summary:We obtain several fuzzy differential subordinations by using a linear operator $\mathcal {I}_{m,\gamma }^{n,\alpha }f(z)=z+\sum \limits _{k=2}^{\infty }(1+\gamma ( k-1))^{n}m^{\alpha }(m+k)^{-\alpha }a_{k}z^{k}$. Using the linear operator $\mathcal {I}_{m,\gamma }^{n,\alpha },$ we also introduce a class of univalent analytic functions for which we give some properties

El-Deeb, Sheza M.

Oros, Georgia I.

Mathematica Bohemica

English

We obtain several fuzzy differential subordinations by using a linear operator $\mathcal{I}_{m,\gamma}^{n,\alpha}f(z)=z+\sum\limits_{k=2}^{\infty}(1+\gamma( k-1))^nm^{\alpha}(m+k)^{-\alpha}a_kz^k$. Using the linear operator $\mathcal{I}_{m,\gamma}^{n,\alpha},$ we also introduce a class of univalent analytic functions for which we give some properties

Sheza M. El-Deeb

Georgia I. Oros

Directory of Open Access Journals

146 (2021) MATHEMATICA BOHEMICA No. 4, 397–406
FUZZY DIFFERENTIAL SUBORDINATIONS CONNECTED WITH
THE LINEAR OPERATOR
Sheza M. El-Deeb, New Damietta, Georgia I. Oros, Oradea
Received November 15, 2019. Published online November 2, 2020.
Communicated by Dagmar Medková
Abstract. We obtain several fuzzy differential subordinations by using a linear operator
I
n,α
m,γf(z) = z +
∞∑
k=2
(1 + γ(k − 1))nmα(m+ k)−αakz
k. Using the linear operator In,αm,γ , we
also introduce a class of univalent analytic functions for which we give some properties.
Keywords: fuzzy differential subordination; fuzzy best dominant; linear operator
MSC 2020 : 30C45
1. Introduction
Let D ⊂ C,H(D) be the class of holomorphic functions on D and denote by Hn(D)
the class of holomorphic and univalent functions on D. In this paper, we denote by
H(U) the class of holomorphic functions in the open unit disk U = {z ∈ C : |z| < 1}
with ∂U = {z ∈ C : |z| = 1} the boundary of the unit disc. For a ∈ C and k ∈ N, we
denote
H[a, k] = {f ∈ H(U) : f(z) = a+ akz
k + ak+1z
k+1 + . . . , z ∈ U},
Ak = {f ∈ H(U) : f(z) = z + ak+1z
k+1 + . . . , z ∈ U} with A1 = A,
and
S ={f ∈ A : f is a univalent function in U}.
Denote by
K =
{
f ∈ A : ℜ
(
1 +
zf ′′(z)
f ′(z)
)
> 0, z ∈ U
}
the class of convex functions in U.
c© The author(s) 2020. This is an open access article under the CC BY-NC-ND licence cbnd
DOI: 10.21136/MB.2020.0159-19 397
Definition 1.1 ([7] and [13]). Let f and g be analytic functions in U. We say
that f is subordinate to g, written f ≺ g, if there exists a Schwarz function w,
which is analytic in U with w(0) = 0 and |w(z)| < 1 for all z ∈ U, such that
f(z) = g(w(z)). Furthermore, if the function g is univalent in U, then we have the
following equivalence:
f(z) ≺ g(z) ⇔ f(0) = g(0) and f(U) ⊂ g(U).
In order to introduce the notion of fuzzy differential subordination, we use the
following definitions and propositions:
Definition 1.2 ([10]). Assume that Y be a nonempty set. An application F :
Y →[0, 1] is called fuzzy subset. A pair (B,FB), where FB : Y →[0, 1] and
(1.1) B = {x ∈ Y : 0 < FB(x) 6 1} = sup(B,FB),
is called fuzzy set. The function FB is called membership function of the fuzzy set
(B,FB).
Proposition 1.1 ([14]).
(i) If (B,FB) = (U ,FU ), then we have B = U , where B = sup(B,FB) and U =
sup(U ,FU).
(ii) If (B,FB) ⊆ (U ,FU ), then we have B ⊆ U , where B = sup(B,FB) and U =
sup(U ,FU).
Let f, g ∈ H(D). We denote
(1.2) f(D) = {f(z) : 0 < Ff(D)f(z) 6 1, z ∈ D} = sup(f(D),Ff(D))
and
(1.3) g(D) = {g(z) : 0 < Fg(D)g(z) 6 1, z ∈ D} = sup(g(D),Fg(D)).
Definition 1.3 ([14]). Let z0 ∈ D and f, g ∈ H(D). The function f is said to be
fuzzy subordinate to g, written f ≺F g or f(z) ≺F g(z), when following conditions
are satisfied:
(i) f(z0) = g(z0),
(ii) Ff(D)f(z) 6 Fg(D)g(z), z ∈ D.
