Journal of Inequalities in Pure and
Applied Mathematics
http://jipam.vu.edu.au/
Volume 5, Issue 1, Article 11, 2004
CONVOLUTION CONDITIONS FOR SPIRALLIKENESS AND CONVEX
SPIRALLIKENESS OF CERTAIN MEROMORPHIC p-VALENT FUNCTIONS
V. RAVICHANDRAN, S. SIVAPRASAD KUMAR, AND K. G. SUBRAMANIAN
D EPARTMENT OF C OMPUTER A PPLICATIONS
S RI V ENKATESWARA C OLLEGE OF E NGINEERING
P ENNALUR , S RIPERUMBUDUR 602 105, I NDIA .
vravi@svce.ac.in
D EPARTMENT OF M ATHEMATICS
S INDHI C OLLEGE
123 P. H. ROAD , N UMBAL , C HENNAI 600 077, I NDIA .
sivpk71@yahoo.com
D EPARTMENT OF M ATHEMATICS
M ADRAS C HRISTIAN C OLLEGE
TAMBARAM , C HENNAI 600 059, I NDIA .
kgsmani@vsnl.net
Received 27 October, 2003; accepted 11 December, 2003
Communicated by H. Silverman
A BSTRACT. In the present investigation, the authors derive necessary and sufficient conditions
for spirallikeness and convex spirallikeness of a suitably normalized meromorphic p-valent function in the punctured unit disk, using convolution. Also we give an application of our result to
obtain a convolution condition for a class of meromorphic functions defined by a linear operator.
Key words and phrases: Meromorphic p-valent functions, Analytic functions, Starlike functions, Convex functions, Spirallike
functions, Convex Spirallike functions, Hadamard product (or Convolution), Subordination, Linear
operator.
2000 Mathematics Subject Classification. Primary 30C45, Secondary 30C80.
1. I NTRODUCTION
Let Σp be the class of meromorphic functions
(1.1)
∞
X
1
f (z) = p +
an z n
z
n=1−p
(p ∈ N := {1, 2, 3, . . .}) ,
which are analytic and p-valent in the punctured unit disk
E∗ := {z : z ∈ C
ISSN (electronic): 1443-5756
c 2004 Victoria University. All rights reserved.
153-03
and 0 < |z| < 1} = E \ {0},
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V. R AVICHANDRAN , S. S IVAPRASAD K UMAR , AND K. G. S UBRAMANIAN
where E := {z : z ∈ C and |z| < 1}.
For two functions f and g analytic in E, we say that the function f (z) is subordinate to g(z)
in E and write
f ≺g
or
f (z) ≺ g(z) (z ∈ E),
if there exists a Schwarz function w(z), analytic in E with
w(0) = 0
and
|w(z)| < 1
(z ∈ E),
such that
f (z) = g(w(z))
(1.2)
(z ∈ E).
In particular, if the function g is univalent in E, the above subordination is equivalent to
f (0) = g(0)
and f (E) ⊂ g(E).
We define two subclasses of meromorphic p-valent functions in the following:
Definition 1.1. Let |λ| < π2 and p ∈ N. Let ϕ be an analytic function in the unit disk E. We
define the classes Spλ (ϕ) and Cpλ (ϕ) by
zf 0 (z)
λ
−iλ
(1.3)
Sp (ϕ) := f ∈ Σp :
≺ −pe
[cos λ ϕ(z) + i sin λ] ,
f (z)
zf 00 (z)
λ
−iλ
(1.4)
Cp (ϕ) := f ∈ Σp : 1 + 0
≺ −pe
[cos λ ϕ(z) + i sin λ] .
f (z)
Analogous to the well known Alexander equivalence [2], we have
1
f ∈ Cpλ (ϕ) ⇔ − zf 0 ∈ Spλ (ϕ)
p
(1.5)
(p ∈ N).
Remark 1.1. For
ϕ(z) =
(1.6)
1 + Az
1 + Bz
(−1 ≤ B < A ≤ 1),
we set
Spλ (ϕ) =: Spλ [A, B]
and
Cpλ (ϕ) =: Cpλ [A, B].
Sp0 (ϕ) =: Sp∗ (ϕ)
and
Cp0 (ϕ) =: Cp (ϕ),
Sp0 [A, B] =: Sp [A, B]
and
Cp0 [A, B] =: Cp [A, B].
