Applications of Riemann–Liouville Fractional Integral of q-Hypergeometric Function for Obtaining Fuzzy Differential Sandwich Results

Alb Lupaş, Alina; Oros, Georgia Irina

doi:10.3390/sym14102097

Open AccessArticle

Applications of Riemann–Liouville Fractional Integral of q-Hypergeometric Function for Obtaining Fuzzy Differential Sandwich Results

by

Alina Alb Lupaş

^*,†

and

Georgia Irina Oros

^†

Department of Mathematics and Computer Science, University of Oradea, 1 Universitatii Street, 410087 Oradea, Romania

^*

Author to whom correspondence should be addressed.

^†

These authors contributed equally to this work.

Symmetry 2022, 14(10), 2097; https://doi.org/10.3390/sym14102097

Submission received: 16 September 2022 / Revised: 25 September 2022 / Accepted: 6 October 2022 / Published: 8 October 2022

(This article belongs to the Special Issue Symmetries of Difference Equations, Special Functions and Orthogonal Polynomials)

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Abstract

:

Studies regarding the two dual notions are conducted in this paper using Riemann–Liouville fractional integral of q-hypergeometric function for obtaining certain fuzzy differential subordinations and superordinations. Fuzzy best dominants and fuzzy best subordinants are given in the theorems investigating fuzzy differential subordinations and superordinations, respectively. Moreover, corollaries are stated by considering particular functions with known geometric properties as fuzzy best dominant and fuzzy best subordinant in the proved results. The study is completed by stating fuzzy differential sandwich theorems followed by related corollaries combining the results previously established concerning fuzzy differential subordinations and superordinations.

Keywords:

Riemann–Liouville fractional integral; q-hypergeometric function; sandwich-type theorem; fuzzy differential subordination; fuzzy differential superordination

MSC:

30C45; 30A10; 33C15

1. Introduction

Fractional calculus and q-hypergeometric function are known to provide notable results when applied in studies regarding geometric function theory. Fuzzy differential subordination notion was introduced in geometric function theory as generalization of the classical concept of differential subordination in 2011. Later, in 2017, the dual notion of fuzzy differential superordination was also introduced generalizing the notion of differential superordination.

The important contribution fractional calculus and q-calculus has in the development of geometric function theory is indisputable and this is comprehensively shown in review article published by Srivastava [1]. In the same article, the importance of hypergeometric functions added to the studies is highlighted and also numerous fractional and q-calculus operators are listed emphasizing their role in the study of univalent functions.

The concept of fuzzy set introduced by Lotfi A. Zadeh in 1965 [2] has tremendous applications in various fields of science and technique. Extensions of many branches of mathematics have developed in the fuzzy context created by introducing the notion of fuzzy set into the studies. Applications of the notion related to integro-differential equations can be seen in [3,4]. Aspects of fuzzy linear fractional programming embedding the concept are presented in [5]. Proportional Integral Derivative (PID) and Fuzzy-PID control are used in [6] and an evolution of the concept of fuzzy normed linear spaces is presented in [7]. Some highlights of the applications of the fuzzy set concept can be read in [8,9].

The famous notion of subordination was interpreted in the fuzzy concept in 2011 [10] and fuzzy differential subordination was introduced in 2012 [11] as an extension of the classical notion of differential subordination due to Miller and Mocanu [12,13]. The theory started to develop in the next years [14,15,16] and the dual notion of fuzzy differential superordination was introduced in 2017 [17]. Later, studies continued to emerge using both theories and different types of operators and various notions familiar to the classical theory of differential subordination [18,19,20]. Some highlights on the development of the theories of fuzzy differential subordination and superordination can be read in [21].

In recent studies, fractional operators have been defined and used in correlation to fuzzy differential subordinations and superordinations. Fractional integral of confluent hypergeometric function is applied for obtaining new results in [22,23,24], fractional integral of Gaussian hypergeometric function is used in [25], fractional derivative is associated to the study of fuzzy differential subordinations in [26], Riemann–Liouville fractional integral of Ruscheweyh and Sălăgean operators helps in obtaining new results in [27] and Wanas operator is associated to fractional calculus aspects for obtaining fuzzy differential subordinations in [28].

Another step in the development of the theories of fuzzy differential subordination and superordination is taken in the present paper by adding quantum calculus aspects alongside fractional calculus.

Quantum calculus has a long history related to mathematical fields. Being known as the study of calculus without limits, it has many applications related to combinatorics [29], orthogonal polynomials [30,31,32], number theory [33] or basic hypergeometric functions [34]. Certain aspects regarding fundamentals of q-calculus and how it was adapted to different theories can be seen in [35,36,37]. In 1911, Jackson introduced the notions of q-derivative [38] and q-integral [39]. The first applications of q-calculus in geometric function theory are seen in [40] where the authors define the class of q-starlike functions. However, the favorable context for the development of studies involving q-calculus in geometric function theory is created by the publication of the book chapter authored by Srivastava [41]. It is the same book chapter which mentions q-hypergeometric function as potentially important for future studies connecting q-calculus to univalent functions.

Geometric function theory uses many q-operators for obtaining geometrical interpretations of certain special classes of functions. For instance, the famous the q-analogue of the Ruscheweyh and Sălăgean differential operators were defined [42,43], and many applications occurred [44,45,46]. A comprehensive indexing of q-operators is listed by Srivastava in a recent review paper [1]. Many applications of q-hypergeometric function in geometric function theory can be listed. A good description of the use of q-hypergeometric function in early studies from geometric function theory is given in [47] and new aspects are shown in [48]. Some results on applications of q-hypergeometric functions could be mentioned as [49,50,51,52]. The research presented in this paper connects an operator defined as Riemann–Liouville fractional integral of q-hypergeometric function with fuzzy differential subordination and superordination theory. Before presenting the operator and the fuzzy context in which the study is conducted, the basic notations and notions familiar to geometric function theory are recalled.

With

U = {z \in C : | z | < 1}

denoting the unit disc of the complex plane, the class of holomorphic functions in U is denoted by

H (U)

. Certain important subclasses of

H (U)

are defined as:

A_{n} = \{f \in H (U) : f (z) = z + a_{n + 1} z^{n + 1} + \dots, z \in U\},

with

A_{1} = A

, and

H (a, n) = \{f \in H (U) : f (z) = a + a_{n} z^{n} + a_{n + 1} z^{n + 1} + \dots, z \in U\},

with

a \in C

,

n \in N^{*}

.

The important tool represented by Riemann–Liouville fractional integral is defined as presented in [53,54]:

Definition 1

([53,54]). The Riemann–Liouville fractional integral of order λ (

λ > 0

) for an analytic function f is defined by

D_{z}^{- λ} f (z) = \frac{1}{Γ (λ)} \int_{0}^{z} \frac{f (t)}{{(z - t)}^{1 - λ}} d t .

The q-hypergeometric function is given by:

Definition 2

([55]). The q-hypergeometric function

ϕ (m, n; q, z)

is defined by

ϕ (m, n; q, z) = \sum_{k = 0}^{\infty} \frac{{(m, q)}_{k}}{{(n, q)}_{k} {(q, q)}_{k}} z^{k},

where

{(m, q)}_{k} = \{\begin{matrix} 1, k = 0, \\ (1 - m) (1 - m q) (1 - m q^{2}) . . . (1 - m q^{k - 1}), k \in N, \end{matrix}

and

0 < q < 1

.

Using Definitions 1 and 2, Riemann–Liouville fractional integral of q-hypergeometric function in introduced as:

Definition 3

([56]). Let m, n be complex numbers with

m \neq 0, - 1, - 2, \dots

and

λ > 0

,

0 < q < 1 .