Proposition 1.2 ([14]). Assume that z0 ∈ D and f, g ∈ H(D). If f(z) ≺F g(z),
z ∈ D, then
(i) f(z0) = g(z0),
(ii) f(D) ⊆ g(D), Ff(D)f(z) 6 Fg(D)g(z), z ∈ D, where f(D) and g(D) are defined
by (1.2) and (1.3), respectively.
398
Definition 1.4 ([16]). Assume thatΦ: C3×U → C and h ∈ S with Φ(a, 0, 0; 0) =
h(0) = a. If p is analytic in U with p(0) = a and satisfies the second order fuzzy
differential subordination
FΦ(C3×U)Φ(p(z), zp
′(z), z2p′′(z); z) 6 Fh(U)h(z),(1.4)
i.e. Φ(p(z), zp′(z), z2p′′(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 for the fuzzy differential
subordination if
Fp(U)p(z) 6 Fq(U)q(z), i.e. p(z) ≺F q(z), z ∈ U
for all p satisfying (1.4). A fuzzy dominant q̃ that satisfies
Fq̃(U)q̃(z) 6 Fq(U)q(z), i.e. q̃(z) ≺F q(z), z ∈ U
for all fuzzy dominants q of (1.4) is said to be the fuzzy best dominant of (1.4).
The integral operator Kαm : A → A is defined for α > 0 and m > −1 as follows
(see Komatu [12] with p = 1):
K0mf(z) = f(z) = z +
∞
∑
k=1
ak+1z
k+1
and
(1.5) Kαmf(z) =
(m+ 1)α
Γ(α)zm
∫ z
0
tm−1
(
log
z
t
)α−1
f(t) dt.
It can be easily verified that
(1.6) Kαmf(z) = z +
∞
∑
k=1
( m+ 1
m+ k + 1
)α
ak+1z
k+1.
El-Ashwah et al. (see [9] with p = 1) introduced the linear multiplier operator In,αm,γ :
A → A given as:
I0,0m,γf(z) = f(z),
I1,αm,γf(z) = I
α
m,γf(z) = (1− γ)K
α
mf(z) + γz(K
α
mf(z))
′
= z +
∞
∑
k=1
(1 + γk)
( m+ 1
m+ k + 1
)α
ak+1z
k+1,
I2,αm,γf(z) = I
α
m,γ(I
1,α
m,γf(z))
= z +
∞
∑
k=1
(1 + γk)2
( m+ 1
m+ k + 1
)α
ak+1z
k+1,
399
in general,
(1.7) In,αm,γf(z) = I
α
m,γ(I
n−1,α
m,γ f(z)) = z +
∞
∑
k=1
(1 + γk)n
( m+ 1
m+ k + 1
)α
ak+1z
k+1,
α > 0, m > −1, γ > 0, n ∈ N0.
It is easily verified from (1.7) that
(1.8) z(In,αm,γf(z))
′ = (m+ 1)In,α−1m,γ f(z)−mI
n,α
m,γf(z)
and
(1.9) γz(In,αm,γf(z))
′ = In+1,αm,γ f(z)− (1− γ)I
n,α
m,γf(z).
By specializing the parameters m, γ, α and n, we obtain the following operators:
I0,δc,γf(z) = K
δ
c,1f(z) (see [11]);
I0,αa−1,γf(z) = L
α
af(z) (see [4] and [5]);
I1,αc,0 f(z) = P
α
c f(z) (see [12] and [17]);
I1,1c,0 f(z) = Lcf(z) (see [6]);
I1,α1,0 f(z) = I
αf(z) (see [8]);
In,0m,γf(z) = D
n
γf(z) (see [1]);
In,0m,1f(z) = D
nf(z) (see [18]).
Also, we note that
(1.10) I−n,αm,γ f(z) = z +
∞
∑
k=1
( 1
1 + γk
)n( m+ 1
m+ k + 1
)α
ak+1z
k+1.
From (1.7) and (1.10) we observe that the operator In,αm,γf(z) is well defined from
n ∈ Z = {0,±1,±2, . . .}.
By using the linear operator In,αm,γf(z) defined by (1.7), we will derive several fuzzy
diferential subordintions for this class.
Definition 1.5. A function f ∈ A belongs to the class MFm,γ(n, α, η) for all
η ∈ [0, 1), n ∈ N0, m > −1, γ > 0 and α > 0 if it satisfies the inequality
F(In,αm,γf)′(U)(I
n,α
m,γf(z))
′ > η, z ∈ U.
400
2. Preliminary
To prove our results, we need the following lemmas.
Lemma 2.1 ([13]). Let ψ ∈ A and G(z) = z−1
∫ z
0
ψ(t) dt, z ∈ U. If
ℜ(1 + zψ′′(z)/ψ′(z)) > −
1
2
, z ∈ U,
then G ∈ K.
