Journal of Inequalities in Pure and
Applied Mathematics
COEFFICIENT INEQUALITIES FOR CERTAIN CLASSES OF
MEROMORPHICALLY STARLIKE AND MEROMORPHICALLY
CONVEX FUNCTIONS
volume 4, issue 1, article 17,
2003.
SHIGEYOSHI OWA AND NICOLAE N. PASCU
Department of Mathematics,
Kinki University
Higashi-Osaka, Osaka 577-8502
JAPAN.
EMail: owa@math.kindai.ac.jp
Department of Mathematics
Transilvania University of Brasov
R-2200 Brasov
ROMANIA.
EMail: pascu@info.unitbv.ro
Received 27 November, 2002;
accepted 17 January, 2003.
Communicated by: H.M. Srivastava
Abstract
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2000
Victoria University
ISSN (electronic): 1443-5756
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Abstract
Let Σr be the class of meromorphic functions f(z) in Dr with a simple pole
at the origin. Two subclasses Tr∗ (α) and Cr (α) of Σr are considered. Some
coefficient properties of functions f(z) to be in the classes Tr∗ (α) and Cr (α) of
Σr are discussed. Also, the starlikeness and the convexity of functions f(z) in
Σr are discussed.
2000 Mathematics Subject Classification: Primary 30C45.
Key words: Meromorphic functions, Univalent functions, Starlike functions, Convex
functions.
Contents
1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2
Coefficient Inequalities for Functions . . . . . . . . . . . . . . . . . . . . 5
3
Starlikeness and Convexity of Functions . . . . . . . . . . . . . . . . . 10
References
Coefficient Inequalities for
Certain Classes of
Meromorphically Starlike and
Meromorphically Convex
Functions
Shigeyoshi Owa and
Nicolae N. Pascu
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1.
Introduction
Let Σr denote the class of functions f (z) of the form:
∞
(1.1)
1 X
f (z) = +
an z n
z n=0
which are analytic in the punctured disk Dr = {z ∈ C : 0 < |z| < r 5 1}. A
function f (z) ∈ Σr is said to be starlike of order α if it satisfies the inequality:
zf 0 (z)
>α
(z ∈ Dr )
(1.2)
Re −
f (z)
for some α (0 5 α < 1). We say that f (z) is in the class Tr∗ (α) for such
functions. A function f (z) ∈ Σr is said to be convex of order α if it satisfies the
inequality:
zf 00 (z)
(1.3)
Re − 1 + 0
>α
(z ∈ Dr )
f (z)
for some α (0 5 α < 1). We say that f (z) is in the class Cr (α) if it is convex
of order α in Dr . We note that f (z) ∈ Cr (α) if and only if −zf 0 (z) ∈ Tr∗ (α).
There are many papers discussing various properties of classes consisting of
univalent, starlike, convex, multivalent, and meromorphic functions in the book
by Srivastava and Owa [3].
Ozaki [2] has shown that the necessary and sufficient condition that f (z) ∈
Σr with an = 0 (n = 1, 2, 3, · · · ) is meromorphic and univalent in Dr is that
Coefficient Inequalities for
Certain Classes of
Meromorphically Starlike and
Meromorphically Convex
Functions
Shigeyoshi Owa and
Nicolae N. Pascu
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there should exist the relation:
∞
X
nan rn+1 5 1
n=1
between its coefficients.
Our results in the present paper are an improvement and extension of the
above theorem by Ozaki [2].
Coefficient Inequalities for
Certain Classes of
Meromorphically Starlike and
Meromorphically Convex
Functions
Shigeyoshi Owa and
Nicolae N. Pascu
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2.
Coefficient Inequalities for Functions
Our first result for the functions f (z) ∈ Σr is contained in
Theorem 2.1. If f (z) ∈ Σr satisfies
(2.1)
∞
X
(n + k + |2α + n − k|)|an |rn+1 5 2(1 − α)
n=0
for some α (0 5 α < 1) and k (α < k 5 1), then f (z) ∈ Tr∗ (α).
Proof. For f (z) ∈ Σr , we know that
|zf 0 (z) + kf (z)| − |zf 0 (z) + (2α − k)f (z)|
∞
1 X
n
= (k − 1) +
(n + k)an z z n=0
∞
X
1
n
− (2α − k − 1) +
(2α + n − k)an z .
z n=0
Therefore, applying the condition of the theorem, we have
r |zf 0 (z) + kf (z)| − r |zf 0 (z) + (2α − k)f (z)|
∞
X
5 (k − 1) +
(n + k)|an |rn+1 − (k + 1 − 2α)
+
n=0
∞
X
|2α + n − k||an |r
n=0
Coefficient Inequalities for
Certain Classes of
Meromorphically Starlike and
Meromorphically Convex
Functions
Shigeyoshi Owa and
Nicolae N. Pascu
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n+1
Page 5 of 14
= 2(α − 1) +
∞
X
(n + k + |2α + n − k|)|an |rn+1
n=0
5 0,
which shows that
∞
X
(n + k + |2α + n − k|)|an |rn+1 5 2(1 − α).