We define the Riemann–Liouville fractional integral of q-hypergeometric function

D_{z}^{- λ} ϕ (m, n; q, z) = \frac{1}{Γ (λ)} \int_{0}^{z} \frac{ϕ (m, n; q, t)}{{(z - t)}^{1 - λ}} d t =

(1)

\frac{1}{Γ (λ)} \sum_{k = 0}^{\infty} \frac{{(m, q)}_{k}}{{(n, q)}_{k} {(q, q)}_{k}} \int_{0}^{z} \frac{t^{k}}{{(z - t)}^{1 - λ}} d t .

The Riemann–Liouville fractional integral of q-hypergeometric function has the following form, after a simple calculation:

D_{z}^{- λ} ϕ (m, n; q, z) = \sum_{k = 0}^{\infty} \frac{{(m, q)}_{k}}{{(n, q)}_{k} {(q, q)}_{k} {(k + 1)}_{λ}} z^{λ + k} .

(2)

We note that

D_{z}^{- λ} ϕ (m, n; q, z) \in H [0, λ] .

Next, the fuzzy context where the research was conducted is shown.

Definition 4

([10]). Fuzzy subset of X is a pair

(A, F_{A})

, with

F_{A} : X \to [0, 1]

and

A = {x \in X : 0 < F_{A} (x) \leq 1}

. The support of the fuzzy set

(A, F_{A})

is the set A and the membership function of

(A, F_{A})

is

F_{A}

. It is denoted

A = supp (A, F_{A})

.

Definition 5

([10]). Consider

D \subset C

, the functions

f, g \in H (D)

and

z_{0} \in D

a fixed point. The function f is fuzzy subordinate to g, written

f ≺_{F} g

, if the following conditions are satisfied:

(1): $f (z_{0}) = g (z_{0}),$
(2): $F_{f (D)} f (z) \leq F_{g (D)} g (z)$ , $z \in D .$

Definition 6

([11], Definition 2.2). Consider h a univalent function in U and

ψ : C^{3} \times U \to C

, such that

h (0) = ψ (a, 0; 0) = a

. When p is analytic in U, such that

p (0) = a

and the fuzzy differential subordination is satisfied

F_{ψ (C^{3} \times U)} ψ (p (z), z p^{'} (z), z^{2} p^{″} (z); z) \leq F_{h (U)} h (z), z \in U,

(3)

then p is a fuzzy solution of the fuzzy differential subordination. The univalent function q is a fuzzy dominant of the fuzzy solutions of the fuzzy differential subordination, if

F_{p (U)} p (z) \leq F_{q (U)} q (z)

,

z \in U

, for all p satisfying (3). A fuzzy dominant

\tilde{q}

that satisfies

F_{\tilde{q} (U)} \tilde{q} (z) \leq F_{q (U)} q (z)

,

z \in U

, for all fuzzy dominants q of (3) is the fuzzy best dominant of (3).

Definition 7

([17]). Consider h an analytic function in U and

φ : C^{3} \times U \to C

. When p and

φ (p (z), z p^{'} (z), z^{2} p^{″} (z); z)

are univalent functions in U and the fuzzy differential superordination is satisfied

F_{h (U)} h (z) \leq F_{φ (C^{3} \times U)} φ (p (z), z p^{'} (z), z^{2} p^{″} (z); z), z \in U,

(4)

then p is a fuzzy solution of the fuzzy differential superordination. An analytic function q is fuzzy subordinant of the fuzzy differential superordination if

F_{q (U)} q (z) \leq F_{p (U)} p (z), z \in U,

for all p satisfying (4). A univalent fuzzy subordinant

\tilde{q}

that satisfies

F_{q (U)} q \leq F_{q (U)} \tilde{q}

for all fuzzy subordinant q of (4) is the fuzzy best subordinant of (4).

Definition 8

([11]). Q represents the set of all analytic and injective functions f on

\bar{U} \ E (f)

, with

f^{'} (ζ) \neq 0

for

ζ \in \partial U \ E (f),

where

E (f) = {ζ \in \partial U : lim_{z \to ζ} f (z) = \infty}

.

The following lemmas are needed as tools in the proof of the theorems stated in the next section.

Lemma 1

([57]). Consider g a univalent function in the unit disc U and

φ, γ

analytic functions in a domain

D \supset g (U),

such that

γ (u) \neq 0

for

u \in g (U)

. Let

G (z) = z g^{'} (z) γ (g (z))

and

h (z) = G (z) + φ (g (z))

, supposing G is starlike univalent in U and

R e (\frac{z h^{'} (z)}{G (z)}) > 0

for

z \in U

.

When p is an analytic function such that

p (0) = g (0)

,

p (U) \subseteq D

and

F_{p (U)} (φ (p (z)) + z p^{'} (z) γ (p (z))) \leq F_{h (U)} (φ (g (z)) + z g^{'} (z) γ (g (z))),

then

F_{p (U)} p (z) \leq F_{g (U)} g (z),

and the fuzzy best dominant is g.

Lemma 2

([58]). Consider g a convex univalent function in U and

φ, γ

analytic functions in a domain

D \supset g (U)

. Assume that

R e (\frac{φ^{'} (g (z))}{γ (g (z))}) > 0

for

z \in U

and

G (z) = z g^{'} (z) γ (g (z))

is a starlike univalent function in U. When

p (z) \in H [g (0), 1] \cap Q

, with

p (U) \subseteq D

and

φ (p (z)) + z p^{'} (z) γ (p (z))

is a univalent function in U and

F_{g (U)} (φ (g (z)) + z g^{'} (z) γ (g (z))) \leq F_{p (U)} (φ (p (z)) + z p^{'} (z) γ (p (z))),

then

F_{g (U)} g (z) \leq F_{p (U)} p (z),

and the fuzzy best subordinant is g.

In the next section, fuzzy differential subordinations and superordinations will be investigated using the operator presented in (1) with his particular form as seen in (2). For the fuzzy differential subordinations and superordinations under investigation, fuzzy best dominant and fuzzy best subordinant are obtained, respectively. As applications of the results contained in the theorems, corollaries emerge when fuzzy best dominant and fuzzy best subordinant are considered particular functions known in geometric function theory to have certain geometric properties very useful in studies related to differential subordinations and superordinations. Two sandwich-type theorems and several corollaries associated are also stated connecting the dual results concerning fuzzy differential subordinations and fuzzy differential superordinations obtained in this paper.

2. Main Results

Considering

φ (u) : = β + ψ u + ε u^{2}

and

γ (u) : = \frac{δ}{u}

in Lemmas 1 and 2, we obtain the following results regarding fuzzy differential subordination:

Theorem 1.

Consider g an analytic and univalent function in U such that

g (z) \neq 0

, for all

z \in U

, and

\frac{z {(D_{z}^{- λ} ϕ (m, n; q, z))}^{'}}{D_{z}^{- λ} ϕ (m, n; q, z)} \in H (U)

, with m, n be complex numbers with

n \neq 0, - 1, - 2, \dots

and

λ > 0

,

0 < q < 1 .

Assume that

\frac{z g^{'} (z)}{g (z)}

is starlike univalent in

U,

R e (\frac{ψ}{δ} g (z) + \frac{2 ε}{δ} g^{2} (z) + 1 + z \frac{g^{″} (z)}{g^{'} (z)} - z \frac{g^{'} (z)}{g (z)}) > 0,

(5)

for

β, ψ, δ, ε \in C

,

δ \neq 0

,

z \in U

and

Ψ_{λ}^{m, n, q} (β, ψ, ε, δ; z) : = β + δ + (ψ - δ) \frac{z {(D_{z}^{- λ} ϕ (m, n; q, z))}^{'}}{D_{z}^{- λ} ϕ (m, n; q, z)} +

(6)

ε {(\frac{z {(D_{z}^{- λ} ϕ (m, n; q, z))}^{'}}{D_{z}^{- λ} ϕ (m, n; q, z)})}^{2} + δ \frac{z {(D_{z}^{- λ} ϕ (m, n; q, z))}^{″}}{{(D_{z}^{- λ} ϕ (m, n; q, z))}^{'}} .