Lemma 2.2 ([15], Theorem 2.6). Assume that h is a convex function with
h(0) = a and ν ∈ C∗ = C \ {0} with ℜ(ν) > 0. If p ∈ H[a, n] with p(0) = a,
Φ: C2 × U → C, Φ(p(z), zp′(z); z) = p(z) + ν−1zp′(z) is analytic function in U and
FΦ(C2×U)
(
p(z) +
1
ν
zp′(z)
)
6 Fh(U)h(z), i.e. p(z) +
1
ν
zp′(z) ≺F h(z), z ∈ U,
then
Fp(U)p(z) 6 Fq(U)q(z) 6 Fh(U)h(z), i.e. p(z) ≺F q(z), z ∈ U,
where
q(z) =
ν
nzν/n
∫ z
0
h(t)t−1+ν/n dt, z ∈ U.
The function q is convex and it is the fuzzy best dominant.
Lemma 2.3 ([15], Theorem 2.7). Let g be a convex function in U and let ψ(z) =
g(z) + nγzg′(z), where z ∈ U, n ∈ N and γ > 0. If
p(z) = g(0) + pnz
n + pn+1z
n+1 + . . .
is holomorphic in U and
Fp(U)(p(z) + γzp
′(z)) 6 Fψ(U)ψ(z), i.e. p(z) + γzp
′(z) ≺F ψ(z), z ∈ U,
then
Fp(U)(p(z)) 6 Fg(U)g(z), i.e. p(z) ≺F g(z), z ∈ U;
this result is sharp.
For the general theory of fuzzy differential subordination and its applications, we
refer the reader to [2], [12]–[14], [16].
The objective of the present section is to obtain several fuzzy differential subor-
dinations associated with the integral operator In,αm,γ by using the method of fuzzy
differential subordination.
401
3. Main results
Assume that η ∈ [0, 1), n ∈ N0, m > 0, α, γ > 0 and z ∈ U are mentioned through
this paper:
Theorem 3.1. Let k be a convex function in U and suppose that h(z) = k(z) +
zk′(z)/(λ+ 2). If f ∈ MFm,γ(n, α, η) and
(3.1) G(z) = Iλf(z) =
λ+ 2
zλ+1
∫ z
0
tλf(t) dt,
then
(3.2) F(In,αm,γf)′(U)(I
n,α
m,γf(z))
′ 6 Fh(U)h(z), i.e. (I
n,α
m,γf(z))
′ ≺F h(z),
implies that
F(In,αm,γG)′(U)(I
n,α
m,γG(z))
′ 6 Fk(U)k(z), i.e. (I
n,α
m,γG(z))
′ ≺F k(z);
this result is sharp.
P r o o f. Since
zλ+1G(z) = (λ+ 2)
∫ z
0
tλf(t) dt,
by differentiating, we obtain
(λ+ 1)G(z) + zG′(z) = (λ+ 2)f(z)
and
(3.3) (λ + 1)In,αm,γG(z) + z(I
n,α
m,γG(z))
′ = (λ+ 2)In,αm,γf(z),
and also by differentiating (3.3) we obtain
(3.4) (In,αm,γG(z))
′ +
1
(λ+ 2)
z(In,αm,γG(z))
′′ = (In,αm,γf(z))
′.
By using (3.4), the fuzzy differential subordination (3.2) becomes
(3.5) F(In,αm,γf)′(U)((I
n,α
m,γG(z))
′ +
1
(λ+ 2)
z(In,αm,γG(z))
′′)
6 Fh(U)
(
k(z) +
1
(λ+ 2)
zk′(z)
)
.
402
We denote
(3.6) q(z) = (In,αm,γG(z))
′ → q ∈ H[1, n].
From (3.6) in (3.5) we have
(3.7) F(In,αm,γf)′(U)
(
q(z) +
1
(λ+ 2)
zq′(z)
)
6 Fh(U)
(
k(z) +
1
(λ+ 2)
zk′(z)
)
,
by applying Lemma 2.3, we have
Fq(U)q(z) 6 Fk(U)k(z), i.e. F(In,αm,γG(z))′(U)(I
n,α
m,γG(z))
′ 6 Fk(U)k(z),
then (In,αm,γG(z))
′ ≺F k(z), k is the fuzzy best dominant. 
Theorem 3.2. Consider h(z) = (1 + (2η − 1)z)/(1 + z), η ∈ [0, 1), λ > 0 and Iλ
is given by (3.1). Then
(3.8) Iλ(MFm,γ(n, α, η)) ⊂ M
F
m,γ(n, α, ζ),
where
(3.9) ζ = 2η − 1 + (λ+ 2)(2− 2η)
∫ 1
0
tλ+2
t+ 1
dt.