For λ = 0, we write
For 0 ≤ α < 1, the classes Spλ [1 − 2α, −1] and Cpλ [1 − 2α, −1] reduces to the classes Spλ (α)
and Cpλ (α) of meromorphic p-valently λ-spirallike functions of order α and meromorphic pvalently λ-convex spirallike functions of order α in E∗ respectively:
0
π
λ
iλ zf (z)
Sp (α) := f ∈ Σp : < e
< −pα cos λ (0 ≤ α < 1; |λ| < ) ,
f (z)
2
00
zf (z)
π
λ
iλ
Cp (α) := f ∈ Σp : < e
1+ 0
< −pα cos λ (0 ≤ α < 1; |λ| < ) .
f (z)
2
The classes Sp0 [1 − 2α, −1] and Cp0 [1 − 2α, −1] reduces to the classes Sp∗ (α) and Cp (α)
of meromorphic p-valently starlike functions of order α and meromorphic p-valently convex
functions of order α in E∗ respectively.
J. Inequal. Pure and Appl. Math., 5(1) Art. 11, 2004
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M EROMORPHIC p- VALENT F UNCTIONS
3
For two functions f (z) given by (1.1) and
g(z) =
(1.7)
∞
X
1
+
bn z n
p
z
n=1−p
(p ∈ N) ,
the Hadamard product (or convolution) of f and g is defined by
∞
X
1
(1.8)
(f ∗ g)(z) := p +
an bn z n =: (g ∗ f )(z).
z
n=1−p
Many important properties of certain subclasses of meromorphic p-valent functions were
studied by several authors including Aouf and Srivastava [1], Joshi and Srivastava [3], Liu and
Srivastava [4], Liu and Owa [5], Liu and Srivastava [6], Owa et al. [7] and Srivastava et al. [9].
Motivated by the works of Silverman et al. [8], Liu and Owa [5] have obtained the following
Theorem 1.2 with λ = 0 for the class Sp∗ (α) and Liu and Srivastava [6] have obtained it for the
classes Spλ (α) (with a slightly different definition of the class).
Theorem 1.2. [6, Theorem 1, p. 14] Let f (z) ∈ Σp . Then
f ∈ Spλ (α)
(0 ≤ α < 1; |λ| <
π
; p ∈ N)
2
if and only if
1 − Ωz
f (z) ∗ p
6 0
=
z (1 − z)2
where
Ω :=
(z ∈ E∗ ),
1 + x + 2p(1 − α) cos λe−iλ
,
2p(1 − α) cos λe−iλ
|x| = 1.
In the present investigation, we extend the Theorem 1.2 for the above defined class Spλ (ϕ). As
a consequence, we obtain a convolution condition for the functions in the class Cpλ (ϕ). Also we
apply our result to obtain a convolution condition for a class of meromorphic functions defined
by a linear operator.
2. C ONVOLUTION CONDITION FOR THE CLASS Spλ (ϕ)
We begin with the following result for the general class Spλ (ϕ):
Theorem 2.1. Let ϕ be analytic in E and be defined on ∂E := {z ∈ C : |z| = 1}. The function
f ∈ Σp is in the class Spλ (ϕ) if and only if
(2.1)
f (z) ∗
1 − Ψz
6= 0
− z)2
(z ∈ E∗ )
z p (1
where
(2.2)
1 + p 1 − e−iλ [cos λ ϕ(x) + i sin λ]
Ψ :=
p {1 − e−iλ [cos λ ϕ(x) + i sin λ]}
(|x| = 1; |λ| <
π
).
2
Proof. In view of (1.3), f (z) ∈ Spλ (ϕ) if and only if
zf 0 (z)
6= −pe−iλ [cos λ ϕ(x) + i sin λ]
f (z)
(z ∈ E∗ ; |x| = 1; |λ| <
π
)
2
or
(2.3)
zf 0 (z) + pe−iλ [cos λ ϕ(x) + i sin λ]f (z) 6= 0
J. Inequal. Pure and Appl. Math., 5(1) Art. 11, 2004
(z ∈ E∗ ; |x| = 1; |λ| <
π
).
2
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4
V. R AVICHANDRAN , S. S IVAPRASAD K UMAR , AND K. G. S UBRAMANIAN
For f ∈ Σp given by (1.1), we have
f (z) = f (z) ∗
(2.4)
1
− z)
z p (1
(z ∈ E∗ )
and
(2.5)
p+1
1
zf (z) = f (z) ∗ p
− p
2
z (1 − z)
z (1 − z)
0
(z ∈ E∗ ).