If the following fuzzy differential subordination is satisfied by g

F_{Ψ_{λ}^{m, n, q} (U)} Ψ_{λ}^{m, n, q} (β, ψ, ε, δ; z) \leq F_{g (U)} (β + ψ g (z) + ε {(g (z))}^{2} + δ \frac{z g^{'} (z)}{g (z)}),

(7)

for

β, ψ, δ, ε \in C

,

δ \neq 0,

then

F_{D_{z}^{- λ} ϕ (U)} (\frac{z {(D_{z}^{- λ} ϕ (m, n; q, z))}^{'}}{D_{z}^{- λ} ϕ (m, n; q, z)}) \leq F_{g (U)} g (z),

(8)

and the fuzzy best dominant is g.

Proof.

Consider

p (z) : = \frac{z {(D_{z}^{- λ} ϕ (m, n; q, z))}^{'}}{D_{z}^{- λ} ϕ (m, n; q, z)}

,

z \in U

,

z \neq 0

. We have

p^{'} (z) = \frac{{(D_{z}^{- λ} ϕ (m, n; q, z))}^{'}}{D_{z}^{- λ} ϕ (m, n; q, z)} + z \frac{{(D_{z}^{- λ} ϕ (m, n; q, z))}^{″}}{D_{z}^{- λ} ϕ (m, n; q, z)} - z {(\frac{{(D_{z}^{- λ} ϕ (m, n; q, z))}^{'}}{D_{z}^{- λ} ϕ (m, n; q, z)})}^{2}

and

\frac{z p^{'} (z)}{p (z)} = 1 + z \frac{{(D_{z}^{- λ} ϕ (m, n; q, z))}^{″}}{{(D_{z}^{- λ} ϕ (m, n; q, z))}^{'}} - z \frac{{(D_{z}^{- λ} ϕ (m, n; q, z))}^{'}}{D_{z}^{- λ} D_{z}^{- λ} ϕ (m, n; q, z)} .

(9)

Let

φ (u) : = β + ψ u + ε u^{2}

and

γ (u) : = \frac{δ}{u}

, it is easy to show that

φ

is analytic in

C

,

γ

is analytic in

C \ {0}

and that

γ (u) \neq 0,

u \in C \ {0} .

Also, setting

G (z) = z g^{'} (z) γ (g (z)) = δ \frac{z g^{'} (z)}{g (z)},

which is starlike univalent in

U,

and

h (z) = G (z) + φ (g (z)) = β + ψ g (z) + ε {(g (z))}^{2} + δ \frac{z g^{'} (z)}{g (z)}

, we obtain by differentiating it

h^{'} (z) = ψ g^{'} (z) + 2 ε g (z) g^{'} (z) + δ \frac{g^{'} (z)}{g (z)} + δ z \frac{g^{″} (z)}{g (z)} - δ z {(\frac{g^{'} (z)}{g (z)})}^{2}

and

\frac{z h^{'} (z)}{G (z)} = \frac{ψ}{δ} g (z) + \frac{2 ε}{δ} g^{2} (z) + 1 + z \frac{g^{″} (z)}{g (z)} - z \frac{g^{'} (z)}{g (z)} .

We get that

R e (\frac{z h^{'} (z)}{G (z)}) = R e (\frac{ψ}{δ} g (z) + \frac{2 ε}{δ} g^{2} (z) + 1 + z \frac{g^{″} (z)}{g (z)} - z \frac{g^{'} (z)}{g (z)}) > 0

.

Using relation (9), we get

β + ψ p (z) + ε {(p (z))}^{2} + δ \frac{z p^{'} (z)}{p (z)} =

β + δ + (ψ - δ) \frac{z {(D_{z}^{- λ} ϕ (m, n; q, z))}^{'}}{D_{z}^{- λ} ϕ (m, n; q, z)} + ε {(\frac{z {(D_{z}^{- λ} ϕ (m, n; q, z))}^{'}}{D_{z}^{- λ} ϕ (m, n; q, z)})}^{2} + δ \frac{z {(D_{z}^{- λ} ϕ (m, n; q, z))}^{″}}{{(D_{z}^{- λ} ϕ (m, n; q, z))}^{'}} .

By using fuzzy differential subordination (7), we obtain

F_{p (U)} (β + ψ p (z) + ε {(p (z))}^{2} + δ \frac{z p^{'} (z)}{p (z)}) \leq F_{g (U)} (β + ψ g (z) + ε {(g (z))}^{2} + δ \frac{z g^{'} (z)}{g (z)}) .

Applying Lemma 1, we have

F_{p (U)} p (z) \leq F_{g (U)} g (z)

,

z \in U,

i.e.,

F_{D_{z}^{- λ} ϕ (U)} (\frac{z {(D_{z}^{- λ} ϕ (m, n; q, z))}^{'}}{D_{z}^{- λ} ϕ (m, n; q, z)}) \leq F_{g (U)} g (z)

,

z \in U

and the fuzzy best dominant is g. □

Example 1.

Let

ϕ (3, 2, q, z) = 1 + \frac{2}{1 - q} z + \frac{2 (1 - 3 q)}{1 - 2 q} z^{2}

for m = 3, n = 2,

z \in U

.

Then

D_{z}^{- λ} ϕ (3, 2, q, z) = \frac{1}{Γ (λ)} \int_{0}^{z} \frac{ϕ (3, 2, q, t)}{{(z - t)}^{1 - λ}} d t =

\frac{1}{Γ (λ)} \int_{0}^{z} (1 + \frac{2}{1 - q} z + \frac{2 (1 - 3 q)}{1 - 2 q} z^{2}) {(z - t)}^{λ - 1} d t

and after changing the variable

z - t = u

and making a simple calculus, we get

D_{z}^{- λ} ϕ (3, 2, q, z) = \frac{z^{λ - 1}}{Γ (λ)},

z \in U .

Consider

g (z) = \frac{1 - z}{1 + z},

z \in U

and differentiating it we obtain

g^{'} (z) = \frac{- 2}{{(1 + z)}^{2}}

and

g^{″} (z) = \frac{4}{{(1 + z)}^{3}}

,

z \in U

.

Choosing

β = 0

,

δ = 1

,

ψ = 1

,

ε = 1

, we obtain after a long calculus that

Ψ_{λ}^{3, 2, q} (0, 1, 1, 1; z) = λ^{2} - λ

.

Using Theorem 1, when Re

\frac{2 z^{3} + 6 z^{2} - 6 z + 4}{{(1 + z)}^{2} (1 - z)} > 0

and

\frac{z g^{'} (z)}{g (z)} = \frac{- 2 z}{1 - z^{2}}

is starlike univalent in U, we obtain

F_{U} (λ^{2} - λ) \leq F_{U} (\frac{- 6 z + 2}{{(1 + z)}^{2} (1 - z)})

,

z \in U,

induce

F_{U} (λ - 1) \leq F_{U} (\frac{1 - z}{1 + z}) .

Corollary 1.

Consider m, n complex numbers with

n \neq 0, - 1, - 2, \dots

;

λ > 0

, 0 < q < 1 and suppose that relation (5) holds. When

F_{Ψ_{λ}^{m, n, q} (U)} Ψ_{λ}^{m, n, q} (β, ψ, ε, δ; z) \leq

F_{g (U)} (β + ψ \frac{M z + 1}{N z + 1} + ε {(\frac{M z + 1}{N z + 1})}^{2} + \frac{δ (M - N) z}{(M z + 1) (N z + 1)}),

with

β, ψ, δ, ε \in C

,

δ \neq 0,

- 1 \leq N < M \leq 1

, and

Ψ_{λ}^{m, n, q}

defined by relation (6), then

F_{D_{z}^{- λ} ϕ (U)} (\frac{z {(D_{z}^{- λ} ϕ (m, n; q, z))}^{'}}{D_{z}^{- λ} ϕ (m, n; q, z)}) \leq F_{g (U)} (\frac{M z + 1}{N z + 1}),

and the fuzzy best dominant is

\frac{M z + 1}{N z + 1} .

