Differential Subordination and Superordination For Analytic Functions Defined
Using A Family Of Generalized Differential Operators
Amnah Shammaky*
Department of Mathematics, Faculty of Science, Jazan University, Jazan, Saudi Arabia
E-mail: [email protected]
(Received on: 07-06-11; Accepted on: 29-06-11)
________________________________________________________________________________________________
ABSTRACT
By making use of the generalized differential operator, the authors derive the subordination and superordination
results for certain normalized analytic functions in the open unit disk. Many of the well-known and new results are shown to follow as special cases of our results.Key Words: Analytic functions, Differential Operators, Differential Subordination, Differential Superordination.
Mathematics Subject Classification: 30C 45.
________________________________________________________________________________________________
1. INTRODUCTION:
Let be the class of functions analytic in the open unit disc
=
{
z
:
|
z
|
<
1
}.
Let(
a
,
n
)
be the subclass ofconsisting of functions of the form
(
)
=
+
+
+1 +1+
⋅
⋅
⋅
⋅
n n n
n
z
a
z
a
a
z
f
Let
∈
=
f
n
{
,
(
)
}
2 2 1
1
+
+
⋅
⋅
⋅⋅
+
=
++ + +
n n n
n
z
a
z
a
z
z
f
and let =
1
. Let the functionsf
andg
be analytic in . We say that the functionf
is subordinate tog
ifthere exists a Schwarz function
w
, analytic in withw
(
0
)
=
0
andw
(
z
)
<
1
such thatf
(
z
)
=
g
(
w
(
z
))
for∈
z
. We denote it byf
(
z
)
g
(
z
).
In particular, if the functiong
is univalent in , the above subordination isequivalent to
f
(
0
)
=
g
(
0
)
andf
(
)
⊂
g
(
) . Letp
,
h
∈
and letφ
(
r
,
s
,
t
;
z
)
:3
×
→
. Ifp
and)
);
(
),
(
),
(
(
p
z
z
p
′
z
z
2p
′′
z
z
φ
are univalent and ifp
satisfies the second order superordination),
);
(
),
(
),
(
(
)
(
z
p
z
z
p
z
z
2p
z
z
h
φ
′
′′
(1.1)then
p
is a solution of the differential superordination (1.1). (Iff
is subordinate toF
, thenF
is called to besuperordinate to
f
) An analytic functionq
is called a subordinant ifq
p
for allp
satisfying (1.1). An univalent subordinantq
that satisfiesp
q
for all subordinantsq
of (1.1) is said to be the best subordinant. Recently Miller and Mocanu [6] obtained conditionsh
,
q
andφ
for which the following implication holds:).
(
)
(
)
);
(
),
(
),
(
(
)
(
2z
p
z
q
z
z
p
z
z
p
z
z
p
z
h
φ
′
′′
With the results of Miller and Mocanu [7], Bulboaca [3] investigated certain classes of first order differential superordinations as well as superordination- preserving integral operators [4]. Ali et al.[2] used the results obtained by Bulboaca [4] and gave the sufficient conditions for certain normalized analytic functions
f
to satisfy* + ,
-. / + ( / ( %
)
(
)
(
)
(
)
(
21
q
z
z
f
z
f
z
z
q
′
where
q
1 andq
2 are given univalent functions in U withq
1(
0
)
=
1
andq
2(
0
)
=
1
. Shanmugam et al. [10] obtainedsufficient conditions for a normalized analytic functions
f
to satisfy)
(
)
(
)
(
)
(
21
q
z
z
f
z
z
f
z
q
′
and)
(
))
(
(
)
(
)
(
2 22
1
q
z
z
f
z
f
z
z
q
′
where
q
1
andq
2
are given univalent functions in U withq
1
(
0
)
=
1
andq
2(
0
)
=
1
.2. DEFINITIONS AND PRELIMINARIES:
For two analytic functions
n 2
z
)
(
∞
=
+
=
n n
a
z
z
f
and
n 2
z
)
(
∞
=
+
=
n n
b
z
z
g
The Hadamard product or convolution of
f
(
z
)
andg
(
z
)
, written as(
f
∗
g
)(
z
)
is defined by.
