General Mathematics Vol. 19, No. 1 (2011), 75–85
Superordination Properties for Certain Analytic
Functions 1
H.A. Al-Kharsani, N.M. Al-Areefi
Abstract
The purpose of the present paper is to derive superordination result
for functions in the class Mµl,m (α, λ, b) of normalized analytic functions
in the open unit disk U . A number of interesting applications of the
superordination result are also considered.
2000 Mathematics Subject Classification: 30C45.
Key words and phrases: Analytic functions, Hadamard product, the
Dziok-Srivastava operator, superordinating factor sequence.
1
Introduction
Let A denote the class of functions of the form
(1)
∞
X
f (z) = z +
an z n
n=2
which are analytic in the unit disc U = {z : |z| < 1}. We also denote by K
the class of functions f ∈ A that are convex in U .
Given two functions f, g ∈ A, where f is given by (1) and g is defined by
(2)
g(z) = z +
∞
X
bn z n .
n=2
1
Received 24 April, 2009
Accepted for publication (in revised form) 17 November, 2009
75
76
H.A. Al-Kharsani, N.M. Al-Areefi
The Hadamard product (or convolution) f ∗ g is defined by
(3)
(f ∗ g)(z) = z +
∞
X
an bn z n
(z ∈ U ).
n=2
For αj ∈ C (j = 1, 2, . . . , l) and βj ∈ C \ {0, −1, −2, . . .} (j = 1, 2, . . . , m), the
generalized hypergeometric function l Fm (α1 , . . . , αl ; β1 , . . . , βm ; z) is defined
by the infinite series
l Fm (α1 , . . . , αl ; β1 , . . . , βm ; z) :=
∞
X
(α1 )n · · · (αl )n z n
(β1 )n · · · (βm )n n!
n=0
(l ≤ m + 1; m ∈ N0 := {0, 1, 2, . . .}),
where (λ)n is the Pochhammer symbol defined by
Γ(λ+n)
1
(n = 0)
(λ)n :=
=
λ(λ+1)(λ+2)
·
·
·
(λ+n−1)
(n
∈ N := {1, 2, 3, . . .}).
Γ(λ)
Corresponding to the function
(4)
h(α1 , . . . , αl ; β1 , . . . , βm ; z) = zl Fm (α1 , αl ; β1 , . . . , βm ; z).
The Dziok-Srivastava operator [4] (see also [11]) H l,m (α1 , . . . , αl ; β1 , . . . , βm )
is defined by the Hadamrd product
(5)
H l,m (α1 , . . . , αl ; β1 , . . . , βm )f (z) := h(α1 , . . . , αl ; β1 , . . . , βm ; z) ∗ f (z).
We note that the linear operator H(α1 , . . . , αl ; β1 , . . . , βm ) includes various
other linear operators which were introduced and studied by Carlson and
Shaffer [3], Hohlov [6], Ruscheweyh [10], and so on [5], [9].
Corresponding to the function h(α1 , . . . , αl ; β1 , . . . , βm ; z), defined by (4),
we introduce a function Fµ (α1 , . . . , αl ; β1 , . . . , βm ; z) given by
(6)
h(α1 , . . . , αl ; β1 , . . . , βm ; z) ∗ Fµ (α1 , . . . , αl ; β1 , . . . , βm ; z)
z
(z ∈ U, µ > 0).
=
(1 − z)µ
Analogous to H(α1 , . . . , αl ; β1 , . . . , βm ), in [2] we define the linear operator
Jµ (α1 , . . . , αl ; β1 , . . . , βm ) on A as follows:
Jµ (α1 , . . . , αl ; β1 , . . . , βm )f (z) = Fµ (α1 , . . . , αl ; β1 , . . . , βm ; z) ∗ f (z)
(7)
(αi ; βj ∈ C \ z 0 ; i = 1, . . . , l; j = 1, . . . , m, µ > 0; z ∈ U ; f ∈ A).
Superordination Properties for Certain Analytic Functions
77
For convenience, we write
(8)
Jµl,m (α1 ) := Jµ (α1 , . . . , αl ; β1 , . . . , βm ).
