The fuzzy differential subordination concept was introduced in 2011, generalizing the concept of differential subordination following a recent trend of adapting fuzzy sets theory to other already-established theories. A prolific tool in obtaining new results related to operators is the fractional integral applied to different functions. The fractional integral of the confluent hypergeometric function was previously investigated using means of the classical theory of subordination. In this paper, we give new applications of this function using the theory of fuzzy differential subordination. Fuzzy differential subordinations are established and their best dominants are also provided. Corollaries are written using particular functions, in which the conditions for the univalence of the fractional integral of the confluent hypergeometric function are given. An example is constructed as a specific application of the results obtained in this paper.

In the research presented in this paper, confluent hypergeometric function is embedded in the theory of strong differential superordinations. In order to proceed with the study, the form of the confluent hypergeometric function is adapted taking into consideration certain classes of analytic functions depending on an extra parameter previously introduced related to the theory of strong differential subordination and superordination. Operators previously defined using confluent hypergeometric function, namely Kummer–Bernardi and Kummer–Libera integral operators, are also adapted to those classes and strong differential superordinations are obtained for which they are the best subordinants. Similar results are obtained regarding the derivatives of the operators. The examples presented at the end of the study are proof of the applicability of the original results.

Gaussian hypergeometric function has been investigated in the context of geometric function theory regarding many aspects. Obtaining univalence conditions for this function is a line of research followed by many scholars. In the present study, methods specific to the differential superordination theory are used for obtaining properties of the Gaussian hypergeometric function regarding convexity of order . Also, a necessary and sufficient condition is proved such that Gaussian hypergeometric function is a close-to-convex function. The applicability of the theoretical findings is demonstrated by a numerical example.