z
)
)(
*
(
n2
∞
=
+
=
n n n
b
a
z
z
g
f
For complex parameters
α
1
,
α
2
,
⋅
⋅
⋅⋅
,
α
q
andβ
1,
β
2,
⋅
⋅
⋅⋅
,
β
sβ
j
∈
\−
,−
=
0
,
−
1
,
−
2
,
⋅
⋅
⋅
,
j
=
1
,
⋅
⋅
⋅
,
s
)
we define the generalized hypergeometric function q
F
s(
α
1,
α
2,
⋅
⋅
⋅
,
α
q,
β
1,
β
2,
⋅
⋅
⋅
,
β
s;
z
)
by∞
=
⋅
⋅
⋅
⋅
⋅
⋅
=
⋅
⋅
⋅
⋅
⋅
⋅
0 1
1 2
1 2
1
,
!
)
(
,
,
)
(
)
(
,
,
)
(
)
;
,
,
,
,
,
,
,
(
n
n
n s n
n q n s
q s
q
n
z
z
F
β
β
α
α
β
β
β
α
α
α
∈
∪
=
∈
+
≤
s
q
s
N
N
z
q
1
,
,
{
0
},
(
where
(
x
)
n
is the Pochhammer symbol defined, in terms of the Gamma functionΓ
,
by⋅
⋅
⋅
=
∈
−
+
⋅
⋅
⋅
+
+
=
Γ
+
Γ
=
}.
,
2
,
1
{
)
1
(
)
2
)(
1
(
0
1
)
(
)
(
)
(
N
n
if
n
x
x
x
x
n
if
x
n
x
x
nCorresponding to the function
h
q,s(
α
1;
z
)
defined by)
;
,
,
,
,
,
,
,
(
)
(
1 1 2 1 2,
z
F
z
h
qsα
=
q sα
α
⋅
⋅
⋅
α
qβ
β
⋅
⋅
⋅
β
swe now define the following operator
D
λm,q,s(
α
1)
f
:
A
1→
A
1 by)
;
(
*
)
(
)
(
)
(
1 , 10 ,
,
f
z
f
z
h
z
D
λqsα
=
qsα
))
;
(
*
)
(
(
)
;
(
*
)
(
)(
1
(
)
(
)
(
1 , 1 , 11 ,
,
f
z
f
z
h
z
z
f
z
h
z
D
λqsα
=
−
λ
qsα
+
λ
qsα
(2.1))).
(
)
(
(
)
(
)
(
1 1, , ,1, 1,
,
f
z
D
D
f
z
D
λmqsα
λqs λmqsα
−* + ,
-If
f
∈
1
, then from (2.1) and (2.2) we may easily deduce that,
)
(
]
)
1
(
1
[
)
(
)
(
12 1
, ,
n n n
n m m
s
q
f
z
z
n
a
z
D
λα
λ
α
∞
=
Γ
−
+
+
=
(2.3)Where
m
∈
0
=∪
{
0
}
andλ
≥
0
and)!
1
(
)
(
,
,
)
(
)
(
,
,
)
(
)
(
1 1
1 1
1 1 1
−
⋅
⋅
⋅
⋅
⋅
⋅
=
Γ
− −
− −
n
n n
n n
s q n
β
β
α
α
α
We remark that, for the choice of the parameter
m
=
0
, the operatorD
λ0,q,s(
α
1)
f
(
z
)
reduces to the well-knownDziok-Srivastava operator [6], for
q
=
2
,
s
=
1
,
α
1=
β
1,
α
2=
1
, we get the operator introduced by Al-oboudi[1], and for
q
=
2
,
s
=
1
,
α
1=
β
1,
α
2=
1
andλ
=
1
, we get the operator introduced by Salagean [8], and for thechoice of the parameter
m
=
0
,α
1=
a
>
0
,
β
1=
c
>
0
,
α
2=
1
,
q
=
2
,
s
=
1
,
we get the operator introduced by Carlson-Shaffer [5]. Also many (well known and new) integral and differential operators can be obtained specializing the parameters.It can be easily verified from the definition of (2.3) that
)
(
)
(
)
1
(
)
(
)
1
(
)
)
(
)
(
(
D
, , 1f
z
1D
, , 1f
z
1D
, , 1f
z
z
λmqsα
′
=
α
λmqsα
+
−
α
−
λmqsα
(2.4) and).