P r o o f. The function h is convex and using the same technique as in the proof
of Theorem 3.1, we obtain from the hypothesis of Theorem 3.2 that
Fq(U)
(
q(z) +
1
(λ+ 2)
zq′(z)
)
6 Fh(U)h(z),
where q(z) is defined in (3.6). By using Lemma 2.2, we obtain that
Fq(U)q(z) 6 Fk(U)k(z) 6 Fh(U)h(z)
implies that
F(In,αm,γG)′(U)(I
n,α
m,γG(z))
′
6 Fk(U)k(z) 6 Fh(U)h(z),
where
k(z) =
λ+ 2
zλ+2
∫ z
0
tλ+1
1 + (2η − 1)t
1 + t
dt = (2η − 1) +
(λ + 2)(2− 2η)
zλ+2
∫ z
0
tλ+1
1 + t
dt.
Since k is convex and k(U) is symmetric with respect to the real axis, we conclude
(3.10) F(In,αm,γG)′(U)(I
n,α
m,γG(z))
′
> min
|z|=1
Fk(U)k(z) = Fk(U)k(1),
and ζ = k(1) = 2η− 1+ (λ+2)(2− 2η)
∫ 1
0
tλ+2/(t+ 1)dt. From (3.10), we conclude
that we have the inclusion in relation (3.8) and hence the proof of the theorem is
complete. 
403
Theorem 3.3. Assume that k is a convex function in U, k(0) = 1, and let h(z) =
k(z) + zk′(z). If f ∈ A and satisfies the fuzzy differential subordination
(3.11) F(In,αm,γf)′(U)(I
n,α
m,γf(z))
′ 6 Fh(U)h(z), i.e. (I
n,α
m,γf(z))
′ ≺F h(z),
then
(3.12) FIn,αm,γf(U)
In,αm,γf(z)
z
6 Fk(U)k(z), i.e.
In,αm,γf(z)
z
≺F k(z),
and this result is sharp.
P r o o f. By denoting
q(z) =
1
z
In,αm,γf(z) =
1
z
(
z +
∞
∑
k=2
(1 + γ(k − 1))n
( m
m+ k
)α
akz
k
)
= 1 +
∞
∑
k=2
(1 + γ(k − 1))n
( m
m+ k
)α
akz
k−1,
we obtain that q(z) + zq′(z) = (In,αm,γf(z))
′. Then
F(In,αm,γf)′(U)(I
n,α
m,γf(z))
′ 6 Fh(U)h(z)
→ Fq(U)(q(z) + zq
′(z)) 6 Fh(U)h(z) = Fk(U)(k(z) + zk
′(z)).
Applying Lemma 2.3, we have
Fq(U)q(z) 6 Fk(U)k(z) → FIn,αm,γf(U)
In,αm,γf(z)
z
6 Fk(U)k(z),
we get
In,αm,γf(z)
z
≺F k(z),
and this result is sharp. 
Theorem 3.4. For h ∈ H(U), h(0) = 1, which satisfies ℜ(1+zh′′(z)/h′(z)) > − 12 ,
if f ∈ A and verifies the fuzzy differential subordination
(3.13) F(In,αm,γf)′(U)(I
n,α
m,γf(z))
′
6 Fh(U)h(z), i.e. (I
n,α
m,γf(z))
′ ≺F h(z),
then
(3.14) FIn,αm,γf(U)
In,αm,γf(z)
z
6 Fk(U)k(z), i.e.
In,αm,γf(z)
z
≺F k(z),
where
k(z) =
1
z
∫ z
0
h(t) dt.
The function k is convex and it is the fuzzy best dominant.
404
P r o o f. Let
q(z) =
In,αm,γf(z)
z
= 1 +
∞
∑
k=2
(1 + γ(k − 1))n
( m
m+ k
)α
akz
k−1, q ∈ H[1, 1],
where ℜ(1 + zh′′(z)/h′(z)) > − 12 . From Lemma 2.1, we have that
k(z) =
1
z
∫ z
0
h(t) dt
is a convex function which satisfies the fuzzy differential subordination (3.13). Since
k(z) + zk′(z) = h(z)
is the fuzzy best dominant, we have q(z)+zq′(z) = (In,αm,γf(z))
′, then (3.13) becomes
Fq(U)(q(z) + zq
′(z)) 6 Fh(U)h(z).
By applying Lemma 2.3, we have
Fq(U)q(z) 6 Fk(U)k(z) → FIn,αm,γf(U)
In,αm,γf(z)
z
6 Fk(U)k(z),
and we obtain that
In,αm,γf(z)
z
≺F k(z).

Putting h(z) = (1 + (2β − 1)z)/(1 + z) in Theorem 3.4, we obtain the following
corollary:
Corollary 3.1. Let h = (1 + (2β − 1)z)/(1 + z) be a convex function in U with
h(0) = 1, 0 6 β < 1. If f ∈ A and verifies the fuzzy differential subordination
F(In,αm,γf)′(U)(I
n,α
m,γf(z))
′ 6 Fh(U)h(z), i.e. (I
n,α
m,γf(z))
′ ≺F h(z),
then
k(z) = 2β − 1 +
2(1− β)
z
ln(1 + z),
and the function k is convex and it is the fuzzy best dominant.