Proof.

Put

g (z) = \frac{M z + 1}{N z + 1}

,

- 1 \leq N < M \leq 1

in Theorem 1 and we obtain the corollary. □

Corollary 2.

Consider m, n complex numbers with

n \neq 0, - 1, - 2, \dots;

λ > 0

,

0 < q < 1

and suppose that relation (5) holds. When

F_{Ψ_{λ}^{m, n, q} (U)} Ψ_{λ}^{m, n, q} (β, ψ, ε, δ; z) \leq F_{g (U)} (β + ψ {(\frac{z + 1}{1 - z})}^{α} + ε {(\frac{z + 1}{1 - z})}^{2 α} + \frac{2 α δ z}{1 - z^{2}}),

with

β, ψ, δ, ε \in C

,

0 < α \leq 1

,

δ \neq 0,

and

Ψ_{λ}^{m, n, q}

defined by relation (6), then

F_{D_{z}^{- λ} ϕ (U)} (\frac{z {(D_{z}^{- λ} ϕ (m, n; q, z))}^{'}}{D_{z}^{- λ} ϕ (m, n; q, z)}) \leq F_{g (U)} {(\frac{z + 1}{1 - z})}^{α},

and the fuzzy best dominant is

{(\frac{z + 1}{1 - z})}^{α} .

Proof.

For

g (z) = {(\frac{z + 1}{1 - z})}^{α}

,

0 < α \leq 1,

corollary follows by using Theorem 1. □

The next theorem and the corollaries associated involve fuzzy differential superordination aspects.

Theorem 2.

Consider g an analytic and univalent function in U with the properties

g (z) \neq 0

and

\frac{z g^{'} (z)}{g (z)}

is starlike univalent in U. Suppose that

R e (\frac{ψ}{δ} g (z) g^{'} (z) + \frac{2 ε}{δ} g^{2} (z) g^{'} (z)) > 0, for ψ, δ, ε \in C, δ \neq 0 .

(10)

For m, n complex numbers with

n \neq 0, - 1, - 2, \dots

;

λ > 0,

0 < q < 1

, when

\frac{z {(D_{z}^{- λ} ϕ (m, n; q, z))}^{'}}{D_{z}^{- λ} ϕ (m, n; q, z)} \in H [g (0), 1] \cap Q

and

Ψ_{λ}^{m, n, q} (β, ψ, ε, δ; z)

defined by relation (6) is univalent in U, then

F_{g (U)} (β + ψ g (z) + ε {(g (z))}^{2} + \frac{δ z g^{'} (z)}{g (z)}) \leq F_{Ψ_{λ}^{m, n, q} (U)} Ψ_{λ}^{m, n, q} (β, ψ, ε, δ; z)

(11)

implies

F_{g (U)} g (z) \leq F_{D_{z}^{- λ} ϕ (U)} (\frac{z {(D_{z}^{- λ} ϕ (m, n; q, z))}^{'}}{D_{z}^{- λ} ϕ (m, n; q, z)}), z \in U,

(12)

and the fuzzy best subordinant is g.

Proof.

Define

p (z) : = \frac{z {(D_{z}^{- λ} ϕ (m, n; q, z))}^{'}}{D_{z}^{- λ} ϕ (m, n; q, z)}

,

z \in U

,

z \neq 0

.

Let

φ (u) : = β + ψ u + ε u^{2}

and

γ (u) : = \frac{δ}{u}

and it is easy to show that

φ

is analytic in

C

,

γ

is analytic in

C \ {0}

with

γ (u) \neq 0

,

u \in C \ {0} .

We can write

\frac{φ^{'} (g (z))}{γ (g (z))} = \frac{g^{'} (z) g (z) [ψ + 2 ε g (z)]}{τ}

, and

R e (\frac{θ^{'} (g (z))}{η (g (z))}) = R e (\frac{ψ}{δ} g (z) g^{'} (z) + \frac{2 ε}{δ} g^{2} (z) g^{'} (z)) > 0

, for

ψ, δ, ε \in C,

δ \neq 0 .

By using relations (9) and (11) we get

F_{g (U)} (β + ψ g (z) + ε {(g (z))}^{2} + \frac{δ z g^{'} (z)}{g (z)}) \leq

F_{p (U)} (β + ψ p (z) + ε {(p (z))}^{2} + \frac{δ z p^{'} (z)}{p (z)}) .

Applying Lemma 2, we obtain

F_{g (U)} g (z) \leq F_{D_{z}^{- λ} ϕ (U)} (\frac{z {(D_{z}^{- λ} ϕ (m, n; q, z))}^{'}}{D_{z}^{- λ} ϕ (m, n; q, z)}), z \in U,

and the fuzzy best subordinant is g. □

Example 2.

Taking the same functions as in Example 1,

ϕ (3, 2, q, z) = 1 + \frac{2}{1 - q} z + \frac{2 (1 - 3 q)}{1 - 2 q} z^{2}

for

m = 3

,

n = 2

,

z \in U

, with

D_{z}^{- λ} ϕ (3, 2, q, z) = \frac{z^{λ - 1}}{Γ (λ)}

,

z \in U,

and

g (z) = \frac{1 - z}{1 + z},

z \in U

with

g^{'} (z) = \frac{- 2}{{(1 + z)}^{2}}

and

g^{″} (z) = \frac{4}{{(1 + z)}^{3}}

,

z \in U

.

Choosing

β = 0

,

δ = 1

,

ψ = 1

,

ε = 1

, we get

Ψ_{λ}^{3, 2, q} (0, 1, 1, 1; z) = λ^{2} - λ

.

Using Theorem 2, when Re

\frac{- 6 z^{2} + 8 z - 6}{{(1 + z)}^{4}} > 0

and

\frac{z g^{'} (z)}{g (z)} = \frac{- 2 z}{1 - z^{2}}

is starlike univalent in U, we obtain

F_{U} (\frac{- 6 z + 2}{{(1 + z)}^{2} (1 - z)}) \leq F_{U} (λ^{2} - λ)

,

z \in U,

induce

F_{U} (\frac{1 - z}{1 + z}) \leq F_{U} (λ - 1) .

Corollary 3.

For m, n complex numbers with

n \neq 0, - 1, - 2, \dots

;

λ > 0

,

0 < q < 1,

suppose that relation (10) holds. When

\frac{z {(D_{z}^{- λ} ϕ (m, n; q, z))}^{'}}{D_{z}^{- λ} ϕ (m, n; q, z)} \in H [g (0), 1] \cap Q

and

F_{g (U)} (β + ψ \frac{M z + 1}{N z + 1} + ε {(\frac{M z + 1}{N z + 1})}^{2} + \frac{δ (M - N) z}{(M z + 1) (N z + 1)})

\leq F_{Ψ_{λ}^{m, n, q} (U)} Ψ_{λ}^{m, n, q} (β, ψ, ε, δ; z),

where

β, ψ, δ, ε \in C

,

δ \neq 0,

- 1 \leq N < M \leq 1

, and

Ψ_{λ}^{m, n, q}

defined by relation (6), then

F_{g (U)} (\frac{M z + 1}{N z + 1}) \leq F_{D_{z}^{- λ} ϕ (U)} (\frac{z {(D_{z}^{- λ} ϕ (m, n; q, z))}^{'}}{D_{z}^{- λ} ϕ (m, n; q, z)}),

and the fuzzy best subordinant is

\frac{M z + 1}{N z + 1}

.

Proof.

Considering

g (z) = \frac{M z + 1}{N z + 1}

,

- 1 \leq N < M \leq 1

in Theorem 2 we have the corollary. □

Corollary 4.