(
)
(
)
1
(
)
(
)
(
)
)
(
)
(
(
1 1 , , 1, , 1
,
,
f
z
D
f
z
D
f
z
D
z
mqsα
mqsα
λ
mqsα
λ
λ′
=
λ−
−
λ+
(2.5)
In order to prove our subordination and superordination results, we make use of the following known results.
Definition: 2-1[7] Denote by
Q
the set of all functionsf
that are analytic and injective on−
E
(
f
)
where∂
∈
=
{
ς
)
(
f
E
lim
(
)
=
∞
},
→
f
z
z ζ And are such that
f
′
(
ς
)
≠
0
forς
∈
∂
-E
(
f
).
Lemma: 2-1 [7] Let the function
q
be univalent in the open unit disc andθ
andφ
be analytic in a domainD
containing
q
( ) withφ
(
w
)
≠
0
whenw
∈
q
( ). setQ
(
z
)
=
z
q
′
(
z
)
φ
(
q
(
z
)),
h
(
z
)
=
θ
(
q
(
z
))
+
Q
(
z
)
. Suppose that(1)
Q
is starlike univalent in and(2)
0
)
(
)
(
Re
′
>
z
Q
z
h
z
for
z
∈
.If
)),
(
(
)
(
))
(
(
))
(
(
)
(
))
(
(
p
z
z
p
z
φ
p
z
θ
q
z
z
q
z
φ
p
z
θ
+
′
+
′
thenp
(
z
)
q
(
z
)
andq
is the best dominant.Lemma: 2-2 [4] Let the function
q
be univalent in the open unit disc andϑ
andφ
be analytic in a domainD
containing
q
( ) suppose that(1)
0
))
(
))
(
(
Re
′
>
z
q
z
q
φ
ϑ
for
z
∈
and(2)
z
q
′
(
z
)
φ
(
q
(
z
))
is starlike univalent in .If
p
∈
[
q
(
0
),
1
]
∩
Q
,
withp
( )⊆
D
,ϑ
(
p
(
z
))
+
z
p
′
(
z
)
φ
(
p
(
z
))
is univalent in and))
(
(
)
(
))
(
(
))
(
(
)
(
))
(
(
q
z
z
q
z
φ
q
z
ϑ
p
z
z
p
z
φ
p
z
* + ,
-. / + ( / ( %
3. SUBORDINATION AND SUPERORDINATION FOR ANALYTIC FUNCTIONS:
We begin with the following
Theorem: 3.1 Let , ,
(
11) ( )
m q s
D
f z
z
µ λ
α
+
∈
H and let the functionq
(
z
)
be analytic and univalent in U such that∈
≠
z
z
q
(
)
0
,
(
, U). Suppose that)
(
)
(
z
q
z
q
z
′
is starlike univalent in U . Let
0
)
(
)
(
)
(
)
(
))
(
(
2
)
(
1
Re
2>
′
′′
+
′
−
+
+
z
q
z
q
z
z
q
z
q
z
z
q
z
q
β
δ
β
ξ
(3.1)∈
β
ξ
δ
α
,
,
,
(
;β
≠
0
)
and µ λ λ
α
ξ
α
δ
β
ξ
µ
α
=
+
+
Ψ
z
z
f
D
z
f
m s q m s q)
(
)
1
(
:
)
)(
,
,
,
,
,
(
1 , , 1, ,
.
1
)
(
)
1
(
)
(
)
2
(
)
1
(
)
(
)
1
(
1 , , 1 , , 1 2 1 , ,−
+
+
+
+
+
+
z
f
D
z
f
D
z
z
f
D
m s q m s q m s qα
α
α
βµ
α
δ
λ λ µ λ (3.2) Ifq
satisfies the following subordination:,
)
(
)
(
))
(
(
)
(
)
)(
,
,
,
,
,
(
1 2, ,
z
q
z
q
z
z
q
z
q
z
f
m s q′
+
+
+
Ψ
λα
µ
ξ
β
δ
α
ξ
δ
β
(
α
,
δ
,
ξ
,
µ
,
β
∈
;β
,
µ
≠
0
),
then
(
;
0
)
)
(
)
(
)
1
(
1 , ,≠
∈
+
µ
µ
α
µ λC
z
q
z
z
f
D
mqsand
q
is the best dominant. (3.3)Proof: Let the function p be defined by
∈
+
=
z
z
z
f
D
z
p
m s q(
)
(
)
1
(
:
)
(
, , 1µ λ
α
∈
≠
f
z
0
;
)By a straightforward computation, we have
.
