Volume 4, Number 1, 2020, Pages 129-137 https://doi.org/10.34198/ejms.4120.129137
Received: March 15, 2020; Accepted: April 8, 2020 2010 Mathematics Subject Classification: 30C45.
Keywords and phrases: fuzzy set, fuzzy differential subordination, λ-pseudo starlike with respect to symmetrical points, λ-pseudo convex with respect to symmetrical points, fuzzy best dominant.
Copyright © 2020 Abbas Kareem Wanas and Dhirgam Allawy Hussein. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution,
Fuzzy Differential Subordinations Results for λ -pseudo Starlike and λ -pseudo Convex Functions with Respect to Symmetrical Points
Abbas Kareem Wanas1 and Dhirgam Allawy Hussein2
1 Department of Mathematics, College of Science, University of Al-Qadisiyah, Iraq e-mail: [email protected]
2 Directorate of Education in Al-Qadisiyah, Diwaniyah, Iraq e-mail: [email protected]
Abstract
In the present work, we establish some fuzzy differential subordination results for λ-pseudo starlike and λ-pseudo convex functions with respect to symmetrical points in the open unit disk.
1. Introduction and Preliminaries
Let T indicate the family of functions f and has the series form:
( ) ∞
=
+
=
2
,
n nzn
a z
z
f (1.1)
which are analytic and univalent in the open unit disk U =
{
z∈C: z <1}
. For functions fj ∈T(
j =1,2)
given by( ) ∞ ( )
=
= +
=
2
, 1, 2 ,
n j n n
j z z a z j
f
the Hadamard product of f1 and f2 is defined by
( )( ) ∞ ( )( )
=
∗
= +
=
∗
2
1 2 2
, 1 , 2
1 .
n
n n
n a z f f z
a z
z f f
A function f ∈T is called starlike with respect to symmetrical points, if (see [8])
( ) ( ) ( )
0, .Re z U
z f z f
z f
z > ∈
−
−
′
The family of all such functions is denote by SS∗.
The family of starlike functions with respect to symmetrical points obviously includes the family of convex functions with respect to symmetrical points, Cs, satisfying the following condition:
( )
( )
( ) ( )
( )
0, .Re z U
z f z f
z f
z > ∈
−
−
′ ′
Recently, Babalola [1] defined the class Lλ of λ-pseudo-starlike functions as follows:
Let f ∈T and λ≥1. Then f ∈Lλ of λ-pseudo-starlike functions in U if and only if
( )
( )
( )
0, .Re z U
z f
z f
z > ∈
′ λ
Definition 1.1 [9]. Let X be a non-empty set. An application F : X →
[ ]
0,1 is called fuzzy subset. An alternate definition, more precise, would be the following:A pair
(
A, FA)
, where FA: X →[ ]
0,1 and A={
x∈ X :0 < FA( )
x ≤1}
=(
A FA)
supp , is called fuzzy subset. The function FA is called membership function of the fuzzy subset
(
A, FA)
.Definition 1.2 [5]. Let two fuzzy subsets of X,
(
M, FM)
and(
N, FN)
. We say that the fuzzy subsets M and N are equal if and only if FM( )
x = FN( )
x , x∈X and we denote this by(
M, FM) (
= N, FN)
. The fuzzy subset(
M, FM)
is contained in thefuzzy subset
(
N, FN)
if and only if FM( )
x ≤ FN( )
x , x∈ X and we denote the inclusion relation by(
M, FM) (
⊆ N, FN)
.Let D ⊆ C and f, g analytic functions. We denote by
( )
D supp( ( )
f D F ( )) { ( )
f z F ( )(
f( )
z)
z D}
f = , f D = :0< f D ≤1, ∈
and
( )
D supp( ( )
g D, F ( )) { ( )
g z :0 F ( )(
g( )
z)
1, z D}
.g = g D = < g D ≤ ∈
Definition 1.3 [5]. Let D⊆ C, z0∈D be a fixed point, and let the functions
( )
., g D
f ∈H The function f is said to be fuzzy subordinate to g and write f pF g or
( )
z g( )
zf pF if the following conditions are satisfied:
1. f
( )
z0 = g( )
z0 ,2. Ff( )D
(
f( )
z)
≤ Fg( )D(
g( )
z)
, z∈D.Definition 1.4 [6]. Let ψ:C3×U →C and let h be univalent in U. If p is analytic in U and satisfies the (second-order) fuzzy differential subordination:
( 3 )
( ( ( )
p z , zp( )
z , z2p( ) ))
z ; z F ( )(
h( )
z)
,F hU
U ψ ′ ′′ ≤
×
ψ C (1.2)
i.e.,
( ( )
p z , zp′( )
z , z2p′′( ) )
z ; z F h( )
z , z∈U,ψ p
then p is called a fuzzy solution of the fuzzy differential subordination. The univalent function is q called a fuzzy dominant of the fuzzy solutions of the fuzzy differential subordination, or more simple a fuzzy dominant, if p
( )
z pF q( )
z , z∈U for all p satisfying (1.2). A fuzzy dominant q~ that satisfies q~( )
z F q( )
z , z∈Up for all fuzzy
dominant q of (1.2) is said to be the fuzzy best dominant of (1.2).