This operator was defined by Cho [7] special cases were studied by Noor [8]
and Alkharsani [1].
Definition 1 Let g be analytic and univalent in U . If f is analytic in U ,
f (0) = g(0), and f (U ) ⊂ g(U ), then one says that f is subordinate to g in U ,
and we write f ≺ g or f (z) ≺ g(z). One also says that g is superordinate to
f in U .
∞
X
Definition 2 Let f =
k=1
plex numbers where
∞
ak z ∈ A. An infinite sequence ak
k
 1

ak
ck =

0
ak 6= 0
ak = 0
will be called superordinating factor if for evey g = z +
∞
X
bk z k in K, one has
k=2
(9)
f−1 ∗ g ≺ g
where f−1 is defined as follows,
f ∗ f−1 ∗ g ≺ f ∗ g,
then
f−1 = z ∗
∞
X
ck z k
k=2
one also says that (11) is equivalent to
∞
X
ck bk z k ≺ g
(z ∈ U ; c1 = 1),
k=1
∞
or the sequence ck
is a superordinating factor if and only if
k=1
(10)
(
of com-
k=1
Re 1 + 2
∞
X
k=1
ck z k
)
>0
(z ∈ U ).
78
H.A. Al-Kharsani, N.M. Al-Areefi
Definition 3 Suppose that f ∈ A. Then the function f−1 is said to be a
member of the class Ll,m
µ (α, λ, b) if it satisfies
l,m
Jµ+1 (α1 )f−1 (z)
Jµl,m (α1 )f−1 (z)
λµ
+ (1 − λµ)
−1
z
z
<1
l,m
Jµ+1 (α1 )f−1 (z)
(11) Jµl,m (α1 )f−1 (z)
+
(1
−
λµ)
+
2b(1
−
α)
−
1
λµ
z
z
(z ∈ U ; 0 ≤ α < 1; λ ≥ 0; b ∈ C \ {0}; µ > 0),
Now, we prove the following lemma which gives a sufficient condition for
functions belonging to the class Ll,m
µ (α, λ, b).
Lemma 1 If the function f−1 satisfies the following conditions:
∞
X
[1 + λ(k − 1)]C(µ, k)|ck | ≤ (1 − α)|b|
k=2
(12)
(0 ≤ α < 1; λ ≥ 0; b ∈ C \ {0}; µ > 0),
where
C(µ, k) =
k
Y
(j + µ − 2)
ψk−1 , ψk−1
(k − 1)!
j=2
(13)
=
(β1 )k−1 · · · (βm )k−1
, (µ > 0, k = 1, 2, 3, . . .),
(α1 )k−1 · · · (αl )k−1
then f−1 ∈ Ll,m
µ (α, λ, b).
Proof. Supposes that the inequality (12) holds. Using the identity
(14)
0
l,m
(α1 )f−1 (z) − (µ − 1)Jµl,m (α1 )f−1 (z),
z Jµl,m (α1 )f−1 (z) = µJµ+1
Superordination Properties for Certain Analytic Functions
79
we have for z ∈ U ,
Jµl,m (α1 )f−1 (z)
+ λ(Jµl,m (α1 )f−1 (z))0 − 1
(1 − λ)
z
Jµl,m (α1 )f−1 (z)
+ λ(Jµl,m (α1 )f−1 (z))0 − 1
− 2b(1 − α) + (1 − λ)
z
∞
X
= (1 + λ(k − 1)C(µ, k)ck z k−1 k=2
∞
X
k−1 (1 + λ(k − 1))C(µ, k)ck z
− 2b(1 − α) +
k=2
≤
∞
X
[1 + λ(k − 1)]C(µ, k)|ck ||z|k−1
k=2
−
(15)
(
≤2
2|b|(1 − α) −
(
∞
X
[1 + λ(k − 1)]C(µ, k)|ck ||z|k−1
k=2
∞
X
[1 + λ(k − 1)]C(µ, k)|ck | − |b|(1 − α)
k=2
)
)
≤ 0,
which sows that f−1 belongs to Ll,m
µ (α, λ, b).
l,m
Let Mµ (α, λ, b) denote the class of functions f in A whose Taylor-Maclaurin
coefficients ak satisfy the condition (12).