n=0
It follows from the above that
zf 0 (z) + kf (z)
zf 0 (z) + (2α − k)f (z) 5 1,
so that
zf 0 (z)
Re −
f (z)
Coefficient Inequalities for
Certain Classes of
Meromorphically Starlike and
Meromorphically Convex
Functions
Shigeyoshi Owa and
Nicolae N. Pascu
>α
(z ∈ Dr ).
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Letting k = 0 in Theorem 2.1, we have
Corollary 2.2. If f (z) ∈ Σr satisfies
(2.2)
∞
X
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(n + α)|an |rn+1 5 1 − α
n=0
for some α ( 12 5 α < 1), then f (z) ∈ Tr∗ (α).
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Theorem 2.1 gives us the following results.
Corollary 2.3. Let the function f (z) ∈ Σr be given by (1.1) with an = |an |e−
then f (z) ∈ Tr∗ (α) if and only if
∞
X
(2.3)
n+1
i
2π
,
(n + α)|an |rn+1 5 1 − α
n=0
for some α ( 12 5 α < 1).
Proof. In view of Theorem 2.1, we see that if the coefficient inequality (2.3)
holds true for some α ( 12 5 α < 1), then f (z) ∈ Tr∗ (α).
Conversely, let f (z) be in the class Tr∗ (α), then
P
1− ∞
nan z n+1
zf 0 (z)
n=0
P
Re −
= Re
>α
n+1
f (z)
1+ ∞
n=0 an z
Coefficient Inequalities for
Certain Classes of
Meromorphically Starlike and
Meromorphically Convex
Functions
Shigeyoshi Owa and
Nicolae N. Pascu
1
for all z ∈ Dr . Letting z = re 2π i , we have that an z n+1 = |an |rn+1 . This implies
that
!
∞
∞
X
X
1−
n|an |rn+1 = α 1 +
|an |rn+1 ,
n=0
n=0
which is equivalent to (2.3).
1
2
5α5
1
1+|a0 |
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Example 2.1. The function f (z) given by
1
1 − α − α|a0 | iθ n
f (z) = + a0 +
e z
z
n+α
belongs to the class Tr∗ (α) for some real θ with
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< 1.
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Remark 2.1. If f (z) ∈ Σr with a0 = 0, then Corollary 2.3 holds true for some
α (0 5 α < 1).
Corollary 2.4. Let the function f (z) ∈ Σr be given by (1.1) with an = 0, then
f (z) ∈ Tr∗ (α) if and only if
∞
X
(n + α)an rn+1 5 1 − α
n=0
for some α ( 21 5 α < 1).
Remark 2.2. If f (z) ∈ Σr with a0 = 0, then Corollary 2.4 holds true for
0 5 α < 1.
Remark 2.3. Juneja and Reddy [1] have given that f (z) ∈ Σ1 with a0 = 0 and
an = 0 belongs to the class T1∗ (α) if and only if
∞
X
Coefficient Inequalities for
Certain Classes of
Meromorphically Starlike and
Meromorphically Convex
Functions
Shigeyoshi Owa and
Nicolae N. Pascu
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(n + α)an 5 1 − α.
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n=1
Theorem 2.5. If f (z) ∈ Σr satisfies
(2.4)
∞
X
n(n + α)|an |rn+1 5 1 − α
n=1
for some α (0 5 α < 1), then f (z) belongs to the class Cr (α).
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Proof. Noting that f (z) ∈ Cr (α) if and only if −zf 0 (z) ∈ Tr∗ (α), and that
∞
1 X
−zf (z) = −
nan z n ,
z n=1
0
we complete the proof of the theorem with the aid of Theorem 2.1.
n+1
Corollary 2.6. Let the function f (z) ∈ Σr be given by (1.1) with an = |an |e− 2π i ,
then f (z) ∈ Cr (α) if and only if the inequality (2.4) holds true for some
α (0 5 α < 1).
Example 2.2. The function f (z) given by
1
1−α
f (z) = + a0 +
eiθ z n
z
n(n + α)
belongs to the class Cr (α) for some real θ with 0 5 α < 1.
Corollary 2.7. Let the function f (z) ∈ Σr be given by (1.1) with an = 0, then
f (z) ∈ Cr (α) if and only if
(2.5)
∞
X
n=1
for some α (0 5 α < 1).
n(n + α)an rn+1 5 1 − α
Coefficient Inequalities for
Certain Classes of
Meromorphically Starlike and
Meromorphically Convex
Functions
Shigeyoshi Owa and
Nicolae N. Pascu
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3.