By making use of the convolutions (2.5) and (2.4) in (2.3), we have
p+1
pe−iλ [cos λ ϕ(x) + i sin λ]
1
f (z) ∗ p
−
+
6= 0
z (1 − z)2 z p (1 − z)
z p (1 − z)
(z ∈ E∗ ; |x| = 1; |λ| <
π
)
2
or
" p e−iλ [cos λϕ(x) + i sin λ] − 1
f (z) ∗
z p (1 − z)2
#
1 + p 1 − e−iλ [cos λϕ(x) + i sin λ] z
+
6= 0
z p (1 − z)2
(z ∈ E∗ ; |x| = 1; |λ| <
π
),
2
which yields the desired convolution condition (2.1) of Theorem 2.1.
By taking λ = 0 in the Theorem 2.1, we obtain the following result for the class Sp∗ (ϕ).
Corollary 2.2. Let ϕ be analytic in E and be defined on ∂E. The function f ∈ Σp is in the class
f ∈ Sp∗ (ϕ) if and only if
f (z) ∗
(2.6)
1 − Υz
6= 0
− z)2
z p (1
(z ∈ E∗ )
where
Υ :=
(2.7)
1 + p (1 − ϕ(x))
p (1 − ϕ(x))
(|x| = 1).
By taking ϕ(z) = (1 + Az)/(1 + Bz), −1 ≤ B < A ≤ 1 in Theorem 2.1, we obtain the
following result for the class Spλ [A, B].
Corollary 2.3. The function f ∈ Σp is in the class Spλ [A, B] if and only if
f (z) ∗
(2.8)
1 − Υz
6= 0
− z)2
z p (1
(z ∈ E∗ )
where
(2.9)
Υ :=
x − B + p(A − B) cos λe−iλ
p(A − B) cos λe−iλ
(|x| = 1).
Remark 2.4. By taking A = 1−2α, B = −1 in the above Corollary 2.3, we obtain Theorem 1.2
of Liu and Srivastava [6].
J. Inequal. Pure and Appl. Math., 5(1) Art. 11, 2004
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M EROMORPHIC p- VALENT F UNCTIONS
5
3. C ONVOLUTION CONDITION FOR THE CLASS Cpλ (ϕ)
By making use of Theorem 2.1, we obtain a convolution condition for functions in the class
in the following:
Cpλ (ϕ)
Theorem 3.1. Let ϕ be analytic in E and be defined on ∂E. The function f ∈ Σp is in the class
Cpλ (ϕ) if and only if
(3.1)
f (z) ∗
p − [2 + p + (p − 1)Ψ]z + (p + 1)Ψz 2
6= 0
z p (1 − z)2
(z ∈ E∗ )
where Ψ is given by (2.2).
Proof. In view of the Alexander-type equivalence (1.5), we find from Theorem 2.1 that f ∈
Cpλ (ϕ) if and only if
0
1 − Ψz
1 − Ψz
0
zf (z) ∗ p
= f (z) ∗ z
6= 0 (z ∈ E∗ )
z (1 − z)2
z p (1 − z)2
which readily yields the desired assertion (3.1) of Theorem 3.1.
By taking λ = 0 in the Theorem 3.1, we obtain the following result of the class Cp (ϕ).
Corollary 3.2. Let ϕ be analytic in E and be defined on ∂E. The function f ∈ Σp is in the class
f ∈ Cp (ϕ) if and only if
(3.2)
f (z) ∗
p − [2 + p + (p − 1)Υ]z + (p + 1)Υz 2
6= 0
z p (1 − z)2
(z ∈ E∗ )
where Υ is given by (2.7).
4. C ONVOLUTION CONDITIONS FOR A CLASS OF FUNCTION DEFINED BY LINEAR
OPERATOR
We begin this section by defining a class Tn+p−1 (ϕ). First of all for a function f (z) ∈ Σp ,
define Dn+p−1 f (z) by
1
n+p−1
D
f (z) = f (z) ∗ p
z (1 − z)n+p
(n+p−1)
(z n+2p−1 f (z))
=
(n + p − 1)!z p
∞
X
1
(m + n + 2p − 1)!
= p+
am z m .
z
(n
+
p
−
1)!(m
+
p)!
m=1−p
By making use of the operator Dn+p−1 f (z), we define the class Tn+p−1 (ϕ) by
Dn+p f (z)
Tn+p−1 (ϕ) = f (z) ∈ Σp : n+p−1
≺ ϕ(z) .
D
f (z)
When
ϕ(z) =
1 + (1 − 2γ)z
1−z
where
γ=
J. Inequal. Pure and Appl. Math., 5(1) Art. 11, 2004
n + p(2 − α)
n+p
(0 ≤ α < 1),
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6
V. R AVICHANDRAN , S. S IVAPRASAD K UMAR , AND K. G. S UBRAMANIAN
the class Tn+p−1 (ϕ) reduces to the following class Tn+p−1 (α) studied by Liu and Owa [5]:
n+p
D f (z)
n + 2p
pα
Tn+p−1 (α) = f (z) ∈ Σp : <
−
<−
.
