Hindawi Publishing Corporation
International Journal of Mathematics and Mathematical Sciences
Volume 2012, Article ID 682162, 12 pages
doi:10.1155/2012/682162
Research Article
A Subclass of Meromorphic Close-to-Convex
Functions of Janowski’s Type
Young Jae Sim and Oh Sang Kwon
Department of Mathematics, Kyungsung University, Busan 608-736, Republic of Korea
Correspondence should be addressed to Oh Sang Kwon, oskwon@ks.ac.kr
Received 9 April 2012; Accepted 19 June 2012
Academic Editor: Aloys Krieg
Copyright q 2012 Y. J. Sim and O. S. Kwon. 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 subclass ΣA, B −1 ≤ B < A ≤ 1 of functions which are analytic in the punctured
unit disk and meromorphically close-to-convex. We obtain some coefficients bounds and some
argument and convolution properties belonging to this class.
1. Introduction
Let HU be the set of all analytic functions on the open unit disk U : {z : |z| < 1}, and let A
be the subclass of HU which contains the functions normalized by f0 0 and f 0 1.
A function fz ∈ A is said to be starlike of order α if and only if
zf z
fz
1.1
> α,
for some α 0 ≤ α < 1 and for all z ∈ U. The class of starlike functions of order α is denoted
by S∗ α.
If f and g are analytic in U, we say that f is subordinate to g, written as follows:
f ≺g
in U or
fz ≺ gz
z ∈ U,
1.2
if there exists a Schwarz function wz, which is analytic in U with
w0 0,
|wz| < 1 z ∈ U
1.3
2
International Journal of Mathematics and Mathematical Sciences
such that
fz gwz
z ∈ U.
1.4
In particular, if the function g is univalent in U, the above subordination is equivalent to
f0 g0,
fU ⊂ gU.
1.5
The Hadamard product or convolution of two series
∞
an zn ,
fz gz n0
∞
bn zn
1.6
n0
is defined by
∞
an bn zn .
f ∗ g z 1.7
n0
For a convex function f, it follows from Alexander’s Theorem that zf z fz ∗ z/1 − z2 is a starlike function. In view of the identity fz fz ∗ z/1 − z, it is then clear that the
classes of convex and starlike functions can be unified by considering functions f satisfying
f ∗ g is starlike for a fixed function g ∈ A.
Though the convolution of two univalent or starlike functions does not need be
univalent, it is well-known that the classes of starlike, convex, and close-to-convex functions
are closed under convolution with convex functions. These results were later extended to
convolution with prestarlike functions. For α < 1, the class Rα of prestarlike functions of
order α is defined by
Rα :
f ∈A:f∗
z
1 − z2−2α
∗
∈ S α ,
1.8
while R1 consists of f ∈ A satisfying fz/z > 1/2.
By using the convex hull method 1, 2 and the method of differential subordination
3, Shanmugam 4 introduced and investigated convolution properties of various subclasses of analytic functions. Ali et al. 5 and Supramaniam et al. 6 investigated these
properties for subclasses of multivalent starlike and convex functions. And Chandrashekar
et al. 7 also investigated these properties for the functions with respect to symmetric points,
conjugate, or symmetric conjugate points. More results using the convex hull method and the
method of differential subordination can be found in 8, 9.
Let Σ denote the class of all univalent meromorphic functions fz normalized by
fz ∞
1 ak zk ,
z k1
1.9
International Journal of Mathematics and Mathematical Sciences
3
which are analytic in the punctured unit disk
U∗ {z : z ∈ C, 0 < |z| < 1} U − {0}.
1.10
We denote by Σ∗ α the subclass of Σ consisting of gz formed by
gz ∞
1 bk zk ,
z k1
1.11
which are meromorphic starlike of order α in U∗ . In particular, we denote by Σ∗ , when α 0.
Also, a function fz of the form 1.9 is said to be meromorphic close-to-convex in U∗ if there
is a gz in Σ∗ such that
zf z
gz
z ∈ U.
<0
1.12
We denote Σk the set of functions close-to-convex in U∗ .
In a recent paper, Gao and Zhou 10 introduced an interesting subclass Ks of the
analytic function class A and the univalent function S, which contains the functions f satisfying the following inequality:
z2 f z
gzg−z
< 0 z ∈ U,
1.13
for some gz ∈ S∗ 1/2. Here, S∗ 1/2 is the class of starlike functions of order 1/2 in U.
After that, many classes related to Ks investigated and studied by some authors. Especially,
Wang et al. 11, 12, Kowalczyk and Leś-Bomba 13, Xu et al. 14, and Seker 15 introduced
the generalized class of the class Ks . And they gave some properties of analytic functions in
each classes.
In this paper, we define a class of meromorphic functions of Janowski’s type, related
to the meromorphically close-to-convex functions as follows.
Definition 1.1. Let f ∈ Σ and be given by 1.9 and −1 ≤ B < A ≤ 1. Then f ∈ ΣA, B, if there
exists gz ∈ Σ∗ 1/2 such that for z ∈ U,
f z
1 Az
≺
.
gzg−z 1 Bz
1.14
Theorem A. If gz ∈ Σ∗ 1/2, then −zgzg−z ∈ Σ∗ .
Proof. At first, we know that
zg z
−
gz
>
1
2
1.15
4
International Journal of Mathematics and Mathematical Sciences
for z ∈ U, since g ∈ Σ∗ 1/2. And let hz −zgzg−z. Then
zg z zg −z
zh z
1
−
.
hz
gz
g−z
1.16
So we have
−
zh z
hz
1−
zg z
gz
−
−zg −z
g−z
> −1 1 1
.