For m, n complex numbers with

n \neq 0, - 1, - 2, \dots

;

λ > 0

,

0 < q < 1,

suppose that relation (10) holds. When

\frac{z {(D_{z}^{- λ} ϕ (m, n; q, z))}^{'}}{D_{z}^{- λ} ϕ (m, n; q, z)} \in H [g (0), 1] \cap Q

and

F_{g (U)} (β + ψ {(\frac{z + 1}{1 - z})}^{α} + ε {(\frac{z + 1}{1 - z})}^{2 α} + \frac{2 α δ z}{1 - z^{2}}) \leq F_{Ψ_{λ}^{m, n, q} (U)} Ψ_{λ}^{m, n, q} (β, ψ, ε, δ; z),

where

β, ψ, δ, ε \in C

,

δ \neq 0,

0 < α \leq 1

, and

Ψ_{λ}^{m, n, q}

defined by relation (6), then

F_{g (U)} {(\frac{z + 1}{1 - z})}^{α} \leq F_{D_{z}^{- λ} ϕ (U)} \frac{z {(D_{z}^{- λ} ϕ (m, n; q, z))}^{'}}{D_{z}^{- λ} ϕ (m, n; q, z)},

and the fuzzy best subordinant is

{(\frac{z + 1}{1 - z})}^{α}

.

Proof.

Letting

g (z) = {(\frac{z + 1}{1 - z})}^{α}

,

0 < α \leq 1

in Theorem 2 we obtain the corollary. □

Combining the results obtained in Theorems 1 and 2, the following sandwich-type result can be established.

Theorem 3.

Consider

g_{1},

g_{2}

analytic and univalent functions in U with the properties

g_{1} (z) \neq 0, g_{2} (z) \neq 0

, for all

z \in U

, and

\frac{z g_{1}^{'} (z)}{g_{1} (z)},

\frac{z g_{2}^{'} (z)}{g_{2} (z)}

are starlike univalent functions. Assume that

g_{1}

satisfies relation (5) and

g_{2}

satisfies relation (10). For m, n complex numbers with

n \neq 0, - 1, - 2, \dots;

λ > 0

,

0 < q < 1,

when

\frac{z {(D_{z}^{- λ} ϕ (m, n; q, z))}^{'}}{D_{z}^{- λ} ϕ (m, n; q, z)} \in H [g (0), 1] \cap Q

and

Ψ_{λ}^{m, n, q} (β, ψ, ε, δ; z)

defined by relation (6) is univalent in U, then

F_{g_{1} (U)} (β + ψ g_{1} (z) + ε {(g_{1} (z))}^{2} + \frac{δ z g_{1}^{'} (z)}{g_{1} (z)}) \leq F_{Ψ_{λ}^{m, n, q} (U)} Ψ_{λ}^{m, n, q} (β, ψ, ε, δ; z)

\leq F_{g_{2} (U)} (β + ψ g_{2} (z) + ε {(g_{2} (z))}^{2} + \frac{δ z g_{2}^{'} (z)}{g_{2} (z)}),

for

β, ψ, δ, ε \in C

,

δ \neq 0

, implies

F_{g_{1} (U)} g_{1} (z) \leq F_{D_{z}^{- λ} ϕ (U)} (\frac{z {(D_{z}^{- λ} ϕ (m, n; q, z))}^{'}}{D_{z}^{- λ} ϕ (m, n; q, z)}) \leq F_{g_{2} (U)} g_{2} (z),

and the fuzzy best subordinant is

g_{1}

and the fuzzy best dominant is

g_{2}

.

Considering

g_{1} (z) = \frac{M_{1} z + 1}{N_{1} z + 1}

and

g_{2} (z) = \frac{M_{2} z + 1}{N_{2} z + 1}

, with

- 1 \leq N_{2} < N_{1} < M_{1} < M_{2} \leq 1

, we get the following corollary.

Corollary 5.

For m, n complex numbers with

n \neq 0, - 1, - 2, \dots

;

λ > 0

,

0 < q < 1,

suppose that relations (5) and (10) hold. When

\frac{z {(D_{z}^{- λ} ϕ (m, n; q, z))}^{'}}{D_{z}^{- λ} ϕ (m, n; q, z)} \in H [g (0), 1] \cap Q

and

F_{g_{1} (U)} (β + ψ \frac{M_{1} z + 1}{N_{1} z + 1} + ε {(\frac{M_{1} z + 1}{N_{1} z + 1})}^{2} + δ \frac{(M_{1} - N_{1}) z}{(M_{1} z + 1) (N_{1} z + 1)})

\leq F_{Ψ_{λ}^{m, n, q} (U)} Ψ_{λ}^{m, n, q} (β, ψ, ε, δ; z) \leq

F_{g_{2} (U)} (β + ψ \frac{M_{2} z + 1}{N_{2} z + 1} + ε {(\frac{M_{2} z + 1}{N_{2} z + 1})}^{2} + δ \frac{(M_{2} - N_{2}) z}{(M_{2} z + 1) (N_{2} z + 1)}),

with

β, ψ, δ, ε \in C

,

τ δ \neq 0,

- 1 \leq N_{2} \leq N_{1} < M_{1} \leq M_{2} \leq 1

, and

Ψ_{λ}^{m, n, q}

defined by relation (6), then

F_{g_{1} (U)} (\frac{M_{1} z + 1}{N_{1} z + 1}) \leq F_{D_{z}^{- λ} ϕ (U)} (\frac{z {(D_{z}^{- λ} ϕ (m, n; q, z))}^{'}}{D_{z}^{- λ} ϕ (m, n; q, z)}) \leq F_{g_{2} (U)} (\frac{M_{2} z + 1}{N_{2} z + 1}),

the fuzzy best subordinant is

\frac{M_{1} z + 1}{N_{1} z + 1}

and the fuzzy best dominant is

\frac{M_{2} z + 1}{N_{2} z + 1}

, respectively.

Let

g_{1} (z) = {(\frac{z + 1}{1 - z})}^{α_{1}}

and

g_{2} (z) = {(\frac{z + 1}{1 - z})}^{α_{2}}

, with

0 < α_{1} < α_{2} \leq 1

, we get the following corollary.

Corollary 6.

For m, n complex numbers with

n \neq 0, - 1, - 2, \dots

;

λ > 0

,

0 < q < 1,

suppose that relations (5) and (10) hold. When

\frac{z {(D_{z}^{- λ} ϕ (m, n; q, z))}^{'}}{D_{z}^{- λ} ϕ (m, n; q, z)}

\in H [g (0), 1] \cap Q

and

F_{g_{1} (U)} (β + ψ {(\frac{z + 1}{1 - z})}^{α_{1}} + ε {(\frac{z + 1}{1 - z})}^{2 α_{1}} + \frac{2 α_{1} δ z}{1 - z^{2}}) \leq F_{Ψ_{λ}^{m, n, q} (U)} Ψ_{λ}^{m, n, q} (β, ψ, ε, δ; z)

\leq F_{g_{2} (U)} (β + ψ {(\frac{z + 1}{1 - z})}^{α_{2}} + ε {(\frac{z + 1}{1 - z})}^{2 α_{2}} + \frac{2 α_{2} δ z}{1 - z^{2}}),

with

β, ψ, δ, ε \in C

,

δ \neq 0,

0 < α_{1} < α_{2} \leq 1

, and

Ψ_{λ}^{m, n, q}

defined by relation (6), then

F_{g_{1} (U)} {(\frac{z + 1}{1 - z})}^{α_{1}} \leq F_{D_{z}^{- λ} ϕ (U)} \frac{z {(D_{z}^{- λ} ϕ (m, n; q, z))}^{'}}{D_{z}^{- λ} ϕ (m, n; q, z)} \leq F_{g_{2} (U)} {(\frac{z + 1}{1 - z})}^{α_{2}},

the fuzzy best subordinant is

{(\frac{z + 1}{1 - z})}^{α_{1}}

and the fuzzy best dominant is

{(\frac{z + 1}{1 - z})}^{α_{2}}

, respectively.

