New Results about Fuzzy Differential Subordinations Associated with Pascal Distribution

: Based upon the Pascal distribution series N r , m

We denote by which is the set of convex functions on Λ.

Definition 2 ([4]
). Let ζ 0 ∈ be a fixed point and let the functions Υ, g ∈ H( ). The function Υ is said to be fuzzy subordinate to g, and we write Υ ≺ F g or Υ(ζ) ≺ F g(ζ), which satisfies the following conditions:

Definition 3 ([5]
). Assume that h ∈ S and Φ : If p satisfies the requirements of the second-order fuzzy differential subordination and is analytic in Λ, with p(0) = α, If q is a fuzzy dominant of the fuzzy differential subordination solutions, then p is said to be a fuzzy solution of the fuzzy differential subordination and satisfies F p(Λ) p(ζ) ≤ F q(Λ) q(ζ), i.e., p(ζ) ≺ F q(ζ), ζ ∈ Λ, for each and every p satisfying (4).

Definition 4. A fuzzy dominant q that satisfies
The fuzzy best dominant of (4) is referred to for all fuzzy dominants.
Assume the function Ω ∈ A m is given by The Hadamard (or convolution) product of Υ and Ω is defined as , . . . , respectively, where q and r are called the parameters, and thus we have the probability formula Now we present a power series whose coefficients are Pascal distribution probabilities, i.e., We easily determine from the ratio test that the radius of convergence of the above power series is at least 1 q ≥ 1; hence, Q r q,m ∈ A m . We define the functions El-Deeb and Bulboacȃ [6] introduced the linear operator N r,m q,λ : where Υ is given by (1), and the symbol " * " stands for the Hadamard (or convolution) product.
Using the operator N r,m q,λ , we create a class of analytical functions and derive several fuzzy differential subordinations for this class.

Definition 5.
If the function Υ ∈ A belongs to the class P F,r,m q,λ (η) for all η ∈ [0, 1) and satisfies the inequality

Preliminary
The following lemmas are needed to show our results.
We define the fuzzy differential subordination general theory and its applications (see [9][10][11][12][13]). The method of fuzzy differential subordination is applied in the next section to obtain a set of fuzzy differential subordinations related to the operator N r,m q,λ .
The fuzzy differential subordination (6) technique is used We denote Putting (10) in (9), we have Using Lemma (3), we obtain and therefore, N r,m q,λ G(ζ) ≺ F k(ζ),where k is the fuzzy best dominant.
Putting m = 1 and λ = 0 in Theorem 1, we obtain the following example since the operator Q r q reduces to Q r q := N r,1 q,0 .
, η ∈ [0, 1), λ > 0 and I ρ is given by (5), then where Proof. A function h belongs to C, and we obtain from the hypothesis of Theorem 2 using the same technique as that in the proof of Theorem 1 that where q(ζ) is defined in (10). By using Lemma 2, we obtain where k(Λ) is symmetric with respect to the real axis, so we have and η * = k(1) = 2η − 1 + (ρ + 2)(2 − 2η) Theorem 3. Assume that k belongs to C in Λ, that k(0) = 1, and that h(ζ) = k(ζ) + ζk (ζ). When Υ ∈ A and the fuzzy differential subordination is satisfied, holds, then Proof. Let and we obtain that Using the Lemma 3, we obtain and we obtain N r,m q,λ Υ(ζ) ζ ≺ F k(ζ).

Theorem 4.
Consider h ∈ H(Λ), which satisfies where k(ζ) = 1 the function k is convex, and it is the fuzzy best dominant.

Proof.
Let belongs to the class C, which satisfies the fuzzy differential subordination (17). Since it is the fuzzy best dominant. We have then (17) becomes By using Lemma 3, we obtain Putting h(ζ) = 1+(2υ−1)ζ 1+ζ in Theorem 4. As a result, we have the following corollary: be a convex function in Λ, with h(0) = 1, 0 ≤ β < 1. If Υ ∈ A and verifies the fuzzy differential subordination the function k is convex and it is the fuzzy best dominant.
Putting m = 1 and λ = 0 in Corollary 1, we obtain the following example.

Conclusions
All of the above results provide information about fuzzy differential subordinations for the operator N r,m q,λ ; we also provide certain properties for the class P F,r,m q,λ (η) of univalent analytic functions. Using these classes and operators, we can create some simple applications.