schwarz--Pick-type estimates are highlighted. An illustrative example is also provided to show how the theoretical conditions can be verified for a particular analytic mapping and how the resulting subordination yields geometric properties such as starlikeness. The proposed approach offers a flexible method for deriving new inclusion, growth, and geometric-property results for analytic functions and may be extended to higher-order differential subordinations, other subclasses of univalent functions, and additional differential and integral operators.