Lemma 1.1 [2]. Let q be univalent in U and let θ and φ be analytic in a domain D containing q
( )
U with φ( )
w ≠0 when w∈q( )
U . Set Q( )
z = zq′( ) ( )
z φ(
q z)
and( )
z(
q( )
z)
Q( )
z .h = θ + Suppose that
(1) Q
( )
z is starlike in U,(2)
( )
( )
0Re >
′ z Q
z h
z for z∈U.
If p is analytic in U, with p
( )
0 =q( ) ( )
0, pU ⊂ D and ψ:C2×U → C,( ( )
p z zp′( ))
z =θ( ( ))
p z + zp′( )
z ⋅φ(
p( )
z)
ψ , is analytic in U, then
( 2 )
[ ( ( ))
p z zp( )
z(
p( )
z)]
F( )h( )
z ,F hU
U θ + ′ ⋅φ ≤
× ψ C
implies
( )p
( )
z F ( )q( )
z ,FpU ≤ qU
i.e., p
( )
z pF q( )
z and q is the fuzzy best dominant, where(
×U)
=supp(
×U F ( U)ψ( ( )
p z zp′( )))
zψ 2 2 , ψ 2× ,
C C
C
{
∈ :0< ( 2 )ψ( ( )
, ′( ))
≤1}
,= z Fψ × p z zp z
U
C C
and
( )
U supp(
U F ( )h( ))
zh = , hU =
{
z∈C:0 < Fh( )U h( ) }
z ≤1.Recently, Oros and Oros [6, 7], Lupaş [3, 4] and Wanas and Majeed [10, 11] have obtained fuzzy differential subordination results for certain classes of analytic functions.
2. Main Results
Theorem 2.1. Suppose that α,β, γ∈C,δ >0,λ ≥1, t∈C\
{ }
0 and q be univalent function in U such that q( )
0 =1, q( )
z ≠0 and( ) ( ) ( ) ( )
( ) ( )
( )
0. 21 1
Re >
′ + ′′
− ′ γ αγ +
+
− β γ
+ q z
z q z z q
z q z z
t q
t (2.1)
Assume that z
(
q( )
z)
γ−2q′( )
z is starlike in U. If f ∈T and Φ(
α,β,γ, λ,δ,t; z)
is analytic in U, where( ) ( ( ) )
( ) ( ) ( ) ( )
( )
( )
′
− β −
+
α
−
−
= ′ δ
λ γ β α Φ
δ λ λ γδ
z f z
z f z f z
f z f
z f z z
t
2
; 2 , , , , ,
( ) ( ) ( )
( )
δ λ
′
− δ −
+
z f z
z f z t f
2
( ) ( ) ( ( ) ( ) )
( ) ( )
,1 2
−
−
− ′
− −
′ λ ′′
+
× f z f z
z f z f z
f z f
z (2.2)
then
( )
[ ( ) ]
( )( ( ) ) ( ) ( )
( ) ( )
α+ β + ′
≤ δ
λ γ β α
Φ ψ × γ
×
ψ 2 , , , , , ; 2 2
z q
z q t z z z q
q F
z t
F C U C U
( )h
( )
z , FhU= (2.3)
implies
( )
( )
( ) ( )
( )
( )
( ) ( )
( )( )
,2 F q z
z f z f
z f
F z qU
U f U f
U
f ≤
−
−
′ λ δ
−
−
′ λ δ
i.e.,
( )
( )
( ) ( )
q( ) (
z z U)
zf z f
z f z
F ∈
−
−
′ λ δ
2 ,
p
and q is the fuzzy best dominant.
Proof. For given f ∈T, define p by
( ) ( ( ) )
( ) ( )
.2 λ δ
−
−
= ′
z f z f
z f z z
p (2.4)
It is clear that the function p is analytic in U and p
( )
0 =1. Simple calculations show that( )
( )
( ) ( )
( )
( )
2(
, , , , ,t; z)
, zp z p t z z z p
p =Φ α β γ λ δ
α+ β + ′
γ (2.5)
where Φ
(
α,β, γ,λ, δ, t; z)
is given by (2.2).In the light of (2.3) and (2.5), we conclude that
( ) ( ( )) ( ) ( )
( ) ( )
α+ β + ′
× γ
ψ 2 2
z p
z p t z z z p
p F C U
( )
( ( ) ) ( ) ( )
( ) ( )
2 .2
α+ β + ′
≤ ψ × γ
z q
z q t z z z q
q
F C U
Define the functions θ and φ by
( ) (
= α +β)
γ−1θ w w w and φ
( )
w =twγ−2.Evidently, the functions θ and φ are analytic in D= C\
{ }
0 and φ( )
w ≠ 0,w∈D. Also, we find that( )
z zq( ) ( )
z(
q z)
tz(
q( )
z)
q( )
zQ = ′ φ = γ−2 ′
and
( ) ( ( ) ) ( ) ( ( ) )
( ) ( )
( ) ( )
2.
