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Journal of Applied Mathematics
Volume 2012, Article ID 494917, 12 pages
doi:10.1155/2012/494917
Research Article
A New Class of Meromorphic Functions Associated
with Spirallike Functions
Lei Shi, Zhi-Gang Wang, and Jing-Ping Yi
School of Mathematics and Statistics, Anyang Normal University, Henan, Anyang 455002, China
Correspondence should be addressed to Lei Shi, shilei 04@yahoo.com.cn
Received 22 September 2012; Revised 26 October 2012; Accepted 31 October 2012
Academic Editor: Alberto Cabada
Copyright q 2012 Lei Shi et al. This is an open access article distributed under the Creative
Commons Attribution License, which permits unrestricted use, distribution, and reproduction in
any medium, provided the original work is properly cited.
We introduce a new class of meromorphic functions associated with spirallike functions. Such
results as subordination property, integral representation, convolution property, and coefficient
inequalities are proved.
1. Introduction
Let Σ denote the class of functions f of the form
fz ∞
1 ak zk ,
z k0
1.1
which are analytic in the punctured open unit disk
U∗ : {z : z ∈ C, 0 < |z| < 1} : U \ {0}.
1.2
Let P denote the class of functions p given by
pz 1 ∞
pk zk
z ∈ U,
1.3
k1
which are analytic in U and satisfy the condition
Re pz > 0
z ∈ U.
1.4
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Let f, g ∈ Σ, where f is given by 1.1 and g is defined by
gz ∞
1 bk zk ,
z k0
1.5
then the Hadamard product or convolution f ∗ g is defined by
∞
1 f ∗ g z : ak bk zk : g ∗ f z.
z k0
1.6
For two functions f and g, analytic in U, we say that the function f is subordinate to g
in U and write
fz ≺ gz
z ∈ U,
1.7
if there exists a Schwarz function ω, which is analytic in U with
ω0 0,
|ωz| < 1 z ∈ U,
1.8
such that
fz gωz
z ∈ U.
1.9
Indeed, it is known that
fz ≺ gz
z ∈ U ⇒ f0 g0,
fU ⊂ gU.
1.10
Furthermore, if the function g is univalent in U, then we have the following equivalence:
fz ≺ gz
z ∈ U ⇐⇒ f0 g0,
fU ⊂ gU.
1.11
A function f ∈ Σ is said to be in the class MS∗ β of meromorphic starlike functions of
order β if it satisfies the inequality
zf z
Re
fz
< −β
z ∈ U; 0 β < 1 .
1.12
For the real number β 0 < β < 1, we know that
fz
zf z
1
1 < −β.
zf z 2β < 2β ⇐⇒ Re fz
1.13
Journal of Applied Mathematics
3
If the complex number α satisfies the condition
α −
1 1
< ,
2 2
1.14
it can be easily verified that
fz
1
1 zf z
1 > 1.
zf z 2α < 2|α| ⇐⇒ Re − α fz
1.15
We now introduce and investigate the following class of meromorphic functions.
Definition 1.1. A function f ∈ Σ is said to be in the class MSα if it satisfies the inequality
1 zf z
>1
Re −
α fz
z ∈ U; α −
1 1
<
.
2 2
1.16
Remark 1.2. For 0 < α < 1, the class MSα is the familiar class of meromorphic starlike
functions of order α.
Remark 1.3. If α |α|eiψ −π/2 < ψ < π/2, then the condition 1.16 is equivalent to
−iψ zf z
< −|α|
Re e
fz
1.17
z ∈ U,
which implies that f belongs to the class of meromorphic spirallike functions. Thus, the class
of meromorphic spirallike functions is a special case of the class MSα .
For some recent investigations on spirallike functions and related functions, see,
for example, the earlier works 1–9 and the references cited in each of these earlier
investigations.
Remark 1.4. The function
fz z−1 1 − z2αRe1/α−1
z ∈ U∗ ; α −
1 1
<
2 2
1.18
belongs to the class MSα .
