Certain sandwich-type results obtained for the Atangana-Baleanu fractional integral applied to the hypergeometric function

The Atangana-Baleanu fractional integral operator is of particular importance due to its nonsingular Mittag-Leffler kernel, which allows it to be used in many branches of applied mathematics for the development and study of mathematical models that involve it. The interest in the study of hypergeometric functions in their connection to the theory of univalent functions has reappeared, as L. de Branges used hypergeometric functions in the proof of the famous Bieberbach conjecture. The confluent (Kummer) hypergeometric function was studied recently from many points of view. Applying the differential subordinations and differential superordinations theories introduced by Miller and Mocanu, interesting inequalities can be established for complex-valued functions, yielding sandwich results. In this paper, we introduce a new operator defined by the Atangana-Baleanu fractional integral applied tothe confluent hypergeometric function
which takes the following form

and we establish several subordination and superordination results, which, taken together, constitute sandwich-type theorems. Interesting corollaries are stated using particular functions as best subordinant and best dominant of the subordinations and superordinations studied in theorems stated in this paper.