Open AccessArticle

New Results about Fuzzy Differential Subordinations Associated with Pascal Distribution

Sheza M. El-Deeb

^1,†

and

Luminiţa-Ioana Cotîrlă

^2,*

Department of Mathematics, Faculty of Science, Damietta University, New Damietta 34517, Egypt

Department of Mathematics, Technical University of Cluj-Napoca, 400114 Cluj-Napoca, Romania

Author to whom correspondence should be addressed.

^†

Current address: Department of Mathematics, College of Science and Arts, Al-Badaya, Qassim University, Buraidah 52571, Saudi Arabia.

Symmetry 2023, 15(8), 1589; https://doi.org/10.3390/sym15081589

Submission received: 21 June 2023 / Revised: 6 August 2023 / Accepted: 10 August 2023 / Published: 15 August 2023

(This article belongs to the Special Issue Symmetry in Pure Mathematics and Real and Complex Analysis)

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Abstract

Based upon the Pascal distribution series

N_{q, λ}^{r, m} Υ (ζ) : = ζ + \sum_{j = m + 1}^{\infty} (\binom{j + r - 2}{r - 1}) [1 + λ (j - 1)] q^{j - 1} {(1 - q)}^{r} a_{j} ζ^{j}

, we can obtain a set of fuzzy differential subordinations in this investigation. We also newly obtain class

P_{q, λ}^{F, r, m} (η)

of univalent analytic functions defined by the operator

N_{q, λ}^{r, m}

, give certain properties for the class

P_{q, λ}^{F, r, m} (η)

and also obtain some applications connected with a special case for the operator. New research directions can be taken on fuzzy differential subordinations associated with symmetry operators.

Keywords:

convolution; fuzzy differential subordination; Pascal distribution

MSC:

30C45; 30C50; 30C80

1. Introduction

Let

H_{m} (ϖ)

represent the class of holomorphic and univalent functions on

ϖ

such that

ϖ \subset C

and let

H (ϖ)

denote the class of holomorphic functions on

ϖ

. The class of holomorphic functions in the open unit disk of the complex plane

Λ = {ζ \in C : |ζ| < 1}

is denoted in this study by a note

H (Λ)

, with

B_{Λ} = {ζ \in C : |ζ| = 1}

standing as the unit disk’s boundary. For

m \in N = \{1, 2, \dots\}

, we define

H_{m} [γ] = \{Υ \in H (Λ) : Υ (ζ) = γ + \sum_{j = m + 1}^{\infty} a_{j} ζ^{j}, ζ \in Λ\},

A_{m} = \{Υ \in H (Λ) : Υ (ζ) = ζ + \sum_{j = m + 1}^{\infty} a_{j} ζ^{j}, ζ \in Λ\} with A_{1} = A,

(1)

and

S = \{Υ \in A_{m} : Υ is a univalent function in Λ\} .

We denote by

C = \{Υ \in A_{m} : ℜ (1 + \frac{ζ Υ^{^{″}} (ζ)}{Υ^{^{'}} (ζ)}) > 0, ζ \in Λ\},

which is the set of convex functions on

Λ

Let

Υ_{1}

and

Υ_{2}

be analytic in

Λ .

Then

Υ_{1}

is subordinate to

Υ_{2}

written as

Υ_{1} ≺ Υ_{2}

if there exists a Schwarz function

ϕ,

which is analytic in

Λ

with

ϕ (0) = 0

and

|ϕ (ζ)| < 1

for all

ζ \in Λ

such that

Υ_{1} (ζ) =

Υ_{2} (ϕ (ζ)) .

Furthermore, if the function

Υ_{2}

is univalent in

Λ,

then we have the following equivalence (see [1,2]):

Υ_{1} (ζ) ≺ Υ_{2} (ζ) \Leftrightarrow Υ_{1} (0) = Υ_{2} (0) and Υ_{1} (Λ) \subset Υ_{2} (Λ) .

