Third Order Differential Subordination Results Associated with Tremblay Fractional Derivative Operator for p-Valent Analytic Functions

This paper studies a class of p-valent analytic functions in the open unit disk using a modified Tremblay operator based on the Riemann–Liouville fractional integral and derivative. We derive a series representation of the operator, which serves as a useful tool to explore its analytic and geometric properties. New third-order differential subordination results are obtained by defining suitable classes of admissible functions. We provide sufficient conditions to ensure subordination relations with a given dominant function, leading to inclusion results in general complex domains. In addition, several applications are given for specific choices of the target domain, showing how the framework is flexible and able to produce unified results in the theory of differential subordination for multivalent analytic functions.