The idea of differential subordination was first presented by S.S. Miller and P.T. Mocanu in two works that were published in 1978 and 1981 in an effort to extend the concept of inequality from the real numbers to the complex plane. S.S. Miller and P.T. Mocanu developed the theory of differential subordination in the context of second-order differential subordinations [4]. In 2011, J.A. Antonino and S.S. Miller [1] developed this idea to include third-order differential subordinations, providing opportunities for further research into the differential subordination theory. This area of research focuses on identifying previously established findings from the theory of second-order differential subordinations that, with the proper extension, are also applicable to the theory of third-order differential subordinations. The initial such extensions could be realized using one of the core concepts of the theory of differential subordinations, the class of admissible functions, as a starting point. The results presented here concerning third-order differential subordinations were obtained considering another fundamental problem in differential subordination theory, which is identifying dominants for the differential subordinations studied and furthermore, finding the best dominant when this is possible [7, 8]. The fractional integral of Gaussian hypergeometric function [6] is used for providing applications of the theoretical results obtained. The next step accomplished in 2014 [9] was to expand the dual theory of differential superordination [3, 2] in order to include third-order differential superordination. New and intriguing findings followed soon considering the idea involving the class of admissible functions [10] and continues to present interest at this time [11]. The extension of the results established for second-order differential superordiantions is now presented following the idea of finding subordinants of the third-order differential superordinations and providing the best subordinant when admitted by the third-order differential superordinations involved in the study.