Considering

φ (u) : = β u

and

γ (u) : = δ

in Lemmas 1 and 2, we obtain the following results regarding fuzzy differential subordination:

Theorem 4.

Consider g a convex and univalent function in U such that

g (0) = λ

,

z \in U

, and

\frac{z {(D_{z}^{- λ} ϕ (m, n; q, z))}^{'}}{D_{z}^{- λ} ϕ (m, n; q, z)} \in H (U),

z \in U

, where

λ > 0

,

0 < q < 1

, m, n are complex numbers with

n \neq 0, - 1, - 2, \dots .

Suppose that

R e (\frac{β + δ}{δ} + z \frac{g^{″} (z)}{g^{'} (z)}) > 0,

(13)

for

β, δ \in C

, δ

\neq 0

,

z \in U,

and

Ψ_{λ}^{m, n, q} (β, δ; q, z) : = (β + δ) \frac{z {(D_{z}^{- λ} ϕ (m, n; q, z))}^{'}}{D_{z}^{- λ} ϕ (m, n; q, z)} + δ \frac{z^{2} {(D_{z}^{- λ} ϕ (m, n; q, z))}^{″}}{D_{z}^{- λ} ϕ (m, n; q, z)}

(14)

- δ {(\frac{z {(D_{z}^{- λ} ϕ (m, n; q, z))}^{'}}{D_{z}^{- λ} ϕ (m, n; q, z)})}^{2} .

If g verifies the fuzzy differential subordination

F_{Ψ_{λ}^{m, n, q} (U)} Ψ_{λ}^{m, n, q} (β, δ; q, z) \leq F_{g (U)} (β g (z) + δ z g^{'} (z)),

(15)

for

β, δ \in C

, δ

\neq 0,

z \in U,

then

F_{D_{z}^{- λ} ϕ (U)} (\frac{z {(D_{z}^{- λ} ϕ (m, n; q, z))}^{'}}{D_{z}^{- λ} ϕ (m, n; q, z)}) \leq F_{g (U)} g (z), z \in U,

(16)

and the fuzzy best dominant is g.

Proof.

Define again

p (z) : = \frac{z {(D_{z}^{- λ} ϕ (m, n; q, z))}^{'}}{D_{z}^{- λ} ϕ (m, n; q, z)}

,

z \in U

,

z \neq 0

, which is analytic in U and

p (0) = λ .

Differentiating it we can write

p^{'} (z) = \frac{{(D_{z}^{- λ} ϕ (m, n; q, z))}^{'}}{D_{z}^{- λ} ϕ (m, n; q, z)} + z \frac{{(D_{z}^{- λ} ϕ (m, n; q, z))}^{″}}{D_{z}^{- λ} ϕ (m, n; q, z)} - z {(\frac{{(D_{z}^{- λ} ϕ (m, n; q, z))}^{'}}{D_{z}^{- λ} ϕ (m, n; q, z)})}^{2}

and

z p^{'} (z) = \frac{z {(D_{z}^{- λ} ϕ (m, n; q, z))}^{'}}{D_{z}^{- λ} ϕ (m, n; q, z)} + \frac{z^{2} {(D_{z}^{- λ} ϕ (m, n; q, z))}^{″}}{D_{z}^{- λ} ϕ (m, n; q, z)} - {(\frac{z {(D_{z}^{- λ} ϕ (m, n; q, z))}^{'}}{D_{z}^{- λ} ϕ (m, n; q, z)})}^{2} .

(17)

Set

φ (u) : = β u

and

γ (u) : = δ

, we can show easily that

φ

is analytic in

C

,

γ

is analytic in

C \ {0}

and

γ (u) \neq 0,

u \in C \ {0} .

Also, we set

G (z) = z g^{'} (z) γ (g (z)) = δ z g^{'} (z),

which is starlike univalent in U.

Let

h (z) = G (z) + φ (g (z)) = β g (z) + δ z g^{'} (z)

.

We get

R e (\frac{z h^{'} (z)}{G (z)}) = R e (\frac{β + δ}{δ} + z \frac{g^{″} (z)}{g^{'} (z)}) > 0

.

Taking account the relation (17), we can write

β p (z) + δ z p^{'} (z) =

(β + δ) \frac{z {(D_{z}^{- λ} ϕ (m, n; q, z))}^{'}}{D_{z}^{- λ} ϕ (m, n; q, z)} + δ \frac{z^{2} {(D_{z}^{- λ} ϕ (m, n; q, z))}^{″}}{D_{z}^{- λ} ϕ (m, n; q, z)} - δ {(\frac{z {(D_{z}^{- λ} ϕ (m, n; q, z))}^{'}}{D_{z}^{- λ} ϕ (m, n; q, z)})}^{2} .

Using fuzzy differential subordination (15), we get

F_{p (U)} (β p (z) + δ z p^{'} (z)) \leq F_{g (U)} (β g (z) + δ z g^{'} (z)) .

Applying Lemma 1, we obtain

F_{p (U)} p (z) \leq F_{g (U)} g (z)

,

z \in U,

i.e.,

F_{D_{z}^{- λ} ϕ (U)} (\frac{z {(D_{z}^{- λ} ϕ (m, n; q, z))}^{'}}{D_{z}^{- λ} ϕ (m, n; q, z)}) \leq F_{g (U)} g (z)

,

z \in U,

and the fuzzy best dominant is g. □

Corollary 7.

Consider

λ > 0,

0 < q < 1,

m, n complex numbers with

n \neq 0, - 1, - 2, \dots

and

g (z) = \frac{M z + 1}{N z + 1}

,

z \in U

,

- 1 \leq N < M \leq 1 .

Suppose that relation (13) holds. When

F_{Ψ_{λ}^{m, n, q} (U)} Ψ_{λ}^{m, n, q} (β, δ; q, z) \leq F_{g (U)} (β \frac{M z + 1}{N z + 1} + \frac{δ (M - N) z}{{(N z + 1)}^{2}}),

with

β, δ \in C

,

δ \neq 0

,

- 1 \leq N < M \leq 1

, and

Ψ_{λ}^{m, n, q}

defined by relation (14), then

F_{D_{z}^{- λ} ϕ (U)} (\frac{z {(D_{z}^{- λ} ϕ (m, n; q, z))}^{'}}{D_{z}^{- λ} ϕ (m, n; q, z)}) \leq F_{g (U)} (\frac{M z + 1}{N z + 1}),

and the fuzzy best dominant is

\frac{M z + 1}{N z + 1} .

Proof.

Consider

g (z) = \frac{M z + 1}{N z + 1}

,

- 1 \leq N < M \leq 1,

in Theorem 4 and we obtain the corollary. □

Corollary 8.

Consider

λ > 0

,

0 < q < 1,

m, n complex numbers with

n \neq 0, - 1, - 2, \dots

and

g (z) = {(\frac{z + 1}{1 - z})}^{α} .

Suppose that relation (13) holds. When

F_{Ψ_{λ}^{m, n, q} (U)} Ψ_{λ}^{m, n, q} (β, δ; q, z) \leq F_{g (U)} (β {(\frac{z + 1}{1 - z})}^{α} + \frac{2 α δ z}{1 - z^{2}} {(\frac{z + 1}{1 - z})}^{α}),

with

β, δ \in C

,

0 < α \leq 1

,

δ \neq 0,

and

Ψ_{λ}^{m, n, q}

defined by relation (14), then

F_{D_{z}^{- λ} ϕ (U)} (\frac{z {(D_{z}^{- λ} ϕ (m, n; q, z))}^{'}}{D_{z}^{- λ} ϕ (m, n; q, z)}) \leq F_{g (U)} {(\frac{z + 1}{1 - z})}^{α}, z \in U,

and the fuzzy best dominant is

{(\frac{z + 1}{1 - z})}^{α} .

Proof.

Corollary follows considering in Theorem 4

g (z) = {(\frac{z + 1}{1 - z})}^{α}

,

0 < α \leq 1 .