1
)
(
)
1
(
)
)
(
)
1
(
(
:
)
(
)
(
1 , , 1 , ,−
+
′
+
=
′
z
f
D
z
f
D
z
z
p
z
p
z
m s q m s qα
α
µ
λ λBy using the identity (2.4), we obtain
.
)
1
(
)
(
)
1
(
)
(
)
2
(
)
1
(
:
)
(
)
(
1 1 , , 1 , ,1
−
+
+
+
+
=
′
α
α
α
α
µ
λ λz
f
D
z
f
D
z
p
z
p
z
m s q m s q By setting 2:
)
(
w
α
ξ
w
δ
w
θ
=
+
+
and(
)
:
,
w
w
β
φ
=
* + ,
-)
(
)
(
))
(
(
)
(
)
(
z
q
z
q
z
z
q
z
q
z
z
Q
=
′
φ
=
β
′
and
.
)
(
)
(
))
(
(
)
(
)
(
))
(
(
)
(
2z
q
z
q
z
z
q
z
q
z
Q
z
q
z
h
=
θ
+
=
α
+
ξ
+
δ
+
β
′
We find that
Q
(
z
)
is starlike univalent in and that.
0
)
(
)
(
)
(
)
(
))
(
(
2
)
(
1
Re
)
(
)
(
Re
2>
′
′′
+
′
−
+
+
=
′
z
q
z
q
z
z
q
z
q
z
z
q
z
q
z
Q
z
h
z
β
δ
β
ξ
The assertion (3.3) of Theorem3.1 now follows by an application of Lemma 2.2.
Using arguments similar to those detailed in the Theorem (3.1) with the equation (2.5), we have the following Theorem.
Theorem: 3.2 Let
∈
µ λα
z
z
f
D
m,q,s(
1)
(
)
and let the function
q
(
z
)
be analytical and univalent in suchthat
q
(
z
)
≠
0
,
(
z
∈
, ). Suppose that)
(
)
(
z
q
z
q
z
′
is starlike univalent in . Let
0
)
(
)
(
)
(
)
(
))
(
(
2
)
(
1
Re
2>
′
′′
+
′
−
+
+
z
q
z
q
z
z
q
z
q
z
z
q
z
q
β
δ
β
ξ
(
α
,
δ
,
ξ
,
β
∈
;β
≠
0
)
(3.4)and µ λ λ
α
ξ
α
δ
β
ξ
µ
α
=
+
Ω
+z
z
f
D
z
f
m s q m s q)
(
)
(
:
)
)(
,
,
,
,
,
(
1 1 , , 1 , ,.
)
2
(
)
(
)
(
)
(
)
(
)
(
)
(
1 1 , , 1 2 , , 2 1 1 , ,−
−
+
+
+ + +λ
α
α
λ
µ
β
α
δ
λ λ µ λz
f
D
z
f
D
z
z
f
D
m s q m s q m s q (3.5)If
q
satisfies the following subordination:)
(
)
(
))
(
(
)
(
)
)(
,
,
,
,
,
(
2 1 , ,z
q
z
q
z
z
q
z
q
z
f
m s q′
+
+
+
Ω
λα
µ
ξ
β
δ
α
ξ
δ
β
(
α
,
δ
,
ξ
,
µ
,
β
∈
;β
,
µ
≠
0
)
then
(
;
0
)
)
(
)
(
)
(
1 1 , ,≠
∈
+µ
µ
α
µ λC
z
q
z
z
f
D
mqs(3.6)
and
q
is the best dominant.For the choices
,
1
1
1
1
)
(
−
≤
<
≤
+
+
=
B
A
Bz
Az
z
q
and,
1
1
)
(
γ−
+
=
z
z
z
q
0
<
γ
≤
1
,
in Theorem 3.1, we get thefollowing results.
