mathematics Article New Applications of S ˘ al˘ agean and Ruscheweyh Operators for Obtaining Fuzzy Differential Subordinations Alina Alb Lupa¸ s* ,† and Georgia Irina Oros †   Citation: Alb Lupa¸ s, A.; Oros, G.I. New Applications of S ˘ al˘ agean and Ruscheweyh Operators for Obtaining Fuzzy Differential Subordinations. Mathematics 2021, 9, 2000. https:// doi.org/10.3390/math9162000 Academic Editor: Alfonso Mateos Caballero Received: 22 July 2021 Accepted: 16 August 2021 Published: 21 August 2021 Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affil- iations. Copyright: © 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/). Department of Mathematics and Computer Science, University of Oradea, 1 Universitatii Street, 410087 Oradea, Romania; georgia_oros_ro@yahoo.co.uk * Correspondence: alblupas@gmail.com † These authors contributed equally to this work. Abstract: The present paper deals with notions from the field of complex analysis which have been adapted to fuzzy sets theory, namely, the part dealing with geometric function theory. Several fuzzy differential subordinations are established regarding the operator L m α , given by L m α : A n →A n , L m α f (z)=(1 − α) R m f (z)+ αS m f (z), where A n = { f ∈H(U), f (z)= z + a n+1 z n+1 + ... , z ∈ U} is the subclass of normalized holomorphic functions and the operators R m f (z) and S m f (z) are Ruscheweyh and S ˘ al˘ agean differential operator, respectively. Using the operator L m α , a certain fuzzy class of analytic functions denoted by SL m F (δ, α) is defined in the open unit disc. Interesting results related to this class are obtained using the concept of fuzzy differential subordination. Examples are also given for pointing out applications of the theoretical results contained in the original theorems and corollaries. Keywords: fuzzy differential subordination; convex function; fuzzy best dominant; differential operator 1. Introduction Fuzzy sets theory epic started in 1965 when Lotfi A. Zadeh published the paper “Fuzzy Sets” [1], received with distrust at first but currently cited by over 95,000 papers. Math- ematicians have been constantly concerned with adapting fuzzy sets theory to different branches of mathematics, and many such connections have been made. The beautiful review paper published in 2017 [2] is a tribute to Lotfi A. Zadeh’s contribution to the scientific world and shows the evolution of the notion of fuzzy set in time and its numerous connections with different topics of mathematics, science, and technique. Another great review article published as part of this Special Issue dedicated to the Centenary of the Birth of Lotfi A. Zadeh [3] gives further details on the development of fuzzy sets theory and highlights the contributions of Professor I. Dzitac who has had Lotfi A. Zadeh as mentor. In 2008, he edited a volume [4], tying his name to that of Lotfi A. Zadeh for posterity. The first applications of fuzzy sets theory in the part of complex analysis studying analytic functions of one complex variable were marked by the introduction of the concept of fuzzy subordination in 2011 [5]. The study was continued, and the notion of fuzzy differential subordination was introduced in 2012 [6]. All the aspects of the classical theory of differential subordination which are synthesized in the monograph published in 2000 [7] by the same authors who have introduced the notion in 1978 [8] and 1981 [9] were then adapted in light of the connection to fuzzy sets theory. At some point, fuzzy differential subordinations began to be studied in connection with different operators with many applications in geometric function theory as it can be seen in the first papers published starting with 2013 [10–12]. The topic is of obvious interest at this time, a fact proved by the numerous papers published in the last 2 years, of which we mention only a few [13–16]. In this paper, fuzzy differential subordinations will be obtained using the differential operator defined and studied in several aspects in [17,18]. Mathematics 2021, 9, 2000. https://doi.org/10.3390/math9162000 https://www.mdpi.com/journal/mathematics