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.

The authors introduce new classes of analytic functions in the open unit disc which are defined by using multiplier transformations. The properties of these classes will be studied by using techniques involving the Briot‐Bouquet differential subordinations. Also an integral transform is established.

Sanford S. Miller and Petru T. Mocanu’s theory of second-order differential subordinations was extended for the case of third-order differential subordinations by José A. Antonino and Sanford S. Miller in 2011. In this paper, new results are proved regarding third-order differential subordinations that extend the ones involving the classical second-order differential subordination theory. A method for finding a dominant of a third-order differential subordination is provided when the behavior of the function is not known on the boundary of the unit disc. Additionally, a new method for obtaining the best dominant of a third-order differential subordination is presented. This newly proposed method essentially consists of finding the univalent solution for the differential equation that corresponds to the differential subordination considered in the investigation; previous results involving third-order differential subordinations have been obtained mainly by investigating specific classes of admissible functions. The fractional integral of the Gaussian hypergeometric function, previously associated with the theory of fuzzy differential subordination, is used in this paper to obtain an interesting third-order differential subordination by involving a specific convex function. The best dominant is also provided, and the example presented proves the importance of the theoretical results involving the fractional integral of the Gaussian hypergeometric function.

In light of the well-established and widely-used theory of differential subordination, recent works incorporating fuzzy elements into Geometric Function Theory have given rise to the concept of fuzzy differential subordination. Second-order fuzzy differential subordinations were taken into consideration for studies up until this point. The research described in this paper aims to expand the concept of fuzzy differential subordination to third-order fuzzy differential subordination, building on an idea first put forth in 2011 by Jos\'{e} A. Antonino and Sanford S. Miller and still being investigated by scholars today. The key concepts and preliminary findings required for the development of this branch of fuzzy differential subordination are introduced. The class of admissible functions is specified, the fundamental theorems are established and the fundamental concepts of the third-order fuzzy subordination approach are presented. Several examples constructed as applications of the new results demonstrate the applicability of the new findings.

The study presented in this paper follows a line of research familiar for Geometric Function Theory, which consists in defining new integral operators and conducting studies for revealing certain geometric properties of those integral operators such as univalence, starlikness, or convexity. The present research focuses on the Bessel function of the first kind and order $\nu$ unveiling the conditions for this function to be univalent and further using its univalent form in order to define a new integral operator on the space of holomorphic functions. For particular values of the parameters implicated in the definition of the new integral operator involving the Bessel function of the first kind, the well-known Alexander, Libera, and Bernardi integral operators can be obtained. In the first part of the study, necessary and sufficient conditions are obtained for the Bessel function of the first kind and order $\nu$ to be a starlike function or starlike of order $\alpha \in [0,1)$. The renowned prolific method of differential subordination due to Sanford S. Miller and Petru T. Mocanu is employed in the reasoning. In the second part of the study, the outcome of the first part is applied in order to introduce the new integral operator involving the form of the Bessel function of the first kind, which is starlike. Further investigations disclose the necessary and sufficient conditions for this new integral operator to be starlike or starlike of order $\frac{1}{2}$.