p-VALENT MEROMORPHIC FUNCTIONS WITH
ALTERNATING COEFFICIENTS BASED ON
INTEGRAL OPERATOR
SH. NAJAFZADEH
Department of Mathematics, Faculty of Science
Maragheh University, Maragheh, Iran
p-valent Meromorphic
Functions
Sh. Najafzadeh, A. Ebadian
and S. Shams
vol. 9, iss. 2, art. 40, 2008
Title Page
EMail: najafzadeh1234@yahoo.ie
Contents
A. EBADIAN AND S. SHAMS
JJ
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EMail: a.ebadian@mail.urmia.ac.ir
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Received:
11 April, 2007
Page 1 of 12
Accepted:
15 January, 2008
Communicated by:
S.S. Dragomir
2000 AMS Sub. Class.:
30C45, 30C50.
Key words:
Meromorphic Functions, Alternating Coefficients, Distortion Bounds.
Abstract:
By using a linear operator, a subclass of meromorphically p−valent functions
with alternating coefficients is introduced. Some important properties of this
class such as coefficient bounds, distortion bounds, etc. are found.
Department of Mathematics, Faculty of Science
Urmia University, Urmia, Iran
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Contents
1
Introduction
3
2
Coefficient Estimates
5
3
Distortion Bounds and Important Properties of σp∗ (β)
8
p-valent Meromorphic
Functions
Sh. Najafzadeh, A. Ebadian
and S. Shams
vol. 9, iss. 2, art. 40, 2008
Title Page
Contents
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1.
Introduction
Let Σp be the class of functions of the form
(1.1)
f (z) = Az
−p
+
∞
X
an z n ,
A≥0
n=p
that are regular in the punctured disk ∆∗ = {z : 0 < |z| < 1} and σp be the subclass
of Σp consisting of functions with alternating coefficients of the type
(1.2)
f (z) = Az
−p
+
∞
X
p-valent Meromorphic
Functions
Sh. Najafzadeh, A. Ebadian
and S. Shams
vol. 9, iss. 2, art. 40, 2008
n−1
(−1)
n
an z ,
an ≥ 0,
A ≥ 0.
n=p
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Let
(1.3)
Contents
z[J (f (z))]0
∗
Σp (β) = f ∈ Σp : Re
< −β, 0 ≤ β < p
J (f (z))
and let σp∗ (β) = Σ∗p (β) ∩ σp where
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Z
(1.4)
JJ
J (f (z)) = (γ − p + 1)
1
(uγ )f (uz)du,
p<γ
0
is a linear operator.
With a simple calculation we obtain

∞
P

−p
n−1 γ−p+1

Az
+
(−1)
an z n , f (z) ∈ σp ;

γ+n+1
n=p
(1.5)
J (f (z)) =
∞ P

γ−p+1
−p

Az
+
an z n ,
f (z) ∈ Σp .