Starlikeness and Convexity of Functions
We consider the radius problems for starlikeness and convexity of functions
f (z) belonging to the class Σr .
Theorem 3.1. A function f (z) ∈ Σr belongs to the class Tr∗ (α) for 0 5 r < r0 ,
where r0 is the smallest positive root of the equation
(3.1)
α|a0 |r3 − (δ + 1 − α)r2 − α|a0 |r + 1 − α = 0,
and
v
v
u∞
u∞
uX
uX 1
n|an |2 + αt
|an |2 .
δ=t
n
n=1
n=1
(3.2)
Coefficient Inequalities for
Certain Classes of
Meromorphically Starlike and
Meromorphically Convex
Functions
Shigeyoshi Owa and
Nicolae N. Pascu
Proof. Using the Cauchy inequality, we see that
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∞
X
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(n + α)|an |rn+1
n=0
= α|a0 |r +
∞
X
|an |rn+1
n=1
v
v
v
v
u∞
u∞
u∞
u∞
uX
uX
uX 1
uX
2
2n+2
2
t
t
t
5 α|a0 |r +
n|an |
nr
+α
|an | t
nr2n+2
n
n=1
n=1
n=1
n=1
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s
= α|a0 |r +
v
v

u∞
u∞
uX 1
uX
t
|an |2 
n|an |2 + αt
n
(1 − r2 )2
n=1
n=1
r4
r2
δ < 1 − α,
1 − r2
where δ is given by (3.2). Therefore, an application of Corollary 2.2 gives us
that f (z) ∈ Tr∗ (α) for 0 5 r < r0 .
= α|a0 |r +
Letting a0 = 0 in Theorem 3.1, we have
Corollary 3.2. A function f (z) ∈ Σr with a0 = 0 belongs to the class Tr∗ (α)
for 0 5 r < r0 , where
r
δ
r0 = 1 −
δ+1−α
and δ is given by (3.2).
Example 3.1. If we consider the function f (z) given by
then f (z) ∈
(θn is real),
Tr∗ (α)
for 0 5 r < r0 with
v
v
u∞
u∞
uX 1
uX 1
t
+
α
δ=t
2
n4
n
n=1
n=1
=
p
p
ζ(2) + α ζ(4) = π
Shigeyoshi Owa and
Nicolae N. Pascu
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∞
1 X 1 iθn n
√ e z
f (z) = +
z n=1 n n
Coefficient Inequalities for
Certain Classes of
Meromorphically Starlike and
Meromorphically Convex
Functions
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6 3 10
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Further, letting α = 0, we have that
π
δ = √ u 1.282550
6
and
s √
6
r0 = √
u 0.661896.
6+π
Finally, for convexity of functions f (z), we derive
Theorem 3.3. A function f (z) ∈ Σr belongs to the class Cr (α) for 0 5 r < r1 ,
where
r
σ
r1 = 1 −
σ+1−α
and
v
v
u∞
u∞
uX
uX
σ=t
n3 |an |2 + αt
n|an |2 .
n=1
n=1
Shigeyoshi Owa and
Nicolae N. Pascu
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Example 3.2. Let us consider the function f (z) given by
∞
1 X 1
√ eiθn z n
f (z) = +
2
z n=1 n n
Coefficient Inequalities for
Certain Classes of
Meromorphically Starlike and
Meromorphically Convex
Functions
(θn is real).
We see that f (z) ∈ Cr (α) for 0 5 r < r0 with
1
πα
.
δ=π √ + √
6 3 10
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Taking α = 0, we obtain
π
δ = √ u 1.282550
6
and
s √
6
r0 = √
u 0.661896.
6+π
Coefficient Inequalities for
Certain Classes of
Meromorphically Starlike and
Meromorphically Convex
Functions
Shigeyoshi Owa and
Nicolae N. Pascu
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References
[1] O.P. JUNEJA AND T.R. REDDY, Meromorphic starlike and univalent
functions with positive coefficients, Ann. Univ. Mariae Curie-Sklodowska,
39(1985), 65–76.
[2] S. OZAKI, Some remarks on the univalency and multivalency of functions,
Sci. Rep. Tokyo Bunrika Daigaku, 2(1934), 41–55.
[3] H.M. SRIVASTAVA AND S. OWA (Ed.), Current Topics in Analytic Function Theory, World Scientific Publishing Company, Singapore, New Jersey,
London, and Hong Kong, 1992.
Coefficient Inequalities for
Certain Classes of
Meromorphically Starlike and
Meromorphically Convex
Functions
Shigeyoshi Owa and
Nicolae N. Pascu
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