In order to introduce the notion of fuzzy differential subordination, we use the following definitions and propositions:

Definition 1

([3]). Assume that

T \neq ⌀

is a Fuzzy subset and

F : T \to [0, 1]

is an application. A pair of

(Λ, F_{Λ}),

where

F_{Λ} : T \to [0, 1],

and

R = \{x \in T : 0 < F_{Λ} (x) \leq 1\} = sup (Λ, F_{Λ}),

a fuzzy subset. The fuzzy set

(Λ, F_{Λ})

is called a function

F_{Λ}

Let

Υ, g \in H (ϖ)

be denoted by

Υ (ϖ) = \{Υ (ζ) : 0 < F_{Υ (ϖ)} Υ (ζ) \leq 1, ζ \in ϖ\} = sup (Υ (ϖ), F_{Υ (ϖ)}),

(2)

and

g (ϖ) = \{g (ζ) : 0 < F_{g (ϖ)} g (ζ) \leq 1, ζ \in ϖ\} = sup (g (ϖ), F_{g (ϖ)}) .

(3)

Proposition 1

([4]). (i) If

(B, F_{B}) = (U, F_{U})

, then we have

B = U,

where

B = sup (B, F_{B})

and

U = sup (U, F_{U}); (i i)

(B, F_{B}) \subseteq (U, F_{U})

, then we have

B \subseteq U,

where

B = sup (B, F_{B})

and

U = sup (U, F_{U}) .

Definition 2

([4]). Let

ζ_{0} \in ϖ

be a fixed point and let the functions

Υ, g \in H (ϖ) .

The function Υ is said to be fuzzy subordinate to g, and we write

Υ ≺_{F} g

Υ (ζ) ≺_{F} g (ζ),

which satisfies the following conditions:

(i): $Υ (ζ_{0}) = g (ζ_{0});$
(ii): $F_{Υ (ϖ)} Υ (ζ) \leq F_{g (ϖ)} g (ζ), ζ \in ϖ .$

Proposition 2

([4]). Assume that

ζ_{0} \in ω

is a fixed point and the functions

Υ, g \in H (ω) .

Υ (ζ) ≺_{F} g (ζ),

ζ \in ω,

then

(i): $Υ (ζ_{0}) = g (ζ_{0})$
(ii): $Υ (ω) \subseteq g (ω), F_{Υ (ω)} Υ (ζ) \leq F_{g (ω)} g (ζ), ζ \in ω,$

where

Υ (ω)

and

g (ω)

are defined by (2) and (3), respectively.

Definition 3

([5]). Assume that

h \in S

and

Φ : C^{3} \times Λ \to C

Φ (α, 0, 0; 0) = h (0) = α

. If p satisfies the requirements of the second-order fuzzy differential subordination and is analytic in Λ, with

p (0) = α

F_{Φ (C^{3} \times Λ)} Φ (p (ζ), ζ p^{^{'}} (ζ), ζ^{2} p^{^{″}} (ζ); ζ) \leq F_{h (Λ)} h (ζ) .

(4)

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) \leq F_{q (Λ)} q (ζ), i . e ., p (ζ) ≺_{F} q (ζ), ζ \in Λ,

for each and every p satisfying (4).

Definition 4.

A fuzzy dominant

\tilde{q}

that satisfies

F_{\tilde{q} (Λ)} \tilde{q} (ζ) \leq F_{q (Λ)} q (ζ),

then

\tilde{q} (ζ) ≺_{F} q (ζ), ζ \in Λ .

The fuzzy best dominant of (4) is referred to for all fuzzy dominants.

Assume the function

Ω \in A_{m}

is given by

Ω (ζ) : = ζ + \sum_{j = m + 1}^{\infty} ψ_{j} ζ^{j}, ζ \in Λ .

The Hadamard (or convolution) product of

Υ

and

Ω

is defined as

(Υ * Ω) (ζ) : = ζ + \sum_{j = m + 1}^{\infty} a_{j} ψ_{j} ζ^{j}, ζ \in Λ .

A variable x is said to have the Pascal distribution if it takes the values

0, 1, 2, 3, \dots

with the probabilities

{(1 - q)}^{r}

\frac{q r {(1 - q)}^{r}}{1!}

\frac{q^{2} r (r + 1) {(1 - q)}^{r}}{2!}

\frac{q^{3} r (r + 1) (r + 2) {(1 - q)}^{r}}{3!}

, …, respectively, where q and r are called the parameters, and thus we have the probability formula

P (X = k) = (\binom{k + r - 1}{r - 1}) q^{k} {(1 - q)}^{r}, k \in N_{0} = N \cup \{0\} .

