ON APPLICATION OF DIFFERENTIAL
SUBORDINATION FOR CERTAIN SUBCLASS OF
MEROMORPHICALLY p-VALENT FUNCTIONS
WITH POSITIVE COEFFICIENTS DEFINED BY
LINEAR OPERATOR
Application Of Differential
Subordination
Waggas Galib Atshan
and S. R. Kulkarni
vol. 10, iss. 2, art. 53, 2009
Title Page
WAGGAS GALIB ATSHAN
S. R. KULKARNI
Department of Mathematics
College of Computer Science And Mathematics
University of Al-Qadisiya, Diwaniya - Iraq
EMail: waggashnd@yahoo.com
Department of Mathematics
Fergusson College,
Pune - 411004, India
EMail: kulkarni_ferg@yahoo.com
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Received:
06 January, 2008
Accepted:
02 May, 2009
Communicated by:
S.S. Dragomir
2000 AMS Sub. Class.:
30C45.
Key words:
Meromorphic functions, Differential subordination, convolution (or Hadamard
product), p-valent functions, Linear operator, δ-Neighborhood, Integral representation, Linear combination, Weighted mean and Arithmetic mean.
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Abstract:
This paper is mainly concerned with the application of differential subordinations for the class of meromorphic multivalent functions with positive
coefficients defined by a linear operator satisfying the following:
−
z p+1 (Ln f (z))0
1 + Az
≺
(n ∈ N0 ; z ∈ U ).
p
1 + Bz
In the present paper, we study the coefficient bounds, δ-neighborhoods and
integral representations. We also obtain linear combinations, weighted and
arithmetic means and convolution properties.
Application Of Differential
Subordination
Waggas Galib Atshan
and S. R. Kulkarni
vol. 10, iss. 2, art. 53, 2009
Title Page
Acknowledgement: The first author, Waggas Galib, is thankful of his wife (Hnd Hekmat Abdulah) for her support of him in his work.
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Contents
1
Introduction
4
2
Coefficient Bounds
7
3
Neighbourhoods and Partial Sums
10
4
Integral Representation
15
5
Linear Combination
16
6
Weighted Mean and Arithmetic Mean
17
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7
Convolution Properties
19
Contents
Application Of Differential
Subordination
Waggas Galib Atshan
and S. R. Kulkarni
vol. 10, iss. 2, art. 53, 2009
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1.
Introduction
Let L(p, m) be a class of all meromorphic functions f (z) of the form:
(1.1) f (z) = z
−p
+
∞
X
ak z k for any m ≥ p,
p ∈ N = {1, 2, . . . },
ak ≥ 0,
k=m
Application Of Differential
Subordination
Waggas Galib Atshan
and S. R. Kulkarni
which are p-valent in the punctured unit disk
U ∗ = {z : z ∈ C, 0 < |z| < 1} = U/{0}.
vol. 10, iss. 2, art. 53, 2009
Definition 1.1. Let f, g be analytic in U . Then g is said to be subordinate to f,
written g ≺ f , if there exists a Schwarz function w(z), which is analytic in U with
w(0) = 0 and |w(z)| < 1 (z ∈ U ) such that g(z) = f (w(z)) (z ∈ U ). Hence
g(z) ≺ f (z) (z ∈ U ), then g(0) = f (0) and g(U ) ⊂ f (U ). In particular, if the
function f (z) is univalent in U , we have the following (e.g. [6]; [7]):
g(z) ≺ f (z)(z ∈ U ) if and only if g(0) = f (0)
and
g(U ) ⊂ f (U ).
Definition 1.2. For functions f (z) ∈ L(p, m) given by (1.1) and g(z) ∈ L(p, m)
defined by
(1.2)
g(z) = z −p +
∞
X
bk z k ,
(bk ≥ 0, p ∈ N, m ≥ p),