4. Conclusion
All the above results give us information about fuzzy differential subordinations
for a linear operator In,αm,γ . We give some properties for the class M
F
m,γ(n, α, η) of
univalent analytic functions.
405
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Authors’ addresses: Sheza Mohammed El-Deeb, Department of Mathematics, Faculty of
Science, Damietta University, New Damietta 34517, Egypt and Department of Mathematics,
College of Science and Arts in Badaya, Qassim University, Buraidah 51452, Saudi Arabia,
e-mail: shezaeldeeb@yahoo.com; Georgia Irina Oros, Department of Mathematics, Univer-
sity of Oradea, Str. Universitatii, No. 1, 410087 Oradea, Romania, e-mail: georgia_oros_ro
@yahoo.co.uk.
406


Fuzzy differential subordinations connected with the linear operator

Institute of Mathematics AS CR, v. v. i.

Mathematica BohemicaSheza M. El-Deeb; Georgia I. OrosFuzzy differential subordinations connected with the linear operatorMathematica Bohemica, Vol. 146 (2021), No. 4, 397–406Persistent URL: http://dml.cz/dmlcz/149256Terms of use:© Institute of Mathematics AS CR, 2021Institute of Mathematics of the Czech Academy of Sciences provides access to digitizeddocuments strictly for personal use. Each copy of any part of this document must contain theseTerms of use.This document has been digitized, optimized for electronic delivery andstamped with digital signature within the project DML-CZ: The Czech DigitalMathematics Library http://dml.cz146 (2021) MATHEMATICA BOHEMICA No. 4, 397–406FUZZY DIFFERENTIAL SUBORDINATIONS CONNECTED WITHTHE LINEAR OPERATORSheza M. El-Deeb, New Damietta, Georgia I. Oros, OradeaReceived November 15, 2019. Published online November 2, 2020.Communicated by Dagmar MedkováAbstract. We obtain several fuzzy differential subordinations by using a linear operatorIn,αm,γf(z) = z +∞∑k=2(1 + γ(k − 1))nmα(m+ k)−αakzk. Using the linear operator In,αm,γ , wealso introduce a class of univalent analytic functions for which we give some properties.Keywords: fuzzy differential subordination; fuzzy best dominant; linear operatorMSC 2020 : 30C451. IntroductionLet D ⊂ C,H(D) be the class of holomorphic functions on D and denote by Hn(D)the class of holomorphic and univalent functions on D. In this paper, we denote byH(U) the class of holomorphic functions in the open unit disk U = {z ∈ C : |z| < 1}with ∂U = {z ∈ C : |z| = 1} the boundary of the unit disc. For a ∈ C and k ∈ N, wedenoteH[a, k] = {f ∈ H(U) : f(z) = a+ akzk + ak+1zk+1 + . . . , z ∈ U},Ak = {f ∈ H(U) : f(z) = z + ak+1zk+1 + . . . , z ∈ U} with A1 = A,andS ={f ∈ A : f is a univalent function in U}.Denote byK ={f ∈ A : ℜ(1 +zf ′′(z)f ′(z))> 0, z ∈ U}the class of convex functions in U.c© The author(s) 2020. This is an open access article under the CC BY-NC-ND licence cbndDOI: 10.21136/MB.2020.0159-19 397Definition 1.1 ([7] and [13]). Let f and g be analytic functions in U. We saythat f is subordinate to g, written f ≺ g, if there exists a Schwarz function w,which is analytic in U with w(0) = 0 and |w(z)| < 1 for all z ∈ U, such thatf(z) = g(w(z)). Furthermore, if the function g is univalent in U, then we have thefollowing equivalence:f(z) ≺ g(z) ⇔ f(0) = g(0) and f(U) ⊂ g(U).In order to introduce the notion of fuzzy differential subordination, we use thefollowing definitions and propositions:Definition 1.2 ([10]). Assume that Y be a nonempty set. An application F :Y →[0, 1] is called fuzzy subset. A pair (B,FB), where FB : Y →[0, 1] and(1.1) B = {x ∈ Y : 0 < FB(x) 6 1} = sup(B,FB),is called fuzzy set. The function FB is called membership function of the fuzzy set(B,FB).Proposition 1.1 ([14]).