New Applications of Sălăgean and Ruscheweyh Operators for Obtaining Fuzzy Differential Subordinations

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 Lαm, given by Lαm:An→An, Lαmf(z)=(1−α)Rmf(z)+αSmf(z), where An={f∈H(U),f(z)=z+an+1zn+1+…,z∈U} is the subclass of normalized holomorphic functions and the operators Rmf(z) and Smf(z) are Ruscheweyh and Sălăgean differential operator, respectively. Using the operator Lαm, a certain fuzzy class of analytic functions denoted by SLFmδ,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.


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. Mathematicians 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][11][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][14][15][16].
In this paper, fuzzy differential subordinations will be obtained using the differential operator defined and studied in several aspects in [17,18].
The basic notions used for conducting the study are denoted as previously established in literature.
Let U = {z ∈ C : |z| < 1} and denote by H(U) the class of holomorphic functions in the unit disc U. Let A n = { f ∈ H(U) : f (z) = z + a n+1 z n+1 + . . . , z ∈ U} be the subclass of normalized holomorphic functions writing A 1 as A. When a ∈ C and n ∈ N * , denote by H[a, n] = { f ∈ H(U) : f (z) = a + a n z n + a n+1 z n+1 + . . . , z ∈ U} writing H 0 = H[0, 1]. The class of convex functions is obtained for α = 0 when 0 < α < 1 the class denoted by K(α) = f ∈ A : Re z f (z) f (z) + 1 > α, z ∈ U contains convex functions of order α. The definitions necessary for using the concept of fuzzy differential subordinations introduced in previously published cited papers are next reminded.
is called fuzzy subset of X. The set A is called the support of the fuzzy set (A, F A ) and F A is called the membership function of the fuzzy set (A, F A ). One can also denote A = supp(A, F A ).
For a fuzzy subset, the real number 0 represents the smallest membership degree of a certain x ∈ X to A and the real number 1 represents the biggest membership degree of a certain x ∈ X to A.
The empty set ∅ ⊂ X is characterized by F ∅ (x) = 0, x ∈ X, and the total set X is characterized by F X (x) = 1, x ∈ X. Definition 2 ([5]). Let D ⊂ C, z 0 ∈ D be a fixed point and let the functions f , g ∈ H(D). The function f is said to be fuzzy subordinate to g and write f ≺ F g or f (z) ≺ F g(z), if the conditions are satisfied: (1) f (z 0 ) = g(z 0 ), Definition 3 ([6] (Definition 2.2)). Let ψ : C 3 × U → C and h univalent in U, with ψ(a, 0; 0) = h(0) = a. If p is analytic in U, with p(0) = a and satisfies the (second-order) fuzzy differential subordination then p is called a fuzzy solution of the fuzzy differential subordination. The univalent function q is called a fuzzy dominant of the fuzzy solutions of the fuzzy differential subordination, or more simple a fuzzy dominant, if F p(U) p(z) ≤ F q(U) q(z), z ∈ U, for all p satisfying (1). A fuzzy dominant q that satisfies F q(U)q (z) ≤ F q(U) q(z), z ∈ U, for all fuzzy dominants q of (1) is said to be the fuzzy best dominant of (1).
then L( f ) = F ∈ K. Lemma 2 ([20]). Let h be a convex function with h(0) = a, and let γ ∈ C * be a complex number with Re γ ≥ 0. If p ∈ H[a, n] with p(0) = a, ψ : C 2 × U → C, ψ(p(z), zp (z); z) = p(z) + 1 γ zp (z) an analytic function in U and The function q is convex and is the fuzzy best dominant.

Lemma 3 ([20]
). Let g be a convex function in U and let h(z) = g(z) + nαzg (z), z ∈ U, where α > 0 and n is a positive integer. If p(z) = g(0) + p n z n + p n+1 z n+1 + . . . , z ∈ U, is holomorphic in U and and this result is sharp.
Sȃlȃgean and Ruscheweyh differential operators are well known in geometric function theory for the nice results obtained by implementing them in the studies. Their definitions and basic properties are given in the next two definitions and remarks.
Definition 4 (Sȃlȃgean [21]). For f ∈ A n , m, n ∈ N, the operator S m is defined by S m : A n → A n , Definition 5 (Ruscheweyh [22]). For f ∈ A n , m, n ∈ N, the operator R m is defined by R m : ...
The next definition shows the operator used for obtaining the original results of this paper, defined in a previously published paper. Two remarks regarding it are also listed.