Now we present a power series whose coefficients are Pascal distribution probabilities, i.e.,

\begin{matrix} Q_{q, m}^{r} (ζ) : = ζ + \sum_{j = m + 1}^{\infty} (\binom{j + r - 2}{r - 1}) q^{j - 1} {(1 - q)}^{r} ζ^{j}, ζ \in Λ, \\ (m \in N, r \geq 1, 0 \leq q \leq 1) . \end{matrix}

We easily determine from the ratio test that the radius of convergence of the above power series is at least

\frac{1}{q} \geq 1

; hence,

Q_{q, m}^{r} \in A_{m}

We define the functions

\begin{matrix} M_{q, λ}^{r, m} (ζ) : = (1 - λ) Q_{q, m}^{r} (ζ) + λ z {(Q_{q, m}^{r} (ζ))}^{'} \\ = ζ + \sum_{j = m + 1}^{\infty} (\binom{j + r - 2}{r - 1}) [1 + λ (j - 1)] q^{j - 1} {(1 - q)}^{r} ζ^{j}, ζ \in Λ, \\ (m \in N, r \geq 1, 0 \leq q \leq 1, λ \geq 0) . \end{matrix}

El-Deeb and Bulboacă [6] introduced the linear operator

N_{q, λ}^{r, m} : A_{m} \to A_{m}

defined by

\begin{matrix} N_{q, λ}^{r, m} Υ (ζ) : = M_{q, λ}^{r, m} (ζ) * Υ (ζ) \\ = ζ + \sum_{j = m + 1}^{\infty} (\binom{j + r - 2}{r - 1}) [1 + λ (j - 1)] q^{j - 1} {(1 - q)}^{r} a_{j} ζ^{j}, ζ \in Λ, \\ (m \in N, r \geq 1, 0 \leq q \leq 1, λ \geq 0), \end{matrix}

where

Υ

is given by (1), and the symbol “*” stands for the Hadamard (or convolution) product.

Remark 1.

(i) For

m = 1,

the operator

N_{q, λ}^{r, m}

reduces to

I_{q, λ}^{r} : = N_{q, λ}^{r, 1}

, introduced and studied by El-Deeb et al. [7]; (ii) for

m = 1

and

λ = 0,

the operator

Q_{q}^{r}

reduces to

Q_{q}^{r} : = N_{q, 0}^{r, 1}

, introduced and studied by El-Deeb et al. [7].

Using the operator

N_{q, λ}^{r, m}

, we create a class of analytical functions and derive several fuzzy differential subordinations for this class.

Definition 5.

If the function

Υ \in A

belongs to the class

P_{q, λ}^{F, r, m} (η)

for all

η \in [0, 1)

and satisfies the inequality

F_{{(N_{q, λ}^{r, m} Υ)}^{^{'}} (Λ)} {(N_{q, λ}^{r, m} Υ (ζ))}^{^{'}} > η, (ζ \in Λ) .

2. Preliminary

The following lemmas are needed to show our results.

Lemma 1

([2]). Assume that

Ϝ \in A

and

G (ζ) = \frac{1}{ζ} \int_{0}^{ζ} Ϝ (t) d t, ζ \in Λ

. If

ℜ \{1 + \frac{ζ Ϝ^{″} (ζ)}{Ϝ^{'} (ζ)}\} > \frac{- 1}{2}, ζ \in Λ

, then

G \in C

Lemma 2

(Theorem 2.6 in [8]). If

Ϝ

is a convex function such that

Ϝ (0) = γ

ν \in C^{*} = C ∖ {0}

with

ℜ (ν) \geq 0 .

p \in

H_{m} [γ]

such that

p (0) = γ, Φ : C^{2} \times Λ \to C

Φ (p (ζ), ζ p^{^{'}} (ζ); ζ) = p (ζ) + \frac{1}{ν} ζ p^{^{'}} (ζ)

is an analytic function in Λ and

F_{Φ (C^{2} \times Λ)} (p (ζ) + \frac{1}{ν} ζ p^{^{'}} (ζ)) \leq F_{h (Λ)} h (ζ) \to p (ζ) + \frac{1}{ν} ζ p^{^{'}} (ζ) ≺_{F} h (ζ), ζ \in Λ,

then

F_{p (Λ)} p (ζ) \leq F_{q (Λ)} q (ζ) \leq F_{h (Λ)} h (ζ) \to p (ζ) ≺_{F} q (ζ), ζ \in Λ,

where

q (ζ) = \frac{ν}{m ζ^{\frac{ν}{m}}} \int_{0}^{ζ} ψ (t) t^{\frac{ν}{m} - 1} d t, ζ \in Λ .