(i) If (B,FB) = (U ,FU ), then we have B = U , where B = sup(B,FB) and U =sup(U ,FU).(ii) If (B,FB) ⊆ (U ,FU ), then we have B ⊆ U , where B = sup(B,FB) and U =sup(U ,FU).Let f, g ∈ H(D). We denote(1.2) f(D) = {f(z) : 0 < Ff(D)f(z) 6 1, z ∈ D} = sup(f(D),Ff(D))and(1.3) g(D) = {g(z) : 0 < Fg(D)g(z) 6 1, z ∈ D} = sup(g(D),Fg(D)).Definition 1.3 ([14]). Let z0 ∈ D and f, g ∈ H(D). The function f is said to befuzzy subordinate to g, written f ≺F g or f(z) ≺F g(z), when following conditionsare satisfied:(i) f(z0) = g(z0),(ii) Ff(D)f(z) 6 Fg(D)g(z), z ∈ D.Proposition 1.2 ([14]). Assume that z0 ∈ D and f, g ∈ H(D). If f(z) ≺F g(z),z ∈ D, then(i) f(z0) = g(z0),(ii) f(D) ⊆ g(D), Ff(D)f(z) 6 Fg(D)g(z), z ∈ D, where f(D) and g(D) are definedby (1.2) and (1.3), respectively.398Definition 1.4 ([16]). Assume thatΦ: C3×U → C and h ∈ S with Φ(a, 0, 0; 0) =h(0) = a. If p is analytic in U with p(0) = a and satisfies the second order fuzzydifferential subordinationFΦ(C3×U)Φ(p(z), zp′(z), z2p′′(z); z) 6 Fh(U)h(z),(1.4)i.e. Φ(p(z), zp′(z), z2p′′(z); z) ≺F h(z), z ∈ U,then p is called a fuzzy solution of the fuzzy differential subordination. The univalentfunction q is called a fuzzy dominant of the fuzzy solutions for the fuzzy differentialsubordination ifFp(U)p(z) 6 Fq(U)q(z), i.e. p(z) ≺F q(z), z ∈ Ufor all p satisfying (1.4). A fuzzy dominant q̃ that satisfiesFq̃(U)q̃(z) 6 Fq(U)q(z), i.e. q̃(z) ≺F q(z), z ∈ Ufor all fuzzy dominants q of (1.4) is said to be the fuzzy best dominant of (1.4).The integral operator Kαm : A → A is defined for α > 0 and m > −1 as follows(see Komatu [12] with p = 1):K0mf(z) = f(z) = z +∞∑k=1ak+1zk+1and(1.5) Kαmf(z) =(m+ 1)αΓ(α)zm∫ z0tm−1(logzt)α−1f(t) dt.It can be easily verified that(1.6) Kαmf(z) = z +∞∑k=1( m+ 1m+ k + 1)αak+1zk+1.El-Ashwah et al. (see [9] with p = 1) introduced the linear multiplier operator In,αm,γ :A → A given as:I0,0m,γf(z) = f(z),I1,αm,γf(z) = Iαm,γf(z) = (1− γ)Kαmf(z) + γz(Kαmf(z))′= z +∞∑k=1(1 + γk)( m+ 1m+ k + 1)αak+1zk+1,I2,αm,γf(z) = Iαm,γ(I1,αm,γf(z))= z +∞∑k=1(1 + γk)2( m+ 1m+ k + 1)αak+1zk+1,399in general,(1.7) In,αm,γf(z) = Iαm,γ(In−1,αm,γ f(z)) = z +∞∑k=1(1 + γk)n( m+ 1m+ k + 1)αak+1zk+1,α > 0, m > −1, γ > 0, n ∈ N0.It is easily verified from (1.7) that(1.8) z(In,αm,γf(z))′ = (m+ 1)In,α−1m,γ f(z)−mIn,αm,γf(z)and(1.9) γz(In,αm,γf(z))′ = In+1,αm,γ f(z)− (1− γ)In,αm,γf(z).By specializing the parameters m, γ, α and n, we obtain the following operators:I0,δc,γf(z) = Kδc,1f(z) (see [11]);I0,αa−1,γf(z) = Lαaf(z) (see [4] and [5]);I1,αc,0 f(z) = Pαc f(z) (see [12] and [17]);I1,1c,0 f(z) = Lcf(z) (see [6]);I1,α1,0 f(z) = Iαf(z) (see [8]);In,0m,γf(z) = Dnγf(z) (see [1]);In,0m,1f(z) = Dnf(z) (see [18]).Also, we note that(1.10) I−n,αm,γ f(z) = z +∞∑k=1( 11 + γk)n( m+ 1m+ k + 1)αak+1zk+1.From (1.7) and (1.10) we observe that the operator In,αm,γf(z) is well defined fromn ∈ Z = {0,±1,±2, . . .