we define the convolution (or Hadamard product) of f (z) and g(z) by
(1.3)
(f ∗ g)(z) = z
+
∞
X
k=m
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k=m
−p
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ak b k z k ,
(p ∈ N, m ≥ p, z ∈ U ).
Definition 1.3 ([9]). Let f (z) be a function in the class L(p, m) given by (1.1). We
define a linear operator Ln by
L0 f (z) = f (z),
L1 f (z) = z −p +
∞
X
(p + k + 1)ak z k =
k=m
(z p+1 f (z))0
zp
and in general
(1.4)
Ln f (z) = L(Ln−1 f (z))
∞
X
−p
=z +
(p + k + 1)n ak z k
=
(z
p+1
k=m
n−1
L
zp
vol. 10, iss. 2, art. 53, 2009
Title Page
f (z))0
,
(n ∈ N).
It is easily verified from (1.4) that
(1.5)
Application Of Differential
Subordination
Waggas Galib Atshan
and S. R. Kulkarni
z(Ln f (z))0 = Ln+1 f (z) − (p + 1)Ln f (z),
(f ∈ L(p, m), n ∈ N0 = N ∪ {0}).
1. Liu and Srivastava [4] introduced recently the linear operator when m = 0,
investigating several inclusion relationships involving various subclasses of
meromorphically p-valent functions, which they defined by means of the linear
operator Ln (see [4]).
2. Uralegaddi and Somanatha [10] introduced the linear operator Ln when p = 1
and m = 0.
3. Aouf and Hossen [2] obtained several results involving the linear operator Ln
when m = 0 and p ∈ N.
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We introduce a subclass of the function class L(p, m) by making use of the principle of differential subordination as well as the linear operator Ln .
Definition 1.4. Let A and B (−1 ≤ B < A ≤ 1) be fixed parameters. We say that a
function f (z) ∈ L(p, m) is in the class L(p, m, n, A, B), if it satisfies the following
subordination condition:
(1.6)
1 + Az
z p+1 (Ln f (z))0
≺
p
1 + Bz
(n ∈ N0 ; z ∈ U ).
By the definition of differential subordination, (1.6) is equivalent to the following
condition:
p+1 n
z (L f (z))0 + p Bz p+1 (Ln f (z))0 + pA < 1, (z ∈ U ).
We can write
2β
L p, m, n, 1 −
, −1 = L(p, m, n, β),
p
where L(p, m, n, β) denotes the class of functions in L(p, m) satisfying the following:
Re{−z p+1 (Ln f (z))0 } > β (0 ≤ β < p; z ∈ U ).
Application Of Differential
Subordination
Waggas Galib Atshan
and S. R. Kulkarni
vol. 10, iss. 2, art. 53, 2009
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2.
Coefficient Bounds
Theorem 2.1. Let the function f (z) of the form (1.1), be in L(p, m). Then the
function f (z) belongs to the class L(p, m, n, A, B) if and only if
(2.1)
∞
X
k(1 − B)(p + k + 1)n ak < (A − B)p,
Application Of Differential
Subordination
Waggas Galib Atshan
and S. R. Kulkarni
k=m
where −1 ≤ B < A ≤ 1, p ∈ N, n ∈ N0 , m ≥ p.
The result is sharp for the function f (z) given by
f (z) = z −p +
(A − B)p
zm,
k(1 − B)(p + k + 1)n
vol. 10, iss. 2, art. 53, 2009
m ≥ p.
Proof. Assume that the condition (2.1) is true. We must show that f ∈ L(p, m, n, A, B),
or equivalently prove that
p+1 n
z (L f (z))0 + p (2.2)
Bz p+1 (Ln f (z))0 + Ap < 1.
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We have
∞
P
p+1
−(p+1)
n
k−1
p+1 n
z (−pz
+
k(p + k + 1) ak z ) + p 0
z (L f (z)) + p k=m
∞
Bz p+1 (Ln f (z))0 + Ap = Bz p+1 (−pz −(p+1) + P k(p + k + 1)n ak z k−1 ) + Ap k=m
∞
P
k(p + k + 1)n ak z k+p
k=m
=
∞
P
n a z k+p (A − B)p + B
k(p
+
k
+
1)
k
k=m
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≤
∞
P