γ+n+1
n=p
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For more details about meromorphic p-valent functions, we can see the recent
works of many authors in [1], [2], [3].
Also, Uralegaddi and Ganigi [4] worked on meromorphic univalent functions
with alternating coefficients.
p-valent Meromorphic
Functions
Sh. Najafzadeh, A. Ebadian
and S. Shams
vol. 9, iss. 2, art. 40, 2008
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2.
Coefficient Estimates
Theorem 2.1. Let
f (z) = Az −p +
∞
X
an z n ∈ Σ p .
n=p
If
∞
X
(2.1)
(n + β)
n=p
γ−p+1
γ+n+1
p-valent Meromorphic
Functions
Sh. Najafzadeh, A. Ebadian
|an | ≤ A(p − β),
and S. Shams
vol. 9, iss. 2, art. 40, 2008
then f (z) ∈ Σ∗p (β).
Proof. It is sufficient to show that
z[J f (z))]0
+p
J f (z))
M = z[J f (z))]0
< 1 for
−
(p
−
2β)
J f (z))
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Contents
|z| < 1.
However, by (1.5)
∞
∞
P
P
γ−p+1
γ−p+1
−pAz −p +
n γ+n+1
an z n + pAz −p +
p γ+n+1
an z n
n=p
n=p
M = ∞
∞
P
P
γ−p+1
γ−p+1
−pAz −p +
n γ+n+1 an z n − (p − 2β)Az −p −
(p − 2β) γ+n+1 an z n n=p
∞ h
P
≤
n=p
(n + p)
n=p
2A(p − β) −
∞
P
γ−p+1
γ+n+1
i
(n − p + 2β)
n=p
|an |
γ−p+1
γ+n+1
.
|an |
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The last expression is less than or equal to 1 provided
∞
∞ X
X
γ−p+1
γ−p+1
|an | ≤ 2A(p−β)− (n−p+2β)
|an |,
(n + p)
γ+n+1
γ+n+1
n=p
n=p
which is equivalent to
∞
X
(n + β)
n=p
γ−p+1
γ+n+1
|an | ≤ A(p − β)
p-valent Meromorphic
Functions
Sh. Najafzadeh, A. Ebadian
and S. Shams
vol. 9, iss. 2, art. 40, 2008
which is true by (2.1) so the proof is complete.
The converse of Theorem 2.1 is also true for functions in σp∗ (β), where p is an
odd number.
Theorem 2.2. A function f (z) in σp is in σp∗ (β) if and only if
(2.2)
∞
X
n=p
(n + β)
γ−p+1
γ+n+1
an ≤ A(p − β).
Proof. According to Theorem 2.1 it is sufficient to prove the “only if” part. Suppose
that
z(J f (z))0
(2.3)
Re
(J f (z))


P∞
−p
n−1 γ−p+1
n
−Apz + n=p n(−1)
a
z
n
γ+n+1
 < −β.
= Re 
P
∞
γ−p+1
−p
n−1
n
Az + n=p (−1)
an z
γ+n+1
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0
f (z))
By choosing values of z on the real axis so that (z(J
is real and clearing the
(J f (z))
denominator in (2.3) and letting z → −1 through real values we obtain
!
∞
∞ X
X
γ−p+1
γ−p+1
Ap −
n
an ≥ β A +
an ,
γ+n+1
γ+n+1
n=p
n=p
p-valent Meromorphic
Functions
Sh. Najafzadeh, A. Ebadian
which is equivalent to
∞
X
(n + β)
n=p
γ−p+1
γ+n+1
an ≤ A(p − β).
and S. Shams
vol. 9, iss. 2, art. 40, 2008
Title Page
Corollary 2.3. If f (z) ∈
(2.4)
an ≤
σp∗ (β)
then
Contents
A(p − β)(γ + n + 1)
(n + β)(γ − p + 1)
for n = p, p + 1, . . . .
The result is sharp for functions of the type
(2.5)
fn (z) = Az −p + (−1)n−1
A(p − β)(γ + n + 1) n
z .
(n + β)(γ − p + 1)
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3.
Distortion Bounds and Important Properties of σp∗ (β)
In this section we obtain distortion bounds for functions in the class σp∗ (β) and prove
some important properties of this class, where p is an odd number.
∞
P
Theorem 3.1. Let f (z) = Az −p +
(−1)n−1 an z n , an ≥ 0 be in the class σp∗ (β)
n=p
and β ≥ γ + 1 then
Ar−p −
(3.1)
A(p − β) p
A(p − β) p
r ≤ |f (z)| ≤ Ar−p +
r .
γ−p+1
γ−p+1
Proof. Since β ≥ γ + 1, so
p-valent Meromorphic
Functions
Sh. Najafzadeh, A. Ebadian
and S. Shams
vol. 9, iss. 2, art. 40, 2008
n+β
γ+n+1
≥ 1. Then
∞
∞ X
X
n+β
(γ − p + 1)
an ≤
(γ − p + 1)an ≤ A(p − β),
γ+n+1
n=p
n=p
and we have
|f (z)| = |Az
−p
+
∞
X
n−1
(−1)
n
an z |
n=p
∞
X
A
A
A(p − β)
≤ p + rp
an ≤ p + r p
.
r
r
(γ
−
p
+
1)
n=p
Similarly,
∞
∞
X
A X
A
A
A(p − β) p
n
p
|f (z)| ≥ p −
an r ≥ p − r
an ≥ p −
r .
r
r
r
γ
−
p
+
1
n=p
n=p
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Theorem 3.2. Let
f (z) = Az −p +
∞
X
an z n
and
g(z) = Az −p +
n=p
∞
X
(−1)n−1 bn z n
n=p
be in the class σp∗ (β). Then the weighted mean of f and g defined by
1
Wj (z) = [(1 − j)f (z) + (1 + j)g(z)]
2
is also in the same class.
p-valent Meromorphic
Functions
Sh. Najafzadeh, A. Ebadian
and S. Shams
vol. 9, iss. 2, art. 40, 2008
Proof. Since f and g belong to σp∗ (β), then by (2.2) we have
 P
∞
γ−p+1