Dn+p−1 f (z)
n+p
n+p
By making use of Corollary 2.2, we prove the following result for the class Tn+p−1 (ϕ):
Theorem 4.1. The function f (z) ∈ Σp is in the class Tn+p−1 (ϕ) if and only if
(4.1)
f (z) ∗
1 + [(n + p)(1 − Ω) − 1]z
6= 0
z p (1 − z)n+p+1
(z ∈ E ∗ ; |x| = 1),
where Ω is given by
(4.2)
Ω :=
1 + (n + p)(ϕ(x) − 1)
(n + p)(ϕ(x) − 1)
(|x| = 1).
Proof. By making use of the familiar identity
z(Dn+p−1 f (z))0 = (n + p)Dn+p f (z) − (n + 2p)Dn+p−1 f (z),
we have
Dn+p f (z)
z(Dn+p−1 f (z))0
=
(n
+
p)
− (n + 2p),
Dn+p−1 f (z)
Dn+p−1 f (z)
and therefore, by using the definition of the class Tn+p−1 (ϕ), we see that f (z) ∈ Tn+p−1 (ϕ) if
and only if
n + 2p n + p
n+p−1
∗
D
f (z) ∈ Sp
−
ϕ(z) .
p
p
Then, by applying Corollary 2.2 for the function Dn+p−1 f (z) , we have
1 − Ωz
(4.3)
Dn+p−1 f (z) ∗ p
6= 0,
z (1 − z)2
where
h
n
oi
n+p
1 + p 1 − n+2p
−
ϕ(x)
p
p
oi
h
n
Ω=
n+p
p 1 − n+2p
−
ϕ(x)
p
p
=
1 + (n + p) (ϕ(x) − 1)
,
(n + p) (ϕ(x) − 1)
Since
D
n+p−1
|x| = 1.
1
f (z) = f (z) ∗ p
,
z (1 − z)n+p
the condition (4.3) becomes
(4.4)
1 − Ωz
f (z) ∗ g(z) ∗ p
z (1 − z)2
6= 0
where
1
.
− z)n+p
By making use of the convolutions (2.5) and (2.4), it is fairly straight forward to show that
1 − Ωz
1 + [(n + p)(1 − Ω) − 1]z
(4.5)
g(z) ∗ p
=
.
2
z (1 − z)
z p (1 − z)n+p+1
By using (4.5) in (4.4), we see that the assertion in (4.1) follows and thus the proof of our
Theorem 4.1 is completed.
g(z) =
J. Inequal. Pure and Appl. Math., 5(1) Art. 11, 2004
z p (1
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M EROMORPHIC p- VALENT F UNCTIONS
7
By taking
ϕ(z) =
1 + (1 − 2γ)z
1−z
where
n + p(2 − α)
(0 ≤ α < 1)
n+p
in our Theorem 4.1, we obtain the following result of Liu and Owa [5]:
γ=
Corollary 4.2. The function f (z) ∈ Σp is in the class Tn+p−1 (α) if and only if
f (z) ∗
1 + [(n + p)(1 − Ω) − 1]z
6= 0
z p (1 − z)n+p+1
(z ∈ E ∗ ; |x| = 1),
where Ω is given by
Ω=
1 + x + 2p(1 − α)
2p(1 − α)
(|x| = 1).
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[3] S.B. JOSHI AND H.M. SRIVASTAVA, A certain family of meromorphically multivalent functions,
Computers Math. Appl. 38(3/4) (1999), 201–211.
[4] J.-L. LIU AND H.M. SRIVASTAVA, A linear operator and associated families of meromorphically
multivalent functions, J. Math. Anal. Appl., 259 (2001), 566–581.
[5] J.-L. LIU AND S. OWA, On a class of meromorphic p-valent functions involving certain linear operators, Internat. J. Math. Math. Sci., 32 (2002), 271–180.
[6] J.-L. LIU AND H.M. SRIVASTAVA, Some convolution conditions for starlikeness and convexity of
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[7] S. OWA, H.E. DARWISH AND M.K. AOUF, Meromorphic multivalent functions with positive and
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[8] H. SILVERMAN, E.M. SILVIA AND D. TELAGE, Convolution conditions for convexity, starlikeness and spiral-likeness, Math. Zeitschr., 162 (1978), 125–130.
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J. Inequal. Pure and Appl. Math., 5(1) Art. 11, 2004
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