n
k(p + k + 1) ak
k=m




< 1.
∞
P


n


k(k + p + 1) ak 
 (A − B)p + B
k=m
The last inequality by (2.1) is true.
Conversely, suppose that f (z) ∈ L(p, m, n, A, B). We must show that the condition (2.1) holds true. We have
p+1 n
z (L f (z))0 + p Bz p+1 (Ln f (z))0 + Ap < 1,
Application Of Differential
Subordination
Waggas Galib Atshan
and S. R. Kulkarni
vol. 10, iss. 2, art. 53, 2009
hence we get
∞
P
k(p + k + 1)n ak z k+p
k=m
< 1.
∞
(A − B)p + B P k(p + k + 1)n ak z k+p Title Page
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k=m
Since Re(z) < |z|, so we have


∞
P


n
k+p


k(p + k + 1) ak z


k=m
Re
< 1.
∞
P


n
k+p


(A
−
B)p
+
B
k(p
+
k
+
1)
a
z


k
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k=m
We choose the values of z on the real axis and letting z → 1− , then we obtain


∞
P


n


k(p + k + 1) ak


k=m
< 1,
∞
P


n


k(p + k + 1) ak 
 (A − B)p + B
k=m
Close
then
∞
X
k(1 − B)(p + k + 1)n ak < (A − B)p
k=m
and the proof is complete.
Corollary 2.2. Let f (z) ∈ L(p, m, n, A, B), then we have
ak ≤
(A − B)p
, k ≥ m.
k(1 − B)(p + k + 1)n
Application Of Differential
Subordination
Waggas Galib Atshan
and S. R. Kulkarni
vol. 10, iss. 2, art. 53, 2009
Corollary 2.3. Let 0 ≤ n2 < n1 , then L(p, m, n2 , A, B) ⊆ L(p, m, n1 , A, B).
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3.
Neighbourhoods and Partial Sums
Definition 3.1. Let −1 ≤ B < A ≤ 1, m ≥ p, n ∈ N0 , p ∈ N and δ ≥ 0. We define
the δ - neighbourhood of a function f ∈ L(p, m) and denote Nδ (f ) such that
(
∞
X
−p
bk z k , and
(3.1) Nδ (f ) = g ∈ L(p, m) : g(z) = z +
k=m
∞
X
k(1 − B)(p + k + 1)n
|ak − bk | ≤ δ
(A
−
B)p
k=m
)
.
Goodman [3], Ruscheweyh [8] and Altintas and Owa [1] have investigated neighbourhoods for analytic univalent functions, we consider this concept for the class
L(p, m, n, A, B).
Theorem 3.2. Let the function f (z) defined by (1.1) be in L(p, m, n, A, B). For
−p
every complex number µ with |µ| < δ, δ ≥ 0, let f (z)+µz
∈ L(p, m, n, A, B), then
1+µ
Nδ (f ) ⊂ L(p, m, n, A, B), δ ≥ 0.
Application Of Differential
Subordination
Waggas Galib Atshan
and S. R. Kulkarni
vol. 10, iss. 2, art. 53, 2009
Title Page
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Page 10 of 24
Proof. Since f ∈ L(p, m, n, A, B), f satisfies (2.1) and we can write for γ ∈ C,
|γ| = 1, that
p+1 n
z (L f (z))0 + p
(3.2)
6= γ.
Bz p+1 (Ln f (z))0 + pA
Equivalently, we must have
(3.3)
(f ∗ Q)(z)
6= 0,
z −p
z ∈ U ∗,
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where
Q(z) = z
−p
+
∞
X
ek z k ,
k=m
γk(1−B)(p+k+1)n
n
such that ek =
, satisfying |ek | ≤ k(1−B)(p+k+1)
and k ≥ m, p ∈ N,
(A−B)p
(A−B)p
n ∈ N0 .
−p
∈ L(p, m, n, A, B), by (3.3),
Since f (z)+µz
1+µ
f (z) + µz −p
1
∗ Q(z) 6= 0,
z −p
1+µ
Application Of Differential
Subordination
Waggas Galib Atshan
and S. R. Kulkarni
vol. 10, iss. 2, art. 53, 2009
and then
1
(f ∗ Q)(z) + µz −p
1+µ
6= 0.