(n
+
β)
an ≤ A(p − β),

γ+n+1
 n=p
(3.2)
∞

P


(n + β) γ−p+1)
bn ≤ A(p − β).

γ+n+1
n=p
After a simple calculation we obtain
∞ X
1−j
1+j
−p
Wj (z) = Az +
an +
bn (−1)n−1 z n .
2
2
n=p
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However,
∞
X
n=p
(n + β)
γ − p + 1)
γ+n+1
1−j
1+j
an +
bn
2
2
∞
∞
1−j X
γ − p + 1)
1+j X
γ − p + 1)
=
(n + β)
an +
(n + β)
bn
2
γ+n+1
2
γ+n+1
n=p
n=p
by (3.2)
1−j
1+j
≤
A(p − β) +
A(p − β)
2
2
= A(p − β).
Hence by Theorem 2.2, Wj (z) ∈
p-valent Meromorphic
Functions
Sh. Najafzadeh, A. Ebadian
σp∗ (β).
and S. Shams
Theorem 3.3. Let
vol. 9, iss. 2, art. 40, 2008
fk (z) = Az −p +
∞
X
(−1)n−1 an,k z n ∈ σp∗ (β),
k = 1, 2, . . . , m
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n=p
Contents
then the arithmetic mean of fk (z) defined by
m
1 X
F (z) =
fk (z)
m k=1
(3.3)
(3.4)
n=p
(n + β)
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Proof. Since fk (z) ∈ σp∗ (β), then by (2.2) we have
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is also in the same class.
∞
X
JJ
γ−p+1
γ+n+1
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an,k ≤ A(p − β)
(k = 1, 2, . . . , m).
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After a simple calculation we obtain
!
∞
m
X
1 X
F (z) =
Az −p +
(−1)n−1 an,k z n
m k=1
n=p
!
∞
m
X
X
1
= Az −p +
(−1)n−1
an,k z n .
m k=1
n=p
However,
∞
X
n=p
and S. Shams
(n + β)
p-valent Meromorphic
Functions
Sh. Najafzadeh, A. Ebadian
γ−p+1
γ+n+1
m
1 X
an,k
m k=1
!
by (3.4))
≤
m
1 X
A(p − β) = A(p − β)
m k=1
vol. 9, iss. 2, art. 40, 2008
Title Page
which in view of Theorem 2.2 yields the proof of Theorem 3.3.
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References
[1] H. IRMAK AND S. OWA, Certain inequalities for multivalent starlike and
meromorphically multivalent functions, Bulletin of the Institute of Mathematics,
Academia Sinica, 31(1) (2001), 11–21.
[2] S.B. JOSHI AND H.M. SRIVASTAVA, A certain family of meromorphically multivalent functions, Computers and Mathematics with Applications, 38
(1999), 201–211.
[3] Sh. NAJAFZADEH, A. TEHRANCHI AND S.R. KULKARNI, Application of
differential operator on p-valent meromorphic functions, An. Univ. Oradea Fasc.
Mat., 12 (2005), 75–90.
[4] B.A. URALEGADDI AND M. D. GANIGI, Meromorphic starlike functions with
alternating coefficients, Rendiconti di Mathematica, Serie VII, 11 (1991), 441–
446.
p-valent Meromorphic
Functions
Sh. Najafzadeh, A. Ebadian
and S. Shams
vol. 9, iss. 2, art. 40, 2008
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