It is clear that
1
α −
>1
Re
α
1 1
<
.
2 2
1.19
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Journal of Applied Mathematics
Then, for the function f given by 1.18, we know that
1
1
1 zf z
z
Re
2 Re
Re −
−1
α fz
α
α
1−z
1
1
− Re
1 1,
> Re
α
α
1.20
which implies that f ∈ MSα .
In this paper, we aim at deriving the subordination property, integral representation,
convolution property, and coefficient inequalities of the function class MSα .
2. Preliminary Results
In order to derive our main results, we need the following lemmas.
Lemma 2.1. Let λ be a complex number. Suppose also that the sequence {Ak }∞
k0 is defined by
A0 2λ,
Ak1
2λ
k2
k
1 Al
k ∈ N0 : N ∪ {0}.
2.1
l0
Then
Ak k
1 2λ j
k ∈ N0 .
k 1! j0
2.2
Proof. From 2.1, we know that
k 2Ak1 2λ 1 k
Al ,
l0
k−1
k 1Ak 2λ 1 Al .
2.3
l0
By virtue of 2.3, we find that
Ak1 k 1 2λ
Ak
k2
k ∈ N0 .
2.4
Thus, for k 1, we deduce from 2.4 that
Ak k
Ak
A3 A2 A1
1 ···
·
·
· A0 2λ j .
Ak−1
A2 A1 A0
k 1! j0
By virtue of 2.1 and 2.5, we get the desired assertion 2.2 of Lemma 2.1.
2.5
Journal of Applied Mathematics
5
Lemma 2.2 Jack’s Lemma 10. Let φ be a nonconstant regular function in U. If |φ| attains its
maximum value on the circle |z| r < 1 at z0 , then
z0 φ z0 tφz0 ,
2.6
for some real number t t 1.
3. Main Results
We begin by deriving the following subordination property of functions belonging to the
class MSα .
Theorem 3.1. A function f ∈ MSα if and only if
−
zf z
1
z
≺ 1 2α Re
−1
fz
α
1−z
z ∈ U∗ ; α −
1 1
<
.
2 2
3.1
Proof. Suppose that
−1/α zf z/fz − 1 − i Im1/α
hz :
Re1/α − 1
z ∈ U; f ∈ MSα .
3.2
We easily know that h ∈ P, which implies that
−1/α zf z/fz − 1 − i Im1/α 1 ωz
Re1/α − 1
1 − ωz
z ∈ U; f ∈ MSα ,
3.3
z ∈ U,
3.4
where ω is analytic in U with ω0 0 and |ωz| < 1 z ∈ U.
It follows from 3.3 that
−
zf z
1
ωz
1 2α Re
−1
fz
α
1 − ωz
which is equivalent to the subordination relationship 3.1.
On the other hand, the above deductive process can be converse. The proof of
Theorem 3.1 is thus completed.
Theorem 3.2. Let f ∈ MSα . Then
fz z
1
ωt
1
· exp −2α Re
−1
dt
z
α
0 t1 − ωt
where ω is analytic in U with ω0 0 and |ωz| < 1 z ∈ U.
z ∈ U∗ ,
3.5
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Journal of Applied Mathematics
Proof. For f ∈ MSα , by Theorem 3.1, we know that 3.1 holds true. It follows that
−
zf z
1
ωz
1 2α Re
−1
fz
α
1 − ωz
z ∈ U,
3.6
where ω is analytic in U with ω0 0 and |ωz| < 1 z ∈ U.
We now find from 3.6 that
f z 1
1
ωz
−2α Re
−1
fz z
α
z1 − ωz
z ∈ U∗ ,
3.7
which, upon integration, yields
z
ωt
1
dt
log zfz −2α Re
−1
α
0 t1 − ωt
z ∈ U.
3.8
The assertion 3.5 of Theorem 3.2 can be easily derived from 3.8.