z −p
Now assume that (f ∗Q)(z)
< δ. Then, by (3.4), we have
z −p (3.4)
1 f ∗Q
µ |µ|
1 (f ∗ Q)(z) |µ| − δ
1 + µ z −p + 1 + µ ≥ |1 + µ| − |1 + µ| > |1 + µ| ≥ 0.
z −p
This is a contradiction as |µ| < δ. Therefore (f ∗Q)(z)
≥ δ.
z −p Letting
∞
X
−p
bk z k ∈ Nδ (f ),
g(z) = z +
k=m
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then
(g ∗ Q)(z) ((f − g) ∗ Q)(z) ≤
δ − z −p
z −p
∞
X
k
≤
(ak − bk )ek z ≤
k=m
∞
X
Application Of Differential
Subordination
Waggas Galib Atshan
and S. R. Kulkarni
|ak − bk ||ek ||z|k
k=m
m
< |z|
∞ X
k(1 − B)(p + k + 1)n
(A − B)p
k=m
vol. 10, iss. 2, art. 53, 2009
|ak − bk |
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≤ δ,
Contents
therefore
(g∗Q)(z)
z −p
6= 0, and we get g(z) ∈ L(p, m, n, A, B), so Nδ (f ) ⊂ L(p, m, n, A, B).
Theorem 3.3. Let f (z) be defined by (1.1) and the partial sums S1 (z) and Sq (z) be
defined by S1 (z) = z −p and
Sq (z) = z
+
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m+q−2
−p
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k
ak z ,
q > m, m ≥ p, p ∈ N.
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k=m
Also suppose that
Close
P∞
k=m Ck ak ≤ 1, where
Ck =
k(1 − B)(p + k + 1)n
.
(A − B)p
Then
f ∈ L(p, m, n, A, B)
f (z)
1
Re
,
>1−
Sq (z)
Cq
(i)
(3.5)
(ii)
(3.6)
Re
Sq (z)
f (z)
Cq
>
,
1 + Cq
Application Of Differential
Subordination
Waggas Galib Atshan
and S. R. Kulkarni
z ∈ U, q > m.
Proof.
−p +µz −p
(i) Since z 1+µ
= z −p ∈ L(p, m, n, A, B), |µ| < 1, then by Theorem 3.2, we
have N1 (z −p ) ⊂ L(p, m, n, A, B), p ∈ N(N1 (z −p ) denoting the 1-neighbourhood).
Now since
∞
X
Ck ak ≤ 1,
k=m
then f ∈ N1 (z −p ) and f ∈ L(p, m, n, A, B).
(ii) Since {Ck } is an increasing sequence, we obtain
m+q−2
(3.7)
X
k=m
∞
X
ak + Cq
ak ≤
k=q+m−1
vol. 10, iss. 2, art. 53, 2009
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∞
X
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Ck ak ≤ 1.
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k=m
Close
Setting
G1 (z) = Cq
1
f (z)
− 1−
Sq (z)
Cq
∞
P
Cq
=
ak z k+p
k=q+m−1
1+
m+q−2
P
k=m
ak z k+p
+ 1,
from (3.7) we get
∞
P
k+p
ak z
Cq
G1 (z) − 1 k=q+m−1
=
G1 (z) + 1 m+q−2
∞
P
P
k+p
k+p
2 + 2
ak z ak z
+ Cq
k=m
k=q+m−1
Cq
∞
P
ak
k=q+m−1
≤
2−2
m+q−2
P
k=m
ak − Cq
∞
P
≤ 1.
vol. 10, iss. 2, art. 53, 2009
ak
k=q+m−1
n
f (z)
Sq (z)
This proves (3.5). Therefore, Re(G1 (z)) > 0 and we obtain Re
Now, in the same manner, we can prove the assertion (3.6), by setting
Sq (z)
Cq
G2 (z) = (1 + Cq )
−
.
f (z)
1 + Cq
This completes the proof.
Application Of Differential
Subordination
Waggas Galib Atshan
and S. R. Kulkarni
o
> 1− C1q .
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4.
Integral Representation
In the next theorem we obtain an integral representation for Ln f (z).
Theorem 4.1. Let f ∈ L(p, m, n, A, B), then
Z z
p(Aψ(t) − 1)
n
L f (z) =
dt,
p+1
(1 − Bψ(t))
0 t
where |ψ(z)| < 1, z ∈ U ∗ .