}.By using the linear operator In,αm,γf(z) defined by (1.7), we will derive several fuzzydiferential subordintions for this class.Definition 1.5. A function f ∈ A belongs to the class MFm,γ(n, α, η) for allη ∈ [0, 1), n ∈ N0, m > −1, γ > 0 and α > 0 if it satisfies the inequalityF(In,αm,γf)′(U)(In,αm,γf(z))′ > η, z ∈ U.4002. PreliminaryTo prove our results, we need the following lemmas.Lemma 2.1 ([13]). Let ψ ∈ A and G(z) = z−1∫ z0ψ(t) dt, z ∈ U. Ifℜ(1 + zψ′′(z)/ψ′(z)) > −12, z ∈ U,then G ∈ K.Lemma 2.2 ([15], Theorem 2.6). Assume that h is a convex function withh(0) = a and ν ∈ C∗ = C \ {0} with ℜ(ν) > 0. If p ∈ H[a, n] with p(0) = a,Φ: C2 × U → C, Φ(p(z), zp′(z); z) = p(z) + ν−1zp′(z) is analytic function in U andFΦ(C2×U)(p(z) +1νzp′(z))6 Fh(U)h(z), i.e. p(z) +1νzp′(z) ≺F h(z), z ∈ U,thenFp(U)p(z) 6 Fq(U)q(z) 6 Fh(U)h(z), i.e. p(z) ≺F q(z), z ∈ U,whereq(z) =νnzν/n∫ z0h(t)t−1+ν/n dt, z ∈ U.The function q is convex and it is the fuzzy best dominant.Lemma 2.3 ([15], Theorem 2.7). Let g be a convex function in U and let ψ(z) =g(z) + nγzg′(z), where z ∈ U, n ∈ N and γ > 0. Ifp(z) = g(0) + pnzn + pn+1zn+1 + . . .is holomorphic in U andFp(U)(p(z) + γzp′(z)) 6 Fψ(U)ψ(z), i.e. p(z) + γzp′(z) ≺F ψ(z), z ∈ U,thenFp(U)(p(z)) 6 Fg(U)g(z), i.e. p(z) ≺F g(z), z ∈ U;this result is sharp.For the general theory of fuzzy differential subordination and its applications, werefer the reader to [2], [12]–[14], [16].The objective of the present section is to obtain several fuzzy differential subor-dinations associated with the integral operator In,αm,γ by using the method of fuzzydifferential subordination.4013. Main resultsAssume that η ∈ [0, 1), n ∈ N0, m > 0, α, γ > 0 and z ∈ U are mentioned throughthis paper:Theorem 3.1. Let k be a convex function in U and suppose that h(z) = k(z) +zk′(z)/(λ+ 2). If f ∈ MFm,γ(n, α, η) and(3.1) G(z) = Iλf(z) =λ+ 2zλ+1∫ z0tλf(t) dt,then(3.2) F(In,αm,γf)′(U)(In,αm,γf(z))′ 6 Fh(U)h(z), i.e. (In,αm,γf(z))′ ≺F h(z),implies thatF(In,αm,γG)′(U)(In,αm,γG(z))′ 6 Fk(U)k(z), i.e. (In,αm,γG(z))′ ≺F k(z);this result is sharp.P r o o f. Sincezλ+1G(z) = (λ+ 2)∫ z0tλf(t) dt,by differentiating, we obtain(λ+ 1)G(z) + zG′(z) = (λ+ 2)f(z)and(3.3) (λ + 1)In,αm,γG(z) + z(In,αm,γG(z))′ = (λ+ 2)In,αm,γf(z),and also by differentiating (3.3) we obtain(3.4) (In,αm,γG(z))′ +1(λ+ 2)z(In,αm,γG(z))′′ = (In,αm,γf(z))′.By using (3.4), the fuzzy differential subordination (3.2) becomes(3.5) F(In,αm,γf)′(U)((In,αm,γG(z))′ +1(λ+ 2)z(In,αm,γG(z))′′)6 Fh(U)(k(z) +1(λ+ 2)zk′(z)).402We denote(3.6) q(z) = (In,αm,γG(z))′ → q ∈ H[1, n].From (3.6) in (3.5) we have(3.7) F(In,αm,γf)′(U)(q(z) +1(λ+ 2)zq′(z))6 Fh(U)(k(z) +1(λ+ 2)zk′(z)),by applying Lemma 2.3, we haveFq(U)q(z) 6 Fk(U)k(z), i.e. F(In,αm,γG(z))′(U)(In,αm,γG(z))′ 6 Fk(U)k(z),then (In,αm,γG(z))′ ≺F k(z), k is the fuzzy best dominant. Theorem 3.2. Consider h(z) = (1 + (2η − 1)z)/(1 + z), η ∈ [0, 1), λ > 0 and Iλis given by (3.1). Then(3.8) Iλ(MFm,γ(n, α, η)) ⊂ MFm,γ(n, α, ζ),where(3.9) ζ = 2η − 1 + (λ+ 2)(2− 2η)∫ 10tλ+2t+ 1dt.P r o o f. The function h is convex and using the same technique as in the proofof Theorem 3.1, we obtain from the hypothesis of Theorem 3.2 thatFq(U)(q(z) +1(λ+ 2)zq′(z))6 Fh(U)h(z),where q(z) is defined in (3.6). By using Lemma 2.2, we obtain thatFq(U)q(z) 6 Fk(U)k(z) 6 Fh(U)h(z)implies thatF(In,αm,γG)′(U)(In,αm,γG(z))′6 Fk(U)k(z) 6 Fh(U)h(z),wherek(z) =λ+ 2zλ+2∫ z0tλ+11 + (2η − 1)t1 + tdt = (2η − 1) +(λ + 2)(2− 2η)zλ+2∫ z0tλ+11 + tdt.Since k is convex and k(U) is symmetric with respect to the real axis, we conclude(3.10) F(In,αm,γG)′(U)(In,αm,γG(z))′> min|z|=1Fk(U)k(z) = Fk(U)k(1),and ζ = k(1) = 2η− 1+ (λ+2)(2− 2η)∫ 10tλ+2/(t+ 1)dt. From (3.10), we concludethat we have the inclusion in relation (3.8) and hence the proof of the theorem iscomplete. 