□

The next theorem and the corollaries associated involve fuzzy differential superordination aspects, too.

Theorem 5.

Consider g a convex and univalent function in U such that

g (0) = λ

, with

0 < q < 1

,

λ > 0,

m, n complex numbers with

n \neq 0, - 1, - 2, \dots

. Suppose that

R e (\frac{β}{δ} g^{'} (z)) > 0, for β, δ \in C, δ \neq 0 .

(18)

When

\frac{z {(D_{z}^{- λ} ϕ (m, n; q, z))}^{'}}{D_{z}^{- λ} ϕ (m, n; q, z)}

\in H [g (0), 1] \cap Q

and

Ψ_{λ}^{m, n, q} (β, δ; q, z)

defined by relation (14) is univalent in U, then

F_{g (U)} (β g (z) + δ z g^{'} (z)) \leq F_{Ψ_{λ}^{m, n, q} (U)} Ψ_{λ}^{m, n, q} (β, δ; q, z)

(19)

implies

F_{g (U)} g (z) \leq F_{D_{z}^{- λ} ϕ (U)} \frac{z {(D_{z}^{- λ} ϕ (m, n; q, z))}^{'}}{D_{z}^{- λ} ϕ (m, n; q, z)}, z \in U,

(20)

and the fuzzy best subordinant is g.

Proof.

Define also

p (z) : = \frac{z {(D_{z}^{- λ} ϕ (m, n; q, z))}^{'}}{D_{z}^{- λ} ϕ (m, n; q, z)}

,

z \in U

,

z \neq 0

, which is analytic in U and

p (0) = λ .

Let

φ (u) : = β u

and

γ (u) : = δ,

it is easy to show that

φ

is analytic in

C

,

γ

is analytic in

C \ {0}

with

γ (u) \neq 0

,

u \in C \ {0} .

Since

\frac{φ^{'} (g (z))}{γ (g (z))} = \frac{β}{δ} g^{'} (z)

, it yields that

R e (\frac{φ^{'} (g (z))}{γ (g (z))}) = R e (\frac{β}{δ} g^{'} (z)) > 0

, for

β, δ \in C,

δ \neq 0 .

The fuzzy differential superordination (19) can be written as

F_{g (U)} (β g (z) + δ z g^{'} (z)) \leq F_{p (U)} (β p (z) + δ z p^{'} (z)), z \in U,

and applying Lemma 2, we obtain

F_{g (U)} g (z) \leq F_{D_{z}^{- λ} ϕ (U)} (\frac{z {(D_{z}^{- λ} ϕ (m, n; q, z))}^{'}}{D_{z}^{- λ} ϕ (m, n; q, z)}), z \in U,

and the fuzzy best subordinant is g. □

Corollary 9.

For

λ > 0

,

0 < q < 1

, m, n complex numbers with

n \neq 0, - 1, - 2, \dots

and

g (z) = \frac{M z + 1}{N z + 1}

,

- 1 \leq N < M \leq 1,

z \in U,

suppose that relation (18) holds. When

\frac{z {(D_{z}^{- λ} ϕ (m, n; q, z))}^{'}}{D_{z}^{- λ} ϕ (m, n; q, z)} \in H [g (0), 1] \cap Q,

and

F_{g (U)} (β \frac{M z + 1}{N z + 1} + \frac{δ (M - N) z}{{(N z + 1)}^{2}}) \leq F_{Ψ_{λ}^{m, n, q} (U)} Ψ_{λ}^{m, n, q} (β, δ; q, z),

with

β, δ \in C,

δ \neq 0,

- 1 \leq N < M \leq 1

, and

Ψ_{λ}^{m, n, q}

defined by relation (14), then

F_{g (U)} (\frac{M z + 1}{N z + 1}) \leq F_{D_{z}^{- λ} ϕ (U)} (\frac{z {(D_{z}^{- λ} ϕ (m, n; q, z))}^{'}}{D_{z}^{- λ} ϕ (m, n; q, z)}),

and the fuzzy best subordinant is

\frac{M z + 1}{N z + 1}

.

Proof.

Considering

g (z) = \frac{M z + 1}{N z + 1}

,

- 1 \leq N < M \leq 1,

in Theorem 5 we obtain the corollary. □

Corollary 10.

For

λ > 0

,

0 < q < 1,

m, n complex numbers with

n \neq 0, - 1, - 2, \dots

and

g (z) = {(\frac{z + 1}{1 - z})}^{α},

suppose that relation (18) holds. When

\frac{z {(D_{z}^{- λ} ϕ (m, n; q, z))}^{'}}{D_{z}^{- λ} ϕ (m, n; q, z)} \in H [g (0), 1] \cap Q

and

F_{g (U)} (β {(\frac{z + 1}{1 - z})}^{α} + \frac{2 α δ z}{1 - z^{2}} {(\frac{z + 1}{1 - z})}^{α}) \leq F_{Ψ_{λ}^{m, n, q} (U)} Ψ_{λ}^{m, n, q} (β, δ; q, z),

with

β, δ \in C

,

0 < α \leq 1

,

δ \neq 0,

and

Ψ_{λ}^{m, n, q}

defined by relation (14), then

F_{g (U)} {(\frac{z + 1}{1 - z})}^{α} \leq F_{D_{z}^{- λ} ϕ (U)} (\frac{z {(D_{z}^{- λ} ϕ (m, n; q, z))}^{'}}{D_{z}^{- λ} ϕ (m, n; q, z)}),

and the fuzzy best subordinant is

{(\frac{z + 1}{1 - z})}^{α} .

Proof.

Corollary follows considering in Theorem 5

g (z) = {(\frac{z + 1}{1 - z})}^{α}

,

0 < α \leq 1 .

□

Combining the results obtained in Theorems 4 and 5, the following sandwich-type result can be established.

Theorem 6.

Consider

g_{1},

g_{2}

convex and univalent functions in U with the properties

g_{1} (z) \neq 0,

g_{2} (z) \neq 0

, for all

z \in U

. Assume that

g_{1}

satisfies relation (13) and

g_{2}

satisfies relation (18). For

λ > 0,

0 < q < 1

, m, n complex numbers with

n \neq 0, - 1, - 2, \dots,

when

\frac{z {(D_{z}^{- λ} ϕ (m, n; q, z))}^{'}}{D_{z}^{- λ} ϕ (m, n; q, z)} \in H [g (0), 1] \cap Q

, and

Ψ_{λ}^{m, n, q} (β, δ; q, z)

defined by (14) is univalent in U, then

F_{g_{1} (U)} (β g_{1} (z) + δ z g_{1}^{'} (z)) \leq F_{Ψ_{λ}^{m, n, q} (U)} Ψ_{λ}^{m, n, q} (β, δ; q, z) \leq F_{g_{2} (U)} (β g_{2} (z) + δ z g_{2}^{'} (z)),

for

β, δ \in C,

δ \neq 0,

implies

F_{g_{1} (U)} g_{1} (z) \leq F_{D_{z}^{- λ} ϕ (U)} (\frac{z {(D_{z}^{- λ} ϕ (m, n; q, z))}^{'}}{D_{z}^{- λ} ϕ (m, n; q, z)}) \leq F_{g_{2} (U)} g_{2} (z), z \in U,

and the fuzzy best subordinant is

g_{1}

and the fuzzy best dominant is

g_{2}

.

Considering

g_{1} (z) = \frac{M_{1} z + 1}{N_{1} z + 1}

and

g_{2} (z) = \frac{M_{2} z + 1}{N_{2} z + 1}

, with

- 1 \leq N_{2} < N_{1} < M_{1} < M_{2} \leq 1

, we get the following corollary.

Corollary 11.