Proof. Let f (z) ∈ L(p, m, n, A, B). Letting − z
y(z) ≺
p+1 (Ln f (z))0
p
= y(z), we have
1 + Az
1 + Bz
Application Of Differential
Subordination
Waggas Galib Atshan
and S. R. Kulkarni
vol. 10, iss. 2, art. 53, 2009
Title Page
y(z)−1 or we can write By(z)−A < 1, so that consequently we have
y(z) − 1
= ψ(z), |ψ(z)| < 1, z ∈ U.
By(z) − A
We can write
−z p+1 (Ln f (z))0
1 − Aψ(z)
=
,
p
1 − Bψ(z)
which gives
p(Aψ(z) − 1)
(L f (z)) = p+1
.
z (1 − Bψ(z))
Hence
Z z
p(Aψ(t) − 1)
n
L f (z) =
dt,
p+1
(1 − Bψ(t))
0 t
and this gives the required result.
n
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5.
Linear Combination
In the theorem below, we prove a linear combination for the class L(p, m, n, A, B).
Theorem 5.1. Let
fi (z) = z
−p
+
∞
X
ak,i z k ,
(ak,i ≥ 0, i = 1, 2, . . . , `, k ≥ m, m ≥ p)
Application Of Differential
Subordination
Waggas Galib Atshan
and S. R. Kulkarni
k=m
belong to L(p, m, n, A, B), then
F (z) =
`
X
vol. 10, iss. 2, art. 53, 2009
ci fi (z) ∈ L(p, m, n, A, B),
i=1
where
P`
i=1 ci
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= 1.
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Proof. By Theorem 2.1, we can write for every i ∈ {1, 2, . . . , `}
∞
X
k(1 − B)(p + k + 1)n
ak,i < 1,
(A − B)p
k=m
therefore
F (z) =
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`
X
ci z −p +
i=1
∞
X
!
ak,i z k
= z −p +
k=m
∞
X
`
X
k=m
i=1
!
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ci ak,i z k .
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However,
∞
X
k(1 − B)(p + k + 1)n
(A − B)p
k=m
Close
`
X
i=1
!
ci ak,i
"
#
`
∞
X
X
k(1 − B)(p + k + 1)n
=
ak,i ci ≤ 1,
(A − B)p
i=1 k=m
then F (z) ∈ L(p, m, n, A, B), so the proof is complete.
6.
Weighted Mean and Arithmetic Mean
Definition 6.1. Let f (z) and g(z) belong to L(p, m), then the weighted mean hj (z)
of f (z) and g(z) is given by
1
hj (z) = [(1 − j)f (z) + (1 + j)g(z)].
2
In the theorem below we will show the weighted mean for this class.
Theorem 6.2. If f (z) and g(z) are in the class L(p, m, n, A, B), then the weighted
mean of f (z) and g(z) is also in L(p, m, n, A, B).
Proof. We have for hj (z) by Definition 6.1,
"
!
!#
∞
∞
X
X
1
−p
k
−p
k
ak z + (1 + j) z +
bk z
hj (z) =
(1 − j) z +
2
k=m
k=m
=z
−p
∞
X
1
+
((1 − j)ak + (1 + j)bk )z k .
2
k=m
Since f (z) and g(z) are in the class L(p, m, n, A, B) so by Theorem 2.1 we must
prove that
∞
X
1
n 1
k(1 − B)(p + k + 1)
(1 − j)ak + (1 + j)bk
2
2
k=m
∞
∞
X
X
1
1
n
= (1 − j)
k(1 − B)(p + k + 1) ak + (1 + j)
k(1 − B)(p + k + 1)n bk
2
2
k=m
k=m
1
1
≤ (1 − j)(A − B)p + (1 + j)(A − B)p.
2
2
Application Of Differential
Subordination
Waggas Galib Atshan
and S. R. Kulkarni
vol. 10, iss. 2, art. 53, 2009
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The proof is complete.
Theorem 6.3. Let f1 (z), f2 (z), . . . , f` (z) defined by
(6.1)
fi (z) = z
−p
+
∞
X
ak,i z k ,
(ak,i ≥ 0, i = 1, 2, . . . , `, k ≥ m, m ≥ p)