We note that
(16)
Mµl,m (α, λ, b) ⊆ Ll,m
µ (α, λ, b).
Example 1 (i) For 0 ≤ α < 1, λ > 0, b ∈ C \ {0} and µ > 0, the following
function defined by
(17)
µ(λ + 1)
1
1
2
f0 (z) = z+
ψ1 z 3 F2 1, 2, 1 + , 2 + , µ+1; z
2b(1 − α)
λ
λ
(z ∈ U )
is in the class Ll,m
µ (α, λ, b).
(ii) For 0 ≤ α < 1, λ > 0, b ∈ C \ {0}, and µ > 0, the following functions
80
H.A. Al-Kharsani, N.M. Al-Areefi
defined by
µ(λ + 1)ψ1 2
z
(z ∈ U ),
(1 − α)|b|
µ(µ + 1)(2λ + 1)ψ2 3
f2 (z) = z ±
z
(z ∈ U ),
(1 − α)|b|
µ(µ + 1)(2λ + 1)ψ2 3
f3 (z) = z + µ(λ + 1)ψ1 z 2 ±
z
2[(1 − α)|b| − 1]
f1 (z) = z ±
(z ∈ U ).
are in the Mµl,m (α, λ, b).
In this paper, we obtain a sharp superordination result associated with the
class Mµl,m (α, λ, b). Some applications of the main result which give important
results of analytic functions are also investigated.
2
Main Theorem
Theorem 1 Let f−1 ∈ Mµl,m (α, λ, b). Then
(18)
µ(λ + 1)ψ1
(f−1 ∗ g)(z) ≺ g(z)
2[µ(λ + 1)ψ1 + |b|(1 − α)]
(z ∈ U )
for every function g in K, and
(19)
Re f−1 (z) > −
µ(λ + 1)ψ1 + |b|(1 − α)
.
µ(λ + 1)ψ1
The constant µ(λ + 1)ψ1 /2[µ(λ + 1)ψ1 + |b|(1 − α)] cannot be replaced by a
larger one.
Proof. Let f−1 ∈ Mµl,m (α, λ, b) and let
(20)
g(z) = z +
∞
X
bk z k
k=2
be any function in the class K. Then we readily have
µ(λ + 1)ψ1
(f−1 ∗ g)(z)
2[µ(λ + 1)ψ1 + |b|(1 − α)]
(21)
µ(λ + 1)ψ1
=
2[µ(λ + 1)ψ1 + |b|(1 − α)]
z+
∞
X
k=2
bk c k z k
!
.
Superordination Properties for Certain Analytic Functions
81
Thus, by Definition 3, the superrordination result (18) will hold true if the
sequence
∞
µ(λ + 1)ck ψ1
(22)
2[µ(λ + 1)ψ1 + |b|(1 − α)] k=1
is a superordinating factor sequence, with c1 = 1. In view of Definition 2, this
is equivalent to the following inequality:
)
(
∞
X
µ(λ + 1)ψ1
ck z k > 0, (z ∈ U ).
(23)
Re 1 +
[µ(λ + 1)ψ1 + |b|(1 − α)]
k=1
Now, since
(24)
[1 + λ(k − 1)]C(µ, k)
(λ ≥ 0, µ > 0)
is an increasing function of K, we have
)
(
∞
X
µ(λ + 1)ψ1
k
Re 1 +
ck z
[µ(λ + 1)ψ1 + |b|(1 − α)]
k=1
µ(λ + 1)ψ1
= Re 1 +
z
µ(λ + 1)ψ1 + |b|(1 − α)
∞
X
1
+
µ(λ + 1)ψ1 ck z k
µ(λ + 1)ψ1 + |b|(1 − α)
k=2
µ(λ + 1)ψ1
>1−
r
µ(λ + 1)ψ1 + |b|(1 − α)
∞
X
1
−
(1 + λ(k − 1)C(µ, k))|ck |rk
µ(λ + 1)ψ1 + |b|(1 − α)
k=2
µ(λ + 1)ψ1
r
>1−
µ(λ + 1)ψ1 + |b|(1 − α)
|b|(1 − a)
r > 0 (|z| = r).