The function q is convex, and it is the fuzzy best dominant.

Lemma 3

(Theorem 2.7 in [8]). Let

g

be a convex function in Λ and

Ϝ (ζ) = g (ζ) + m γ ζ g^{^{'}} (ζ),

where

ζ \in Λ, m \in N

and

γ > 0,

p (ζ) = g (0) + p_{m} ζ^{m} + p_{m + 1} ζ^{m + 1} + . . . \in H (Λ),

and

F_{p (Λ)} (p (ζ) + γ ζ p^{^{'}} (ζ)) \leq F_{ψ (Λ)} ψ (ζ) \to p (ζ) + γ ζ p^{^{'}} (ζ) ≺_{F} ψ (ζ), ζ \in Λ .

Then

F_{p (Λ)} (p (ζ)) \leq F_{g (Λ)} g (ζ) \to p (ζ) ≺_{F} g (ζ), ζ \in Λ .

This result is sharp.

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_{q, λ}^{r, m}

3. Main Results

Assume that

η \in [0, 1)

m \in N, r \geq 1, 0 \leq q \leq 1, λ \geq 0

and

ζ \in Λ

are mentioned throughout this paper.

Theorem 1.

Let k belong to

C

in Λ, and

h (ζ) = k (ζ) + \frac{1}{ρ + 2} ζ k^{^{'}} (ζ) .

Υ \in P_{q, λ}^{F, r, m} (η)

and

G (ζ) = I^{ρ} Υ (ζ) = \frac{ρ + 2}{ζ^{ρ + 1}} \int_{0}^{ζ} t^{ρ} Υ (t) d t,

(5)

then

F_{{(N_{q, λ}^{r, m} Υ)}^{^{'}} (Λ)} {(N_{q, λ}^{r, m} Υ (ζ))}^{^{'}} \leq F_{h (Λ)} h (ζ) \to {(N_{q, λ}^{r, m} Υ (ζ))}^{^{'}} ≺_{F} h (ζ),

(6)

implies

F_{{(N_{q, λ}^{r, m} G)}^{^{'}} (Λ)} {(N_{q, λ}^{r, m} G (ζ))}^{^{'}} \leq F_{k (Λ)} k (ζ) \to {(N_{q, λ}^{r, m} G (ζ))}^{^{'}} ≺_{F} k (ζ) .

Proof.

Since

ζ^{ρ + 1} G (ζ) = (ρ + 2) \int_{0}^{ζ} t^{ρ} Υ (t) d t,

by differentiating, we obtain

(ρ + 1) G (ζ) + ζ G^{^{'}} (ζ) = (ρ + 2) Υ (ζ),

and

(ρ + 1) N_{q, λ}^{r, m} G (ζ) + ζ {(N_{q, λ}^{r, m} G (ζ))}^{^{'}} = (ρ + 2) N_{q, λ}^{r, m} Υ (ζ),

(7)

and also, by differentiating (7), we obtain

{(N_{q, λ}^{r, m} G (ζ))}^{^{'}} + \frac{1}{(ρ + 2)} ζ {(N_{q, λ}^{r, m} G (ζ))}^{^{″}} = {(N_{q, λ}^{r, m} Υ (ζ))}^{^{'}} .

(8)

The fuzzy differential subordination (6) technique is used

F_{{(N_{q, λ}^{r, m} Υ)}^{^{'}} (Λ)} ({(N_{q, λ}^{r, m} G (ζ))}^{^{'}} + \frac{1}{(ρ + 2)} ζ {(N_{q, λ}^{r, m} G (ζ))}^{^{″}}) \leq

F_{h (Λ)} (k (ζ) + \frac{1}{(ρ + 2)} ζ k^{^{'}} (ζ)) .

(9)

We denote

q (ζ) = {(N_{q, λ}^{r, m} G (ζ))}^{^{'}}, so q \in H_{1} [n] .

(10)

Putting (10) in (9), we have

F_{{(N_{q, λ}^{r, m} Υ)}^{^{'}} (Λ)} (q (ζ) + \frac{1}{(ρ + 2)} ζ q^{^{'}} (ζ)) \leq F_{h (Λ)} (k (ζ) + \frac{1}{(ρ + 2)} ζ k^{^{'}} (ζ)) .