k=m
be in the class L(p, m, n, A, B), then the arithmetic mean of fi (z) (i = 1, 2, . . . , `)
defined by
`
h(z) =
(6.2)
1X
fi (z)
` i=1
Application Of Differential
Subordination
Waggas Galib Atshan
and S. R. Kulkarni
vol. 10, iss. 2, art. 53, 2009
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is also in the class L(p, m, n, A, B).
Proof. By (6.1), (6.2) we can write
h(z) =
`
∞
1 X −p X
z +
ak,i z k
` i=1
k=m
Contents
!
= z −p +
∞
X
k=m
`
!
1X
ak,i z k .
` i=1
Since fi (z) ∈ L(p, m, n, A, B) for every i = 1, 2, . . . , `, so by using Theorem 2.1,
we prove that
!
`
∞
X
1X
n
ak,i
k(1 − B)(p + k + 1)
` i=1
k=m
!
`
∞
`
1X X
1X
n
=
k(1 − B)(p + k + 1) ak,i ≤
(A − B)p.
` i=1 k=m
` i=1
The proof is complete.
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7.
Convolution Properties
Theorem 7.1. If f (z) and g(z) belong to L(p, m, n, A, B) such that
(7.1)
f (z) = z
−p
+
∞
X
k
ak z ,
g(z) = z
−p
+
k=m
∞
X
bk z k ,
k=m
then
T (z) = z −p +
∞
X
Application Of Differential
Subordination
Waggas Galib Atshan
and S. R. Kulkarni
(a2k + b2k )z k
vol. 10, iss. 2, art. 53, 2009
k=m
is in the class L(p, m, n, A1 , B1 ) such that A1 ≥ (1 − B1 )µ2 + B1 , where
√
2(A − B)
.
µ= p
m(m + 2)n (1 − B)
Proof. Since f, g ∈ L(p, m, n, A, B), Theorem 2.1 yields
2
∞ X
k(1 − B)(p + k + 1)n
ak ≤ 1
(A
−
B)p
k=m
and
∞ X
k(1 − B)(p + k + 1)n
k=m
(A − B)p
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2
bk
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≤ 1.
We obtain from the last two inequalities
2
∞
X
1 k(1 − B)(p + k + 1)n
(7.2)
(a2k + b2k ) ≤ 1.
2
(A
−
B)p
k=m
Close
However, T (z) ∈ L(p, m, n, A1 , B1 ) if and only if
∞ X
k(1 − B1 )(p + k + 1)n
(7.3)
(a2k + b2k ) ≤ 1,
(A
1 − B1 )p
k=m
where −1 ≤ B1 < A1 ≤ 1, but (7.2) implies (7.3) if
2
k(1 − B1 )(p + k + 1)n
1 k(1 − B)(p + k + 1)n
<
.
(A1 − B1 )p
2
(A − B)p
Hence, if
1 − B1
k(p + k + 1)n 2
1−B
<
α , where α =
.
A1 − B1
2p
A−B
In other words,
k(k + 2)n 2
1 − B1
<
α .
A1 − B1
2
This is equivalent to
A1 − B1
2
>
.
1 − B1
k(k + 2)n α2
So we can write
A1 − B1
2(A − B)2
(7.4)
>
= µ2 .
1 − B1
m(m + 2)n (1 − B)2
Hence we get A1 ≥ (1 − B1 )µ2 + B1 .
Theorem 7.2. Let f (z) and g(z) of the form (7.1) belong to L(p, m, n, A, B). Then
the convolution (or Hadamard product) of two functions f and g belong to the class,
that is, (f ∗ g)(z) ∈ L(p, m, n, A1 , B1 ), where A1 ≥ (1 − B1 )v + B1 and
(A − B)2
v=
.
m(1 − B)2 (m + 2)n
Application Of Differential
Subordination
Waggas Galib Atshan
and S. R. Kulkarni
vol. 10, iss. 2, art. 53, 2009
Title Page
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Proof. Since f, g ∈ L(p, m, n, A, B), by using the Cauchy-Schwarz inequality and
Theorem 2.1, we obtain
∞
X
k(1 − B)(p + k + 1)n p
(7.5)
ak b k
(A − B)p
k=m
! 12
! 12
∞
∞
X
X
k(1 − B)(p + k + 1)n
k(1 − B)(p + k + 1)n
ak
bk
≤1.
≤
(A − B)p
(A − B)p
k=m
k=m
We must find the values of A1 , B1 so that