−
[µ(λ + 1)ψ1 + |b|(1 − α)]
This proves the inequality (23), and hence also subordination result (18) asserted by Theorem 1. The inequality (19) follows from (18) by taking
(25)
g(z) =
z
∈ K.
1−z
82
H.A. Al-Kharsani, N.M. Al-Areefi
Next, we consider the function
(26)
f1 (z) = z −
µ(λ + 1)ψ1 2
z (0 ≤ α < 1; λ ≥ 0; b ∈ C \ {0}, µ > 0)
(1 − α)|b|
which is a member of the class Mµl,m (α, λ, b). Then by using (18), we have
(27)
z
µ(λ + 1)ψ1
f−1 (z) ≺
2[µ(λ + 1)ψ1 + |b|(1 − α)]
1−z
(z ∈ U ).
It can be easily verified for the function f1 (z) defined by (26) that
µ(λ + 1)ψ1
1
(28)
inf Re
f−1 (z)
=−
(z ∈ U )
z∈U
2[µ(λ + 1)ψ1 + |b|(1 − α)]
2
which completes the proof of Theorem 1.
3
Some Applications
Taking µ = 1 in Theorem 1, we obtain the following:
Corollary 1 If the function f−1 satisfies
(29)
∞
X
[1 + λ(k − 1)]ψk−1 |ck | ≤ m
(λ ≥ 0, m > 0),
k=2
then for every function g in K, one has
(30)
The constant
(λ + 1)ψ1
(f−1 ∗ g)(z) ≺ g(z)
2[(λ + 1)ψ1 + m]
m
Re f−1 (z) > − 1 +
.
(λ + 1)ψ1
(z ∈ U )
(λ + 1)ψ1
cannot be replaced by larger one.
2[(λ + 1)ψ1 + m]
Putting λ = 0 in Theorem 1, we have the following corollary.
Corollary 2 If the function f−1 satisfies
(31)
∞
X
k=2
C(µ, k)|ck | ≤ m,
m > 0,
Superordination Properties for Certain Analytic Functions
83
where C(µ, k) is defined by (13), then for every function g in K, one has
µψ1
(f−1 ∗ g)(z) ≺ g(z)
2[µψ1 + m]
The constant
(z ∈ U ),
m
Re f−1 (z) > − 1 +
µψ1
.
µψ1
cannot be replaced by larger one.
2[µψ1 + m]
Next, letting λ = 1 and µ = 1, in Theorem 1, we obtain the following
corollary.
Corollary 3 If the function f−1 satisfies
∞
X
(32)
k|ck |ψk−1 ≤ m
(m > 0),
k=2
then for every funcion g in K, one has
ψ1
(f−1 ∗ g)(z) ≺ g(z)
2ψ1 + m
The constant
(z ∈ U ),
m
Re f−1 (z) > − 1 +
.
2ψ1
ψ1
cannot be replaced by larger one.
2ψ1 + m
Also, by taking λ = 0 and µ = 1, in Theorem 1, we have the following:
Corollary 4 If the function f satisfies
∞
X
(33)
ψk−1 |ck | ≤ m
(m > 0),
k=2
then for every function g in K, one has
(34)
ψ1
(f−1 ∗ g)(z) ≺ g(z)
2(ψ1 + m)
The constant
(z ∈ U ),
Re f−1
m
<− 1+
ψ1
ψ1
cannot be replaced by larger one.
2(ψ1 + m)
.
84
H.A. Al-Kharsani, N.M. Al-Areefi
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Superordination Properties for Certain Analytic Functions
H.A. Al-Kharsani, N.M. Al-Areefi
Faculty of Science
Department of Mathematics
P.O. Box 838, Dammam 31113, Saudi Arabia
e-mail: najarifi@hotmail.com
85