′
β + + α
= + θ
= γ
z q
z q t z z z q
q z Q z q z h
Since z
(
q( )
z)
γ−2q′( )
z is starlike univalent in U, we observe that Q is starlike univalent in U.( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
( )
. 21 1
Re
Re
′ + ′′
− ′ γ αγ +
+
− β γ +
=
′
z q
z q z z q
z q z z
t q t
z Q
z h
z (2.6)
Now from (2.1) and (2.6), it is evident that
( ) ( )
0.Re >
′ z Q
z h z
On the application of Lemma 1.1, yields Fp( )U p
( )
z ≤ Fq( )U q( )
z . By using (2.4), we obtain( )
( )
( ) ( )
( )
( )
( ) ( )
( )( )
,2 F q z
z f z f
z f
F z qU
U f U f
U
f ≤
−
−
′ λ δ
−
−
′ λ δ
i.e.
( ( ) )
( ) ( )
q( )
z zf z f
z f z
pF λ δ
−
−
′
2 and q is the fuzzy best dominant.
By putting the fuzzy dominant
( )
, 1 11 γ = =
−
= + t
z z z
q and α =β=0 in Theorem
2.1, we obtain the following results:
Corollary 2.1. Let 0. 1
Re 1
2 2 >
− + z
z If f ∈T and
( ) ( ) ( ( ) ( ) ) ( ) ( )
−
−
− ′
− −
′ λ ′′
+
δ f z f z
z f z f z z f
z f 1 z
is analytic in U, then
( )
( )
( ) ( ( ) ( ) ) ( ) ( )
−
−
− ′
− −
′ λ ′′
+
× δ
ψ f z f z
z f z f z z f
z f F z
U 1
C2 ( ) ,
1 2
2 2
≤ ψ × −
z F z
U C
implies
( )
( )
( ) ( )
1 ,1 2
z z z
f z f
z f z
F −
+
−
−
′ λ δ
p
(
z∈U)
and
( )
z z z
q −
= + 1
1 is the fuzzy best dominant.
Theorem 2.2. Suppose that α,β, γ∈C, δ>0, λ≥1,t∈C\
{ }
0 and q is univalent function in U such that q( )
0 =1, q( )
z ≠ 0 and let q satisfy (2.1). Assume that( )
(
q z)
q( )
zz γ−2 ′ is starlike in U. If f ∈T and Ψ
(
α,β, γ,λ,δ,t; z)
is analytic in U, where( ) ( ( ( ) ) )
( ) ( )
( )
( ) ( )
( )
( ( ( ) ) )
′ ′
− ′ β −
+
α
− ′
−
′ ′
= δ
λ γ β α Ψ
δ λ λ γδ
z f z
z f z f z
f z f
z f z z
t
2
; 2 , , , , ,
( ) ( )
( )
( ( ( ) ) )
δ λ
′ ′
− ′ δ −
+
z f z
z f z t f
2
( ( ) ( ))
( ) ( ) ( ( ) ( ) )
( ) ( )
( )
,2
− ′
−
− ″
− − + ′
′′
+ ′
′′
λ ′
×
z f z f
z f z f z z
f z f z
z f z f z
z (2.7)
then
( )
[ ( ) ]
( )( ( ) )
( ) ( )
( ) ( )
′
β + + α
≤ δ
λ γ β α
Ψ ψ × γ
×
ψ 2 , , , , , ; 2 2
z q
z q t z z z q
q F
z t
F C U C U
( )h
( )
z , FhU= (2.8)
implies
( ( )( ) ) ( ) ( )
( )
( ( ( ) ) )
( ) ( )
(
2)
F ( )q( )
z , zf z f
z f
F z qU
U f U f
U
f ≤
− ′
−
′ ′ λ δ
− ′
−
′ ′ λ δ
i.e.,
( ( ( ) ) )
( ) ( )
( )
q( ) (
z z U)
z f z f
z f z
F ∈
− ′
−
′ ′ λ δ 2 ,
p
and q is the fuzzy best dominant.
Proof. For given f ∈T, define p by
( ) ( ( ( ) ) )
( ) ( )
( )
.2 ′ λ δ
− ′
−
= ′
z f z f
z f z z
p (2.9)
It is obvious that the function p is analytic in U and p
( )
0 =1. After some computations, we deduce that( )
( )
( ) ( )
( )
( )
2(
, , , , ,t; z)
, zp z p t z z z p
p = Ψ α β γ λ δ
α+ β + ′
γ (2.10)
where Ψ
(
α,β, γ,λ, δ,t; z)
is given by (2.7).By making use of (2.8) and (2.10), it follows that
( )
( ( ) ) ( ) ( )
( ) ( )
α+ β + ′
× γ
ψ 2 2
z p
z p t z z z p
p
F C U
( )
( ( ) ) ( ) ( ) ( ) ( )
2 .2
α+ β + ′
≤ ψ × γ
z q
z q t z z z q
q
F C U
The remaining part of Theorem 2.2 is similar to that of Theorem 2.1 and thus we omit it.
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