Definition 6 ([17]
). Let α ≥ 0, m, n ∈ N. Denote by L m α the operator given by L m α : A n → A n , The membership function of the fuzzy set ( f + g)(D) associated to the function f + g coincide with the half of the sum of the membership functions of the fuzzy sets f (D), respectively g(D), associated to the function f , respectively g, i.e.,

Main Results
First, a new fuzzy class of analytic functions is defined using the operator given by Definition 6.
The first result for this class is related to its convexity.
Proof. Consider the functions For obtaining the required conclusion, the function A fuzzy subordination result is given in the next theorem and a related example follows.

Theorem 2.
Considering the convex function in U denoted by g and defining h(z) = g(z) z 0 t c f (t)dt, z ∈ U, then the fuzzy differential subordination implies , z ∈ U, and this result is sharp.
Replacing (9) in (8) we obtain Using Lemma 3, we have and g is the best dominant. We have obtained (L m α G(z)) ≺ F g(z), z ∈ U.
Several fuzzy subordination results are contained in the next theorems and corollaries. Some are followed by examples.

Proof.
As function h given in the theorem is convex, we can use the same arguments as in the proof of Theorem 2. Interpreting the hypothesis of Theorem 3, we read that where p(z) is given by (9). By applying Lemma 2, the following fuzzy inequality is obtained: Using the hypothesis of convexity for function g, it is known that g(U) is symmetric with respect to the real axis and we can write and β * = g(1) = 2δ − 1 + (c+2)(2−2δ) n 1 0 t c+2 n −1 1+t dt. From (11) we deduce inclusion (10). Theorem 4. Let g be a convex function with g(0) = 1 and define the function h(z) = g(z) + zg (z), z ∈ U.
If a function f ∈ A n satisfies for α ≥ 0 and m, n ∈ N, then we obtain the fuzzy differential subordination and this result is sharp.
Using this result in (12), we can write We can now apply Lemma 3 and obtain Therefore, and this result is sharp.

Theorem 5.
Let h be a convex function of order − 1 2 with h(0) = 1. If a function f ∈ A n satisfies for α ≥ 0 and m, n ∈ N, then n −1 dt is convex and is the fuzzy best dominant.

Proof.
Let As Re 1 + zh (z) h (z) > − 1 2 , z ∈ U, from Lemma 1, we obtain that q(z) = 1 nz 1 n z 0 h(t)t 1 n −1 dt is a convex function and verifies the differential equation associated to the fuzzy differential subordination (13) q(z) + zq (z) = h(z), therefore it is the fuzzy best dominant.

Using Theorem 5 we obtain
Theorem 6. Define the function h(z) = g(z) + zg (z), z ∈ U, using g a convex function in U with g(0) = 1. If a function f ∈ A n satisfies for α ≥ 0 and m, n ∈ N, then we obtain the sharp fuzzy differential subordination Proof. For f ∈ A n , f (z) = z + ∑ ∞ j=n+1 a j z j we have . Relation (15) becomes By using Lemma 3, we have We obtain L m+1 Theorem 7. Given a convex function g with g(0) = 1, define function h(z) = g(z) + zg (z), z ∈ U. If we take α ≥ 0 and m, n ∈ N and a function f ∈ A n satisfying then the sharp fuzzy differential subordination results Proof. Using the definition of operator L m α , we get Using this result in (16), we obtain which can be easily transformed into αj m+1 + (1 − α)jC m m+j−1 a j z j−1 = 1 + p n z n + p n+1 z n+1 + ....
Using the notation in (18), the fuzzy differential subordination becomes By using Lemma 3, we have and this result is sharp.

Theorem 8.
Let h be a convex function of order − 1 2 which satisfies h(0) = 1. If a function f ∈ A n satisfies for α ≥ 0 and m, n ∈ N, then the fuzzy differential subordination can be written as n −1 dt being convex and the best fuzzy dominant.

Proof.
As h is a convex function of order − 1 2 , Lemma 1 can be applied and we have that q(z) = 1 nz 1 n z 0 h(t)t 1 n −1 dt is a convex function and verifies the differential equation associated to the fuzzy differential subordination (19) q(z) + zq (z) = h(z), therefore it is the fuzzy best dominant.
If α ≥ 0, m, n ∈ N, f ∈ A n and satisfies the differential subordination then F L m α f (U) (L m α f (z)) ≤ F q(U) q(z), i.e., (L m α f (z)) ≺ F q(z), z ∈ U, where q is given by q(z) = 2β − 1 + 2(1−β) nz Further studies on the newly introduced class can be conducted for obtaining results that give coefficient estimates, distortion theorems, or closure theorems, as it is usual in geometric function theory. Furthermore, the way this class is introduced can inspire research for introducing other interesting fuzzy classes and studying their properties. The limit imposed on δ ∈ (0, 1] could be further investigated so that other possible values of δ for correct definitions of fuzzy classes could be found.

Conflicts of Interest:
The authors declare no conflict of interest.