(11)

Using Lemma (3), we obtain

F_{q (Λ)} q (ζ) \leq F_{k (Λ)} k (ζ), i . e . F_{{(N_{q, λ}^{r, m} G (ζ))}^{^{'}} (Λ)} {(N_{q, λ}^{r, m} G (ζ))}^{^{'}} \leq F_{k (Λ)} k (ζ),

and therefore,

{(N_{q, λ}^{r, m} 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_{q}^{r}

reduces to

Q_{q}^{r} : = N_{q, 0}^{r, 1}

Example 1.

Let k be an element of

C

in Λ and

h (ζ) = k (ζ) + \frac{1}{ρ + 2} ζ k^{^{'}} (ζ) .

Υ \in P_{q, λ}^{F, r, m} (η)

and

G

is given by (5), then

F_{{(Q_{q}^{r} Υ)}^{^{'}} (Λ)} {(Q_{q}^{r} Υ (ζ))}^{^{'}} \leq F_{h (Λ)} h (ζ) \to {(Q_{q}^{r} Υ (ζ))}^{^{'}} ≺_{F} h (ζ),

implies

F_{{(Q_{q}^{r} G)}^{^{'}} (Λ)} {(Q_{q}^{r} G (ζ))}^{^{'}} \leq F_{k (Λ)} k (ζ) \to {(Q_{q}^{r} G (ζ))}^{^{'}} ≺_{F} k (ζ) .

Theorem 2.

Assume that

h (ζ) = \frac{1 + (2 η - 1) ζ}{1 + ζ}, η \in [0, 1),

λ > 0

and

I^{ρ}

is given by (5), then

I^{ρ} [P_{q, λ}^{F, r, m} (η)] \subset P_{q, λ}^{F, r, m} (η^{*}),

(12)

where

η^{*} = 2 η - 1 + (ρ + 2) (2 - 2 η) \int_{0}^{1} \frac{t^{ρ + 2}}{t + 1} dt .

(13)

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

F_{q (Λ)} (q (ζ) + \frac{1}{(ρ + 2)} ζ q^{^{'}} (ζ)) \leq F_{h (Λ)} h (ζ),

where

q (ζ)

is defined in (10). By using Lemma 2, we obtain

F_{q (Λ)} q (ζ) \leq F_{k (Λ)} k (ζ) \leq F_{h (Λ)} h (ζ),

which implies

F_{{(N_{q, λ}^{r, m} G)}^{^{'}} (Λ)} {(N_{q, λ}^{r, m} G (ζ))}^{^{'}} \leq F_{k (Λ)} k (ζ) \leq F_{h (Λ)} h (ζ),

where

\begin{matrix} k (ζ) & = & \frac{ρ + 2}{ζ^{ρ + 2}} \int_{0}^{ζ} t^{ρ + 1} \frac{1 + (2 η - 1) t}{1 + t} d t \\ = & (2 η - 1) + \frac{(ρ + 2) (2 - 2 η)}{ζ^{ρ + 2}} \int_{0}^{ζ} \frac{t^{ρ + 1}}{1 + t} d t \in C, \end{matrix}

where

k (Λ)

is symmetric with respect to the real axis, so we have

F_{{(N_{q, λ}^{r, m} G)}^{^{'}} (Λ)} {(N_{q, λ}^{r, m} G (ζ))}^{^{'}} \geq min_{|ζ| = 1} F_{k (Λ)} k (ζ) = F_{k (Λ)} k (1),

(14)

and

η^{*} = k (1) = 2 η - 1 + (ρ + 2) (2 - 2 η) \int_{0}^{1} \frac{t^{ρ + 2}}{t + 1} d t .

□

Theorem 3.

Assume that k belongs to

C

in Λ, that

k (0) = 1,

and that

h (ζ) = k (ζ) + ζ k^{^{'}} (ζ) .

When

Υ \in A

and the fuzzy differential subordination is satisfied,

F_{{(N_{q, λ}^{r, m} Υ)}^{^{'}} (Λ)} {(N_{q, λ}^{r, m} Υ (ζ))}^{^{'}} \leq F_{h (Λ)} h (ζ) \to {(N_{q, λ}^{r, m} Υ (ζ))}^{^{'}} ≺_{F} h (ζ),

(15)

holds, then

F_{N_{q, λ}^{r, m} Υ (Λ)} \frac{N_{q, λ}^{r, m} Υ (ζ)}{ζ} \leq F_{k (Λ)} k (ζ) \to \frac{N_{q, λ}^{r, m} Υ (ζ)}{ζ} ≺_{F} k (ζ) .

