On Fuzzy Differential Subordination

The theory of differential subordination was introduced by S.S.Miller and P.T.Mocanu in [2], then developed in many papers. In [1] the authors investigate various subordination results for some subclasses of analytic functions in the unit disc. G.I.Oros and G.Oros define the notion of fuzzy subordination and in [3, 4, 5] they define the notion of fuzzy differential subordination. In this paper, we determine sufficient conditions for a multivalent function to be a dominant of the fuzzy differential subordination.


Introduction
We introduce some basic notions and results that are used in the sequel.   We also need the following notations and results from the classical complex analysis [5].
For D ⊂ C, we denote by H(D) the class of holomorphic functions on D, and by H n (D) the class of holomorphic and univalent functions on D.
In this paper, we denote by H(U ) the set of holomorphic functions in the unit disc U = {z ∈ C : |z| < 1} with ∂U = {z ∈ C : |z| = 1} the boundary of the unit disc.
For a ∈ C and n ∈ N we denote Let D ⊂ C and f, g ∈ H(D) holomorphic functions. We denote by ).

Definition 1.3 ([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 (z) > 0, z ∈ U } the class of normalized starlike functions in U, f (z) + 1 > 0, z ∈ U } the class of normalized convex functions in U and by ϕ (z) > 0, z ∈ U } the class of normalized close-to-convex functions in U [5].
, z ∈ U, for α real number and f ∈ A p [2]. Let

Definition 1.4 ([4]
). Let ψ : C 3 ×U → C and let h be univalent in U . If p is analytic in U and satisfies the (second-order) fuzzy differential subordination i.e. ψ(p(z), zp (z), z 2 p (z); z) < F h(z), z ∈ U , 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 p(z) < F g(z), z ∈ U , for all p satisfying (1). A fuzzy dominantq that satisfiesq(z) < F q(z), z ∈ U , for all fuzzy dominant q of (1) is said to be the fuzzy best dominant of (1).
. Let h be convex in U and let P : where q(z) = γ nz γ/n z 0 h(t)t γ n −1 dt. The function q is convex and is the fuzzy best (a, n)-dominant.

Main Results
Proposition 2.1. Let q be univalent in U and let θ and φ be analytic in a domain D containing q(U ), with φ(w) = 0, when w ∈ q(U ). Set Q(z) = zq (z).φ[q(z)] and h(z) = θ[q(z)] + Q(z) and suppose that either (i) Q is starlike, or (ii) h is convex. In addition, assume that Proof. The proof of Proposition is similar to Theorem 1.1 [5].
For p ∈ H[p, p] with p(z) = 0 in U and is analytic in U, then i.e. p(z) < F q(z), z ∈ U , and q is the best dominant.
By the (i)and (ii), we obtained that Q is starlike in U and Re( zh (z) Q(z) ) > 0 for all z ∈ U . Since it satisfies preconditions of Proposition 2.1, it follows Proposition 2.1, and q is the best dominant. Theorem 2.1. Let q ∈ H[1, 1] be univalent and satisfies the following conditions: (i) q(z) is convex, ψ(q(z), zq (z)) = (1 − α + αρ)q(z) + µzq (z), then , z ∈ U and q is the best dominant.