Theorem 3.3. Let f ∈ MSα . Then
fz ∗
1 − eiθ z 2αRe1/α − 1eiθ 1 − z
z1 − z
2
∗
/ 0 z ∈ U ; 0 < θ < 2π.
3.9
Proof. Assume that f ∈ MSα . By Theorem 3.1, we know that 3.1 holds, which implies that
−
iθ
zf z
1
e
1
2α
Re
−
1
/
fz
α
1 − eiθ
z ∈ U∗ ; 0 < θ < 2π.
3.10
It is easy to see that the condition 3.10 can be written as follows:
1−e
iθ
1
zf z 1 − e 2α Re
− 1 eiθ fz /
0.
α
iθ
3.11
We note that
z
1
1
1
,
fz ∗
z
1−z
z1 − z
1 − 2z
z
1
−zf z fz ∗
fz ∗
−
.
2
z 1 − z
z1 − z2
fz fz ∗
3.12
Thus, by substituting 3.12 into 3.11, we get the desired assertion 3.9 of Theorem 3.3.
Journal of Applied Mathematics
7
Theorem 3.4. Let λ Re1/α − 1|α|. If f ∈ MSα , then
|ak | k
1 2λ j
k ∈ N0 .
k 1! j0
3.13
The inequality 3.13 is sharp for the function given by
fz 1
z1 − z2−2α
0 < α < 1.
3.14
Proof. Suppose that
−1/α zf z/fz − 1 − i Im1/α
.
hz :
Re1/α − 1
3.15
We easily know that h ∈ P.
If we put
hz 1 h1 z h2 z2 · · · ,
3.16
|hk | 2 k ∈ N.
3.17
it is known that
From 3.15, we have
−
1
1 zf z
1
− 1 − i Im
Re
− 1 hz.
α fz
α
α
3.18
We now set
1
A : 1 i Im
,
α
1
B : Re
− 1.
α
3.19
−zf z αA αBhzfz.
3.20
It follows from 3.18 that
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Journal of Applied Mathematics
Combining 1.1, 3.16, and 3.20, we obtain
1
− z − 2 a1 2a2 z · · · kak zk−1 · · ·
z
1
k
k
a0 a1 z · · · ak z · · · .
1 αBh1 z · · · αBhk z · · ·
z
3.21
In view of 3.21, we get
a0 αBh1 0,
3.22
−kak ak αBak−1 h1 ak−2 h2 · · · a0 hk hk1 k ∈ N.
3.23
From 3.17 and 3.22, we obtain
|a0 | 2|α|B 2λ.
3.24
Moreover, we deduce from 3.17 and 3.23 that
2|α|B
|ak | k1
1
k−1
|al |
l0
2λ
k1
1
k−1
k ∈ N.
|al |
3.25
l0
Next, we define the sequence {Ak }∞
k0 as follows:
A0 2λ,
Ak1
2λ
k2
k
1 Al
k ∈ N0 .
3.26
l0
In order to prove that
3.27
|ak | Ak ,
we make use of the principle of mathematical induction. By noting that
|a0 | 2λ A0 .
3.28
l 0, 1, 2, . . . , k; k ∈ N0 .
3.29
Therefore, assuming that
|al | Al
Combining 3.25 and 3.26, we get
2λ
|ak1 | k2
1
k
l0
|al |
2λ
k2
k
1 Al
l0
Ak1 .
3.30
Journal of Applied Mathematics
9
Hence, by the principle of mathematical induction, we have
k ∈ N0 |ak | Ak
3.31
as desired.
By means of Lemma 2.1 and 3.26, we know that
Ak k
1 2λ j
k ∈ N0 .
k 1! j0
3.32
Combining 3.31 and 3.32, we readily get the coefficient estimates asserted by Theorem 3.4.
For the sharpness, we consider the function f given by 3.14. A simple calculation
shows that
zf z
Re −
fz
1 2α − 3z
Re
1−z
2 − α > α.
3.33
Thus, the function f belongs to the class MSα . Since 0 < α < 1, we have
λ 1 − α.