403Theorem 3.3. Assume that k is a convex function in U, k(0) = 1, and let h(z) =k(z) + zk′(z). If f ∈ A and satisfies the fuzzy differential subordination(3.11) F(In,αm,γf)′(U)(In,αm,γf(z))′ 6 Fh(U)h(z), i.e. (In,αm,γf(z))′ ≺F h(z),then(3.12) FIn,αm,γf(U)In,αm,γf(z)z6 Fk(U)k(z), i.e.In,αm,γf(z)z≺F k(z),and this result is sharp.P r o o f. By denotingq(z) =1zIn,αm,γf(z) =1z(z +∞∑k=2(1 + γ(k − 1))n( mm+ k)αakzk)= 1 +∞∑k=2(1 + γ(k − 1))n( mm+ k)αakzk−1,we obtain that q(z) + zq′(z) = (In,αm,γf(z))′. ThenF(In,αm,γf)′(U)(In,αm,γf(z))′ 6 Fh(U)h(z)→ Fq(U)(q(z) + zq′(z)) 6 Fh(U)h(z) = Fk(U)(k(z) + zk′(z)).Applying Lemma 2.3, we haveFq(U)q(z) 6 Fk(U)k(z) → FIn,αm,γf(U)In,αm,γf(z)z6 Fk(U)k(z),we getIn,αm,γf(z)z≺F k(z),and this result is sharp. Theorem 3.4. For h ∈ H(U), h(0) = 1, which satisfies ℜ(1+zh′′(z)/h′(z)) > − 12 ,if f ∈ A and verifies the fuzzy differential subordination(3.13) F(In,αm,γf)′(U)(In,αm,γf(z))′6 Fh(U)h(z), i.e. (In,αm,γf(z))′ ≺F h(z),then(3.14) FIn,αm,γf(U)In,αm,γf(z)z6 Fk(U)k(z), i.e.In,αm,γf(z)z≺F k(z),wherek(z) =1z∫ z0h(t) dt.The function k is convex and it is the fuzzy best dominant.404P r o o f. Letq(z) =In,αm,γf(z)z= 1 +∞∑k=2(1 + γ(k − 1))n( mm+ k)αakzk−1, q ∈ H[1, 1],where ℜ(1 + zh′′(z)/h′(z)) > − 12 . From Lemma 2.1, we have thatk(z) =1z∫ z0h(t) dtis a convex function which satisfies the fuzzy differential subordination (3.13). Sincek(z) + zk′(z) = h(z)is the fuzzy best dominant, we have q(z)+zq′(z) = (In,αm,γf(z))′, then (3.13) becomesFq(U)(q(z) + zq′(z)) 6 Fh(U)h(z).By applying Lemma 2.3, we haveFq(U)q(z) 6 Fk(U)k(z) → FIn,αm,γf(U)In,αm,γf(z)z6 Fk(U)k(z),and we obtain thatIn,αm,γf(z)z≺F k(z).Putting h(z) = (1 + (2β − 1)z)/(1 + z) in Theorem 3.4, we obtain the followingcorollary:Corollary 3.1. Let h = (1 + (2β − 1)z)/(1 + z) be a convex function in U withh(0) = 1, 0 6 β < 1. If f ∈ A and verifies the fuzzy differential subordinationF(In,αm,γf)′(U)(In,αm,γf(z))′ 6 Fh(U)h(z), i.e. (In,αm,γf(z))′ ≺F h(z),thenk(z) = 2β − 1 +2(1− β)zln(1 + z),and the function k is convex and it is the fuzzy best dominant.4. ConclusionAll the above results give us information about fuzzy differential subordinationsfor a linear operator In,αm,γ . We give some properties for the class MFm,γ(n, α, η) ofunivalent analytic functions.405References[1] F.M.Al-Oboudi: On univalent functions defined by a generalized Sălăgean operator. Int.J. Math. Math. Sci. 25-27 (2004), 1429–1436. zbl MR doi[2] A.Alb Lupaş: On special fuzzy differerential subordinations using convolution productof Sălăgean operator and Ruscheweyh derivative. J. Comput. Anal. Appl. 15 (2013),1484–1489. zbl MR[3] A.Alb Lupaş, G.Oros: On special fuzzy differerential subordinations using Sălăgean andRuscheweyh operators. Appl. Math. 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Fuzzy differential subordinations connected with the linear operator

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