Corollary: 3.3 Assume that (3.1) holds. If
f
∈
and)
1
)(
1
(
)
(
1
1
1
1
)
)(
,
,
,
,
,
(
2 1 , ,Bz
Az
z
B
A
Bz
Az
Bz
Az
z
f
m s q+
+
−
+
+
+
+
+
+
+
Ψ
λα
µ
ξ
β
δ
α
ξ
δ
β
(
α
,
δ
,
ξ
,
µ
,
β
∈
;),
0
,
µ
≠
* + ,
-. / + ( / ( % 1
(
;
0
)
1
1
)
(
)
1
(
1, ,
≠
∈
+
+
+
µ
µ
α
µλ
C
Bz
Az
z
z
f
D
mqsand
Bz
Az
+
+
1
1
is the best dominant.
Corollary: 3.4 Assume that (3.1) holds. If
f
∈
and) 1 (
2 1
1 1
1 )
)( , , , , ,
( 2
2
1 , ,
z z z
z z
z z
f
m s q
+ + + + + + + +
Ψ
α
µ
ξ
β
δ
α
ξ
δ
β
γ
γ γ
λ
∈
β
µ
ξ
δ
α
,
,
,
,
(
;β
,
µ
≠
0
),
whereΨ
λm,q,s(
α
1,
µ
,
ξ
,
β
,
δ
,
f
)
is as defined in (3.2), then(
;
0
)
1
1
)
(
)
1
(
1, ,
≠
∈
−
+
+
µ
µ
α
µ γλ
C
z
z
z
z
f
D
mqsand
γ
−
+
z
z
1
1
is the best dominant.
For special case when
∈
−
=
b
z
z
q
b(
)
1
(
1
)
(
2 \{0}){
0
}),
m
=
0
;
α
1=
a
=
1
,
β
1=
c
=
1
,
1
,
0
,
1
,
2
,
1
2
=
=
=
δ
=
ξ
=
µ
=
α
=
α
q
s
and1
,
b
=
β
Theorem 3.1 reduces at once to the followingknown result obtained by Srivastava and Lashin [11].
Corollary: 3.5 Let b be a non zero complex number. If
f
∈
, and,
1
1
)
(
)
(
1
1
z
z
z
f
z
f
z
b
−
+
′
′′
+
thenb
z
z
f
2)
1
(
1
)
(
−
′
and bz
)
21
(
1
−
is the best dominant.Remark: 3.1 We remark that Theorem 3.2 can be restated, for different choices of the function
q
.Next, by appealing to Lemma 2.2 of the preceding section, we prove the following:
Theorem: 3.6 Let
q
be analytic and univalent in such thatq
(
z
)
≠
0
and)
(
)
(
z
q
z
q
z
′
be starlike univalent in .
Further, let us assume that
,
0
)
(
))
(
(
2
Re
q
z
2+
q
z
>
β
ξ
β
δ
∈
β
ξ
δ
,
,
(
;β
≠
0
).
(3.7)If
f
∈
∈
+
≠
µ λ
α
z
z
f
D
ms
q
(
1
)
(
)
0
, , 1[
q
(
0
),
1
]
∩
Q
,
and
Ψ
λm,q,s(
α
1,
µ
,
ξ
,
β
,
δ
,
f
)
is univalent in , then)
,
,
,
,
,
(
)
(
)
(
))
(
(
)
(
2 , , 1f
z
q
z
q
z
z
q
z
q
δ
β
mqsα
µ
ξ
β
δ
ξ
α
Ψ
λ′
+
+
+
implies
(
;
0
)
)
(
)
1
(
)
(
, ,α
1+
µ
∈
µ
≠
µ λ
C
z
z
f
D
z
q
m s q
(3.8)
* + ,
-Proof: By setting
2
:
)
(
w
α
ξ
w
δ
w
ϑ
=
+
+
and(
)
:
,
w
w
β
φ
=
it can be easily verified that
ϑ
is analytic in ,φ
is analytic in\
{
0
}
and thatφ
(
w
)
≠
0
(
w
∈
\
{
0
}
). Sinceq
is convex (univalent) function it follows that,2
( ( ))
2
Re
Re
( ( ))
( )
0,
( ( ))
q z
q z
q z
q z
ϑ
δ
ξ
ϕ
β
β
′
=
+
>
(
δ
,
ξ
,
β
∈
;β
≠
0
).