(16)

Proof.

Let

\begin{matrix} q (ζ) & = & \frac{N_{q, λ}^{r, m} Υ (ζ)}{ζ} = \frac{ζ + \sum_{j = m + 1}^{\infty} (\binom{j + r - 2}{r - 1}) [1 + λ (j - 1)] q^{j - 1} {(1 - q)}^{r} a_{j} ζ^{j}}{ζ} \\ = & 1 + \sum_{j = m + 1}^{\infty} (\binom{j + r - 2}{r - 1}) [1 + λ (j - 1)] q^{j - 1} {(1 - q)}^{r} a_{j} ζ^{j - 1}, \end{matrix}

and we obtain that

q (ζ) + ζ q^{^{'}} (ζ) = {(N_{q, λ}^{r, m} Υ (ζ))}^{^{'}},

F_{{(N_{q, λ}^{r, m} Υ)}^{^{'}} (Λ)} {(N_{q, λ}^{r, m} Υ (ζ))}^{^{'}} \leq F_{h (Λ)} h (ζ)

implies

F_{q (Λ)} (q (ζ) + ζ q^{^{'}} (ζ)) \leq F_{h (Λ)} h (ζ) = F_{k (Λ)} (k (ζ) + ζ k^{^{'}} (ζ)) .

Using the Lemma 3, we obtain

F_{q (Λ)} q (ζ) \leq F_{k (Λ)} k (ζ) \to F_{N_{q, λ}^{r, m} Υ (Λ)} \frac{N_{q, λ}^{r, m} Υ (ζ)}{ζ} \leq F_{k (Λ)} k (ζ),

and we obtain

\frac{N_{q, λ}^{r, m} Υ (ζ)}{ζ} ≺_{F} k (ζ) .

□

Theorem 4.

Consider

h \in H (Λ)

, which satisfies

ℜ (1 + \frac{ζ h^{^{″}} (ζ)}{h^{^{'}} (ζ)})

> \frac{- 1}{2}

when

h (0) = 1

. If the fuzzy differential subordination

F_{{(N_{q, λ}^{r, m} Υ)}^{^{'}} (Λ)} {(N_{q, λ}^{r, m} Υ (ζ))}^{^{'}} \leq F_{h (Λ)} h (ζ) \to {(N_{q, λ}^{r, m} Υ (ζ))}^{^{'}} ≺_{F} h (ζ),

(17)

then

F_{N_{q, λ}^{r, m} Υ (Λ)} \frac{N_{q, λ}^{r, m} Υ (ζ)}{ζ} \leq F_{k (Λ)} k (ζ) i . e . \frac{N_{q, λ}^{r, m} Υ (ζ)}{ζ} ≺_{F} k (ζ),

(18)

where

k (ζ) = \frac{1}{ζ} \int_{0}^{ζ} h (t) dt,

the function k is convex, and it is the fuzzy best dominant.

Proof.

Let

q (ζ) = \frac{N_{q, λ}^{r, m} Υ (ζ)}{ζ} = 1 + \sum_{j = d + 1}^{\infty} \frac{{[j]}_{q}!}{{[λ + 1]}_{q, j - 1}} a_{j} ψ_{j} ζ^{j - 1}, q \in H_{1} [1],

where

ℜ (1 + \frac{ζ h^{^{″}} (ζ)}{h^{^{'}} (ζ)}) > \frac{- 1}{2} .

From Lemma 1, we have

k (ζ) = \frac{1}{ζ} \int_{0}^{ζ} h (t) d t \in

belongs to the class

C,

which satisfies the fuzzy differential subordination (17). Since

k (ζ) + ζ k^{^{'}} (ζ) = h (ζ),

it is the fuzzy best dominant. We have

q (ζ) + ζ q^{^{'}} (ζ) = {(N_{q, λ}^{r, m} Υ (ζ))}^{^{'}},

then (17) becomes

F_{q (Λ)} (q (ζ) + ζ q^{^{'}} (ζ)) \leq F_{h (Λ)} h (ζ) .

By using Lemma 3, we obtain

F_{q (Λ)} q (ζ) \leq F_{k (Λ)} k (ζ), i . e . F_{N_{q, λ}^{r, m} Υ (Λ)} \frac{N_{q, λ}^{r, m} Υ (ζ)}{ζ} \leq F_{k (Λ)} k (ζ),

then

\frac{N_{q, λ}^{r, m} Υ (ζ)}{ζ} ≺_{F} k (ζ) .