For m, n complex numbers with

n \neq 0, - 1, - 2, \dots

;

λ > 0

,

0 < q < 1,

suppose that relations (13) and (18) hold for

g_{1} (z) = \frac{M_{1} z + 1}{N_{1} z + 1}

and

g_{2} (z) = \frac{M_{2} z + 1}{N_{2} z + 1}

, respectively. When

\frac{z {(D_{z}^{- λ} ϕ (m, n; q, z))}^{'}}{D_{z}^{- λ} ϕ (m, n; q, z)} \in H [g (0), 1] \cap Q

and

F_{g_{1} (U)} (β \frac{M_{1} z + 1}{N_{1} z + 1} + \frac{δ (M_{1} - N_{1}) z}{{(N_{1} z + 1)}^{2}}) \leq F_{Ψ_{λ}^{m, n, q} (U)} Ψ_{λ}^{m, n, q} (β, δ; q, z)

\leq F_{g_{2} (U)} (β \frac{M_{2} z + 1}{N_{2} z + 1} + \frac{δ (M_{2} - N_{2}) z}{{(N_{2} z + 1)}^{2}}), z \in U,

with

β, δ \in C,

δ \neq 0,

- 1 \leq N_{2} \leq N_{1} < M_{1} \leq M_{2} \leq 1

, and

Ψ_{λ}^{m, n, q}

defined by relation (14), then

F_{g_{1} (U)} (\frac{M_{1} z + 1}{N_{1} z + 1}) \leq F_{D_{z}^{- λ} ϕ (U)} (\frac{z {(D_{z}^{- λ} ϕ (m, n; q, z))}^{'}}{D_{z}^{- λ} ϕ (m, n; q, z)}) \leq F_{g_{2} (U)} (\frac{M_{2} z + 1}{N_{2} z + 1}), z \in U,

the fuzzy best subordinant is

\frac{M_{1} z + 1}{N_{1} z + 1}

and the fuzzy best dominant is

\frac{M_{2} z + 1}{N_{2} z + 1}

, respectively.

Considering

g_{1} (z) = {(\frac{z + 1}{1 - z})}^{α_{1}}

and

g_{2} (z) = {(\frac{z + 1}{1 - z})}^{α_{2}}

, with

0 < α_{1} < α_{2} \leq 1

, we obtain the following corollary.

Corollary 12.

For m, n complex numbers with

n \neq 0, - 1, - 2, \dots

;

λ > 0

,

0 < q < 1

, suppose that relations (13) and (18) hold for

g_{1} (z) = {(\frac{z + 1}{1 - z})}^{α_{1}}

and

g_{2} (z) = {(\frac{z + 1}{1 - z})}^{α_{2}}

, respectively. When

\frac{z {(D_{z}^{- λ} ϕ (m, n; q, z))}^{'}}{D_{z}^{- λ} ϕ (m, n; q, z)} \in H [g (0), 1] \cap Q

and

F_{g_{1} (U)} (β {(\frac{z + 1}{1 - z})}^{α_{1}} + \frac{2 α_{1} δ z}{1 - z^{2}} {(\frac{z + 1}{1 - z})}^{α_{1}}) \leq F_{Ψ_{λ}^{m, n, q} (U)} Ψ_{λ}^{m, n, q} (β, δ; q, z)

\leq F_{g_{2} (U)} (β {(\frac{z + 1}{1 - z})}^{α_{2}} + \frac{2 α_{2} δ z}{1 - z^{2}} {(\frac{z + 1}{1 - z})}^{α_{2}}), z \in U,

with

β, δ \in C,

δ \neq 0,

0 < α_{1} < α_{2} \leq 1

, and

Ψ_{λ}^{m, n, q}

defined by relation (14), then

F_{g_{1} (U)} {(\frac{z + 1}{1 - z})}^{α_{1}} \leq F_{D_{z}^{- λ} ϕ (U)} (\frac{z {(D_{z}^{- λ} ϕ (a, b; q, z))}^{'}}{D_{z}^{- λ} ϕ (a, b; q, z)}) \leq F_{g_{2} (U)} {(\frac{z + 1}{1 - z})}^{α_{2}}, z \in U,

the fuzzy best subordinant is

{(\frac{z + 1}{1 - z})}^{α_{1}}

and the fuzzy best dominant is

{(\frac{z + 1}{1 - z})}^{α_{2}}

, respectively.

3. Conclusions

This paper presents new fuzzy differential subordinations and superordinations obtained by involving in the studies the operator presented in (1). The first theorem refers to a new fuzzy differential subordination for which the fuzzy best dominant is provided. Two corollaries follow as applications of the new result by considering two particular functions with nice geometric properties as fuzzy best subordinant of the first fuzzy differential subordination investigated in this study. Next, similar results are presented regarding a fuzzy differential superordination for which the fuzzy best dominant is obtained and two corollaries follow as applications. The sandwich-type theorem connecting Theorems 1 and 2 is stated as Theorem 3 and the corollaries formulated as a consequence of this theorem appear naturally by combining the results presented in the corollaries related to Theorems 1 and 2. Finally, a special fuzzy differential subordination is investigated in Theorem 4 and the dual result concerning a fuzzy differential superordination is contained in Theorem 5. Naturally, specific corollaries also accompany these theorems.

As future applications of the results presented in this paper, they could serve as inspiration for Riemann–Liouville fractional integral to be applied on other q-calculus functions and operators. Symmetry properties might be investigated regarding the operator shown in (1) considering the fuzzy subordinations and superordinations obtained here. In addition, classes of analytic functions could be introduced due to the geometric properties which can be interpreted from the corollaries. Those classes could also be investigated related to symmetry properties given by the use of a fractional operator in their definition.

Keeping in mind the notable applications of fuzzy sets theory and fractional calculus in real life contexts as it can be seen for example in [59,60,61], hopefully, this new fractional q-operator will find applications in future studies.

Author Contributions

Conceptualization, A.A.L. and G.I.O.; methodology, G.I.O.; software, A.A.L.; validation, A.A.L. and G.I.O.; formal analysis, A.A.L. and G.I.O.; investigation, A.A.L.; resources, G.I.O.; data curation, G.I.O.; writing—original draft preparation, A.A.L.; writing—review and editing, A.A.L. and G.I.O.; visualization, A.A.L.; supervision, G.I.O.; project administration, A.A.L.; funding acquisition, G.I.O. All authors have read and agreed to the published version of the manuscript.

Funding

This research received no external funding.

Institutional Review Board Statement

Not applicable.

Informed Consent Statement

Not applicable.

Data Availability Statement

Not applicable.

Conflicts of Interest

The authors declare no conflict of interest.

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Alb Lupaş, A.; Oros, G.I. Applications of Riemann–Liouville Fractional Integral of q-Hypergeometric Function for Obtaining Fuzzy Differential Sandwich Results. Symmetry 2022, 14, 2097. https://doi.org/10.3390/sym14102097

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Alb Lupaş A, Oros GI. Applications of Riemann–Liouville Fractional Integral of q-Hypergeometric Function for Obtaining Fuzzy Differential Sandwich Results. Symmetry. 2022; 14(10):2097. https://doi.org/10.3390/sym14102097

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Alb Lupaş, Alina, and Georgia Irina Oros. 2022. "Applications of Riemann–Liouville Fractional Integral of q-Hypergeometric Function for Obtaining Fuzzy Differential Sandwich Results" Symmetry 14, no. 10: 2097. https://doi.org/10.3390/sym14102097

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Alb Lupaş, A., & Oros, G. I. (2022). Applications of Riemann–Liouville Fractional Integral of q-Hypergeometric Function for Obtaining Fuzzy Differential Sandwich Results. Symmetry, 14(10), 2097. https://doi.org/10.3390/sym14102097

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Applications of Riemann–Liouville Fractional Integral of q-Hypergeometric Function for Obtaining Fuzzy Differential Sandwich Results

Abstract

1. Introduction

2. Main Results

3. Conclusions

Author Contributions

Funding

Institutional Review Board Statement

Informed Consent Statement

Data Availability Statement

Conflicts of Interest

References

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