The assertion (3.8) of Theorem3.6 follows by an application of Lemma 2.2.
4. SANDWICH THEOREMS:
Combining Theorem 3.1 and Theorem 3.6, we get the following sandwich theorem.
Theorem: 4.1 Let
q
1 andq
2 be univalent in U such thatq
1≠
0
andq
2≠
0
,
(
z
∈
U) with)
(
)
(
1 1
z
q
z
q
z
′
and
)
(
)
(
2 2
z
q
z
q
z
′
being starlike univalent. Suppose that
q
1 satisfies (3.7) andq
2 satisfies (3.1). Iff
∈
A ,∈
+
µλ
α
z
z
f
D
m,q,s(
11
)
(
)
H
[
q
(
0
),
1
]
∩
Q
,
andΨ
λm,q,s(
α
1,
µ
,
ξ
,
β
,
δ
,
f
)
is univalent in U, then)
)(
,
,
,
,
,
(
)
(
)
(
))
(
(
)
(
, , 11 1 2
1
1
f
z
z
q
z
q
z
z
q
z
q
δ
β
mqsα
µ
ξ
β
δ
ξ
α
Ψ
λ′
+
+
+
)
(
)
(
))
(
(
)
(
2 2 2
2 2
z
q
z
q
z
z
q
z
q
+
+
′
+
ξ
δ
β
α
∈
β
µ
ξ
δ
α
,
,
,
,
(
C;β
,
µ
≠
0
)
implies(
;
0
)
)
(
)
(
)
1
(
)
(
, , 1 21
∈
≠
+
µ
µ
α
µλ
C
z
q
z
z
f
D
z
q
m s q
and
q
1 andq
2 are respectively the best subordinant and the dominant.Remark: 4.1 By taking
m
=
0
andα
1=
β
1 in Theorem 4.1, we have the result obtained by Shanmugam et al. [10].Remark: putting
α
1=
b
>
0
,
α
2=
1
,
β
1=
c
>
0
,
q
=
2
,
s
=
1
in the above results we obtain the results obtained by Selvaraj and Karthikeyan [9].REFERENCES:
[1] F. M. Al-Oboudi, On univalent functions defined by a generalized Salagean operator, Int. J. Math. Math. Sci., 25-28(2004), 1429–1436.
[2] R. M. Ali et al., Differential sandwich theorems for certain analytic functions, Far East J. Math. Sci., 15 (2004), no. 1, 87–94.
[3] T. Bulboaca, Classes of first-order differential superordinations, Demonstratio Math., 35 (2002), no. 2, 287–292.
[4] T. Bulboaca, A class of superordination-preserving integral operators, Indag. Math. (N.S.), 13 (2002), no. 3, 301–311.
* + ,
-. / + ( / ( %
-[6] J. Dziok and H. M. Srivastava, Classes of analytic functions associated with the generalized hypergeometric function, Appl. Math.Comp. 103 (1999), 1-13.
[7] S. S. Miller and P. T. Mocanu, Subordinants of differential superordina-tions, Complex Var. Theory Appl., 48 (2003), no. 10, 815–826. MR2014390 (2004i:30015).
[8] G. S. Salagean, Subclasses of univalent functions, in Complex analysis—fifth Romanian-Finnish seminar, Part 1 (Bucharest, 1981), Lecture Notes in Math., 1013, Springer, Berlin, 362–372.
[9] C. Selvaraj and K.R. Karthikeyan, Differential Subordination and Superordination for analytic functions defined using a family of generalized differential operators, An. St.Univ.Ovidius Constanta, 17(1) (2009), 201-201
[10] T. N. Shanmugam, C. Ramachandran, M. Darus and S. Sivasubramanian, Differential sandwich theorems for some subclasses of analytic functions involving a linear operator, Acta Math. Univ. Comenianae, 16 (2007), no. 2, 287–294.
[11] H. M. Srivastava and A. Y. Lashin, Some applications of the Briot-Bouquet differential subordination, JIPAM. J. Inequal. Pure Appl. Math., 6 (2005), no. 2, Article 41, 7 pp. (electronic).
