Volume 9 (2008), Issue 2, Article 40, 5 pp.
p-VALENT MEROMORPHIC FUNCTIONS WITH ALTERNATING
COEFFICIENTS BASED ON INTEGRAL OPERATOR
SH. NAJAFZADEH, A. EBADIAN, AND S. SHAMS
D EPARTMENT OF M ATHEMATICS , FACULTY OF S CIENCE
M ARAGHEH U NIVERSITY, M ARAGHEH , I RAN
najafzadeh1234@yahoo.ie
D EPARTMENT OF M ATHEMATICS , FACULTY OF S CIENCE
U RMIA U NIVERSITY, U RMIA , I RAN
a.ebadian@mail.urmia.ac.ir
Received 11 April, 2007; accepted 15 January, 2008
Communicated by S.S. Dragomir
A BSTRACT. 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.
Key words and phrases: Meromorphic Functions, Alternating Coefficients, Distortion Bounds.
2000 Mathematics Subject Classification. 30C45, 30C50.
1. I NTRODUCTION
Let Σp be the class of functions of the form
f (z) = Az
(1.1)
−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
∞
X
(1.2)
f (z) = Az −p +
(−1)n−1 an z n , an ≥ 0, A ≥ 0.
n=p
Let
(1.3)
Σ∗p (β)
=
f ∈ Σp : Re
z[J (f (z))]0
J (f (z))
< −β, 0 ≤ β < p
and let σp∗ (β) = Σ∗p (β) ∩ σp where
Z
(1.4)
J (f (z)) = (γ − p + 1)
0
121-07
1
(uγ )f (uz)du,
p<γ
2
S H . NAJAFZADEH , A. E BADIAN , AND S. S HAMS
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
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.
2. C OEFFICIENT E STIMATES
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
|an | ≤ A(p − β),
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 |z| < 1.
−
(p
−
2β)
J f (z))
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 = ∞
∞
−pAz −p + P n γ−p+1 an z n − (p − 2β)Az −p − P (p − 2β) γ−p+1 an z n γ+n+1
γ+n+1
n=p
n=p
∞ h
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 |
The last expression is less than or equal to 1 provided
∞ ∞
X
X
γ−p+1
γ−p+1
(n + p)
|an | ≤ 2A(p − β) −
(n − p + 2β)
|an |,
γ+n+1
γ+n+1
n=p
n=p
which is equivalent to
∞
X
n=p
(n + β)
γ−p+1
γ+n+1
which is true by (2.1) so the proof is complete.
|an | ≤ A(p − β)
The converse of Theorem 2.1 is also true for functions in σp∗ (β), where p is an odd number.
J. Inequal. Pure and Appl. Math., 9(2) (2008), Art. 40, 5 pp.
http://jipam.vu.edu.au/
p- VALENT M EROMORPHIC F UNCTIONS
3
Theorem 2.2. A function f (z) in σp is in σp∗ (β) if and only if
∞
X
γ−p+1
an ≤ A(p − β).
(2.2)
(n + β)
γ+n+1
n=p
Proof. According to Theorem 2.1 it is sufficient to prove the “only if” part. Suppose that


∞
P
n
n−1 γ−p+1
−p
an z
n(−1)
−Apz +
γ+n+1


z(J f (z))0
n=p

 < −β.
(2.3)
Re
= Re 
∞

P
(J f (z))
γ−p+1
Az −p +
(−1)n−1 γ+n+1
an z n
n=p
0
f (z))
is real and clearing the denominator
By choosing values of z on the real axis so that (z(J
(J f (z))
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
which is equivalent to
∞
X
(n + β)
n=p
γ−p+1
γ+n+1
an ≤ A(p − β).
Corollary 2.3. If f (z) ∈ σp∗ (β) then
(2.4)
an ≤
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)
3. D ISTORTION B OUNDS AND I MPORTANT P ROPERTIES 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.
Theorem 3.1. Let f (z) = Az −p +
∞
P
(−1)n−1 an z n , an ≥ 0 be in the class σp∗ (β) and β ≥ γ +1
n=p
then
(3.1)
Ar−p −
Proof. Since β ≥ γ + 1, so
A(p − β) p
A(p − β) p
r ≤ |f (z)| ≤ Ar−p +
r .
γ−p+1
γ−p+1
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
J. Inequal. Pure and Appl. Math., 9(2) (2008), Art. 40, 5 pp.
http://jipam.vu.edu.au/
4
S H . NAJAFZADEH , A. E BADIAN , AND S. S HAMS
and we have
|f (z)| = |Az
≤
−p
+
∞
X
(−1)n−1 an z n |
n=p
∞
X
A
A
A(p − β)
+ rp
an ≤ p + r p
.
p
r
r
(γ − p + 1)
n=p
Similarly,
∞
∞
X
A X
A(p − β) p
A
A
n
p
|f (z)| ≥ p −
an r ≥ p − r
an ≥ p −
r .
r
r
r
γ
−
p
+
1
n=p
n=p
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.
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

γ−p+1)

(n + β) γ+n+1 bn ≤ A(p − β).

n=p
After a simple calculation we obtain
Wj (z) = Az
−p
+
∞ X
1−j
n=p
1+j
an +
bn (−1)n−1 z n .
2
2
However,
∞
X
1−j
1+j
γ − p + 1)
(n + β)
an +
bn
γ+n+1
2
2
n=p
∞
∞
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) ∈ σp∗ (β).
Theorem 3.3. Let
fk (z) = Az
−p
+
∞
X
(−1)n−1 an,k z n ∈ σp∗ (β),
k = 1, 2, . . . , m
n=p
J. Inequal. Pure and Appl. Math., 9(2) (2008), Art. 40, 5 pp.
http://jipam.vu.edu.au/
p- VALENT M EROMORPHIC F UNCTIONS
5
then the arithmetic mean of fk (z) defined by
m
1 X
F (z) =
fk (z)
m k=1
(3.3)
is also in the same class.
Proof. Since fk (z) ∈ σp∗ (β), then by (2.2) we have
∞
X
γ−p+1
(3.4)
(n + β)
an,k ≤ A(p − β)
γ
+
n
+
1
n=p
(k = 1, 2, . . . , m).
After a simple calculation we obtain
!
m
∞
X
1 X
Az −p +
(−1)n−1 an,k z n
F (z) =
m k=1
n=p
!
∞
m
X
1 X
−p
n−1
= Az +
(−1)
an,k z n .
m k=1
n=p
However,
∞
X
n=p
(n + β)
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
which in view of Theorem 2.2 yields the proof of Theorem 3.3.
R EFERENCES
[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.
J. Inequal. Pure and Appl. Math., 9(2) (2008), Art. 40, 5 pp.
http://jipam.vu.edu.au/