∞
X
k(1 − B1 )(p + k + 1)n
ak bk < 1.
(7.6)
(A1 − B1 )p
k=m
Therefore, by (7.5), (7.6) holds true if
p
(1 − B)(A1 − B1 )
(7.7)
ak b k ≤
, k ≥ m, m ≥ p, ak 6= 0, bk 6= 0.
(1 − B1 )(A − B)
√
(A−B)p
By (7.5), we have ak bk < k(1−B)(p+k+1)
n , therefore (7.7) holds true if
2
k(1 − B)(p + k + 1)n
k(1 − B1 )(p + k + 1)n
≤
,
(A1 − B1 )p
(A − B)p
Application Of Differential
Subordination
Waggas Galib Atshan
and S. R. Kulkarni
vol. 10, iss. 2, art. 53, 2009
Title Page
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which is equivalent to
(1 − B1 )
k(1 − B)2 (p + k + 1)n
<
.
(A1 − B1 )
(A − B)2 p
Alternatively, we can write
(1 − B1 )
k(1 − B)2 (k + 2)n
<
,
(A1 − B1 )
(A − B)2
Close
to obtain
A1 − B1
(A − B)2
>
= v.
1 − B1
m(1 − B)2 (m + 2)n
Hence we get A1 > v(1 − B1 ) + B1 .
Application Of Differential
Subordination
Waggas Galib Atshan
and S. R. Kulkarni
vol. 10, iss. 2, art. 53, 2009
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References
[1] O. ALTINTAS AND S. OWA, Neighborhoods of certain analytic functions with
negative coefficients, IJMMS, 19 (1996), 797–800.
[2] M.K. AOUF AND H.M. HOSSEN, New criteria for meromorphic p-valent starlike functions, Tsukuba J. Math., 17 (1993), 481–486.
[3] A.W. GOODMAN, Univalent functions and non-analytic curves, Proc. Amer.
Math. Soc., 8 (1957), 598–601.
Application Of Differential
Subordination
Waggas Galib Atshan
and S. R. Kulkarni
vol. 10, iss. 2, art. 53, 2009
[4] J.-L. LIU AND H.M. SRIVASTAVA, Classes of meromorphically multivalent functions associated with the generalized hypergeometric functions, Math.
Comput. Modelling, 39 (2004), 21–34.
[5] J.-L. LIU AND H.M. SRIVASTAVA, Subclasses of meromorphically multivalent functions associated with a certain linear operator, Math. Comput. Modelling, 39 (2004), 35–44.
[6] S.S. MILLER AND P.T. MOCANU, Differential subordinations and univalent
functions, Michigan Math. J., 28 (1981), 157–171.
[7] S.S. MILLER AND P.T. MOCANU, Differential Subordinations : Theory and
Applications, Series on Monographs and Textbooks in Pure and Applied Mathematics, Vol. 225, Marcel Dekker, New York and Basel, 2000.
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[8] St. RUSCHEWEYH, Neighborhoods of univalent functions, Proc. Amer. Math.
Soc., 81 (1981), 521–527.
[9] H.M. SRIVASTAVA AND J. PATEL, Applications of differential subordination to certain subclasses of meromorphically multivalent functions, J. Ineq.
Pure and Appl. Math., 6(3) (2005), Art. 88. [ONLINE: http://jipam.vu.
edu.au/article.php?sid=561]
[10] B.A. URALEGADDI AND C. SOMANATHA, New criteria for meromorphic
starlike univalent functions, Bull. Austral. Math. Soc., 43 (1991), 137–140.
Application Of Differential
Subordination
Waggas Galib Atshan
and S. R. Kulkarni
vol. 10, iss. 2, art. 53, 2009
Title Page
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