3.34
Then f becomes
−1
fz z 1 − z
−2λ
−1
z
∞
−2λ
n0
n
−z
n
∞
1 2λ2λ 1 · · · 2λ n n
z .
z n0
n 1!
3.35
This completes the proof of Theorem 3.4.
Theorem 3.5. If f ∈ Σ satisfies the inequality
∞
k |k 2α||ak | 1 − |2α − 1|
k0
α −
1 1
<
,
2 2
3.36
then f ∈ MSα .
Proof. To prove f ∈ MSα , it suffices to show that
fz
1
1 zf z 2α < 2|α|
z ∈ U,
3.37
zf z 2αfz <1
zf z
z ∈ U∗ .
3.38
which is equivalent to
10
Journal of Applied Mathematics
From 3.36, we know that
1−
∞
k|ak | |2α − 1| k0
∞
|k 2α||ak | > 0.
3.39
k0
Now, by the maximum modulus principle, we deduce from 1.1 and 3.39 that
zf z 2αfz zf z
2α − 1 ∞ k 2αa zk1 k
k0
k1
−1 ∞
ka
z
k
k0
|2α − 1| 1−
∞
k0 |k
∞
k0
2α||ak ||z|k1
k|ak ||z|k1
|k 2α||ak |
|2α − 1| ∞
k0
<
1− ∞
k0 k|ak |
3.40
1,
which implies that the assertion of Theorem 3.5 holds.
Theorem 3.6. If f ∈ Σ satisfies the condition
1 zf z − zf z < 1 − α
f z
fz 2α
1
<α<1 ,
2
3.41
then f ∈ MSα .
Proof. Define the function ϕ by
zf z/fz 1
ϕz : zf z/fz 2α − 1
z ∈ U.
3.42
Then we see that ϕ is analytic in U with ϕ0 0.
It follows from 3.42 that
−
zf z 1 1 − 2αϕz
.
fz
1 − ϕz
3.43
By differentiating both sides of 3.43 logarithmically, we obtain
1
21 − αzϕ z
zf z zf z
−
.
f z
fz
1 1 − 2αϕz 1 − ϕz
3.44
Journal of Applied Mathematics
11
From 3.41 and 3.44, we find that
1−α
21
−
αzϕ
zf
zf
z
z
z
1 .
−
<
1 1 − 2αϕz 1 − ϕz f z
fz
2α
3.45
Next, we claim that |ϕz| < 1. Indeed, if not, there exists a point z0 ∈ U such that
max ϕz ϕz0 1.
3.46
|z||z0 |
By Lemma 2.2, we have
ϕz0 eiθ ,
z0 ϕ z0 teiθ
t 1.
3.47
Moreover, for z z0 , we find from 3.44 and 3.47 that
1 z0 f z0 − z0 f z0 f z0 fz0 21 − αteiθ
1 1 − 2αeiθ 1 − eiθ 21 − αt
1 21 − 2α cos θ 1 − 2α2 ·
1
1−α
<α<1 .
2α
2
3.48
√
2 − 2 cos θ
But 3.48 contradicts to 3.45. Therefore, we conclude that |ϕz| < 1, that is
zf z/fz 1 < 1,
zf z/fz 2α − 1 3.49
which shows that f ∈ MSα .
Acknowledgments
The present investigation was supported by the National Natural Science Foundation under
Grants 11101053 and 11226088, the Key Project of Chinese Ministry of Education under Grant
211118, the Excellent Youth Foundation of Educational Committee of Hunan Province under
Grant 10B002, the Open Fund Project of the Key Research Institute of Philosophies and Social
Sciences in Hunan Universities under Grants 11FEFM02 and 12FEFM02, and the Key Project
of Natural Science Foundation of Educational Committee of Henan Province under Grant
12A110002 of the People’s Republic of China. The authors would like to thank the referees
for their careful reading and valuable suggestions which essentially improved the quality of
this paper.
12
Journal of Applied Mathematics
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