□

Putting

h (ζ) = \frac{1 + (2 υ - 1) ζ}{1 + ζ}

in Theorem 4. As a result, we have the following corollary:

Corollary 1.

Let

h = \frac{1 + (2 υ - 1) ζ}{1 + ζ}

be a convex function in Λ, with

h (0) = 1,

0 \leq β < 1 .

Υ \in A

and verifies the fuzzy differential subordination

F_{{(N_{q, λ}^{r, m} Υ)}^{^{'}} (Λ)} {(N_{q, λ}^{r, m} Υ (ζ))}^{^{'}} \leq F_{h (Λ)} (\frac{1 + (2 υ - 1) ζ}{1 + ζ}), i . e . {(N_{q, λ}^{r, m} Υ (ζ))}^{^{'}} ≺_{F} (\frac{1 + (2 υ - 1) ζ}{1 + ζ}),

then

F_{N_{q, λ}^{r, m} Υ (Λ)} \frac{N_{q, λ}^{r, m} Υ (ζ)}{ζ} \leq F_{k (Λ)} k (ζ),

(19)

then

\frac{N_{q, λ}^{r, m} Υ (ζ)}{ζ} ≺_{F} k (ζ),

where

k (ζ) = 2 υ - 1 + \frac{2 (1 - υ)}{ζ} ln (1 + ζ),

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.

Example 2.

Let

h = \frac{1 + (2 υ - 1) ζ}{1 + ζ}

be a convex function in Λ, with

h (0) = 1,

0 \leq β < 1 .

f \in A

and verifies the fuzzy differential subordination

F_{{(Q_{q}^{r} Υ)}^{^{'}} (Λ)} {(Q_{q}^{r} Υ (ζ))}^{^{'}} \leq F_{h (Λ)} (\frac{1 + (2 υ - 1) ζ}{1 + ζ}), i . e . {(Q_{q}^{r} Υ (ζ))}^{^{'}} ≺_{F} (\frac{1 + (2 υ - 1) ζ}{1 + ζ})

then

F_{Q_{q}^{r} Υ (Λ)} \frac{Q_{q}^{r} Υ (ζ)}{ζ} \leq F_{k (Λ)} k (ζ) i . e . \frac{Q_{q}^{r} Υ (ζ)}{ζ} ≺_{F} k (ζ),

(20)

where

k (ζ) = 2 υ - 1 + \frac{2 (1 - υ)}{ζ} ln (1 + ζ) .

4. Conclusions

All of the above results provide information about fuzzy differential subordinations for the operator

N_{q, λ}^{r, m}

; we also provide certain properties for the class

P_{q, λ}^{F, r, m} (η)

of univalent analytic functions. Using these classes and operators, we can create some simple applications.

Author Contributions

Conceptualization, S.M.E.-D. and L.-I.C.; Formal analysis, S.M.E.-D. and L.-I.C.; Investigation, S.M.E.-D. and L.-I.C. All authors have read and agreed to the published version of the manuscript.

Funding

This research received no external funding.

Data Availability Statement

Not applicable.

Conflicts of Interest

The authors declare no conflict of interest.

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El-Deeb, S.M.; Cotîrlă, L.-I. New Results about Fuzzy Differential Subordinations Associated with Pascal Distribution. Symmetry 2023, 15, 1589. https://doi.org/10.3390/sym15081589

AMA Style

El-Deeb SM, Cotîrlă L-I. New Results about Fuzzy Differential Subordinations Associated with Pascal Distribution. Symmetry. 2023; 15(8):1589. https://doi.org/10.3390/sym15081589

Chicago/Turabian Style

El-Deeb, Sheza M., and Luminiţa-Ioana Cotîrlă. 2023. "New Results about Fuzzy Differential Subordinations Associated with Pascal Distribution" Symmetry 15, no. 8: 1589. https://doi.org/10.3390/sym15081589

APA Style

El-Deeb, S. M., & Cotîrlă, L. -I. (2023). New Results about Fuzzy Differential Subordinations Associated with Pascal Distribution. Symmetry, 15(8), 1589. https://doi.org/10.3390/sym15081589

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New Results about Fuzzy Differential Subordinations Associated with Pascal Distribution

Abstract

1. Introduction

2. Preliminary

3. Main Results

4. Conclusions

Author Contributions

Funding

Data Availability Statement

Conflicts of Interest

References

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