2 2
1.17
Hence h ∈ Σ∗ .
Therefore, we know that ΣA, B is a subclass of Σk , by Theorem A.
We will obtain some coefficient bounds and some argument and convolution properties belonging to this class.
2. On the Coefficient Estimates of Functions in ΣA, B
Now we give the coefficient estimates of functions in ΣA, B. We need the following Lemma
to estimate of coefficient of functions in ΣA, B.
Lemma 2.1. Let g ∈ Σ∗ 1/2 and be given by 1.11, then one has
1
2 ≤ .
2b2n−1 − 2b0 b2n−2 2b1 b2n−3 − · · · −1n−1 2bn−2 bn −1n bn−1
n
2.1
Proof. According to the Theorem A, we have −zgzg−z ∈ Σ∗ , and if let
Gz −zgzg−z,
2.2
we know that G−z −Gz, so Gz is an odd meromorphic starlike function. If we let
∞
1 C2n−1 z2n−1 ,
z n1
Gz 2.3
it’s well-known 16, Volume 2, page 232 that
|C2n−1 | ≤
1
.
n
2.4
Substituting the series expressions of gz, Gz in 2.2 and comparing the coefficients of
two side of this equation, using 2.3 we can get an equality:
2
C2n−1 2b2n−1 − 2b0 b2n−2 2b1 b2n−3 − · · · −1n−1 2bn−2 bn −1n bn−1
,
and the proof of Lemma 2.1 is completed.
2.5
International Journal of Mathematics and Mathematical Sciences
5
Theorem 2.2. Let fz given by 1.9 and −1 ≤ B < A ≤ 1. If
∞ 1
≤ A − B,
1 |B|n|an | 1 |A|
n
n1
2.6
then f ∈ ΣA, B.
Proof. Let the functions fz and gz be given by 1.9 and 1.11, respectively. Furthermore,
we let gz ∈ Σ∗ 1/2. Then
Gz −zgzg−z ∞
1 C2n−1 z2n−1 ,
z n1
2.7
where C2n−1 is given by 2.5 and |C2n−1 | ≤ 1/n. Now we obtain
Δ : zf z − zgzg−z − Azf z − Bzgzg−z
∞
∞
∞
∞
1 n
2n−1 2n−1
n
nan z C2n−1 z
− Bnan z .
− B − A − AC2n−1 z
n1
z n1
n1
n1
2.8
Thus, for |z| r 0 ≤ r < 1, we have from 2.6
Δ≤
∞
∞
n|an |r |C2n−1 |r 2n−1 −
n
n1
≤
∞
n1
1 |B|n|an |r n ∞
n1
n1
∞
∞
1 |B|n|an |r n n1
∞
∞
1 A − B − |AC2n−1 |r 2n−1 − n|Ban |r n
r n1
n1
1 |A||C2n−1 |r 2n−1 − A − B
1
r
2.9
1
1
1 |A| r 2n−1 − A − B
n
r
n1
≤ 0.
Thus we have
zf z − zgzg−z < Azgzg−z − Bzf z,
2.10
which is equivalent to
f z
f z gzg−z − 1 < A − B gzg−z ,
which implies that f ∈ ΣA, B.
2.11
6
International Journal of Mathematics and Mathematical Sciences
Theorem 2.3. Let fz ∈ ΣA, B −1 ≤ B < A ≤ 1 and gz is given by 1.9 and 1.11, respectively. Then for k ≥ 1 one has
4k2 |a2k |2 k−1 4 1 − B2 n2 |a2n |2
n1
k 1 − B2 2n − 12 |a2n−1 |2 21 − AB2n − 1 a2n−1 C2n−1 1 − A2 |C2n−1 |2
n1
≤ A − B2 ,
2.12
where C2k−1 is given by 2.5.
Proof. Let fz ∈ ΣA, B. Then we have
−
zf z 1 Awz
,
Gz
1 Bwz
2.13
where w is an analytic function in U, |wz| < 1 for z ∈ U and Gz −zgzg−z. Then,
−zf z − Gz A · Gz Bzf z wz.
2.14
Thus, putting
wz ∞
tn zn ,
2.15
n1
we obtain
−
∞
∞
nan zn − C2n−1 z2n−1
n1
n1
∞
∞
1 A − B A · C2n−1 z2n−1 nBan zn
z n1
n1
∞
n
tn z .
2.16
n1
With comparing the coefficients, we can write
−
2k
nan zn −
n1
k
∞
C2n−1 z2n−1 cn zn
n1
n2k
1
k
2k−1
1 nBan zn
A − B A · C2n−1 z2n−1 z n1
n1
∞
n1
tn zn .
2.17
International Journal of Mathematics and Mathematical Sciences
7
Then, squaring the modulus of the both sides of the above equality and integrating along
|z| r and using the fact the |wz| < 1, we obtain
k
|2n − 1a2n−1 C2n−1 |2 r 4n−2 n1
k
|2na2n |2 r 4n
n1
k
k−1
1 < A − B2 2 |AC2n−1 B2n − 1a2n−1 |2 r 4n−2 |2Bna2n |2 r 4n .
r
n1
n1
2.18
Letting r → 1 on the both sides, we obtain
k
|2n − 1a2n−1 C2n−1 |2 −
n1
k
|AC2n−1 B2n − 1a2n−1 |2
n1
k
|2na2n |2 −
n1
k−1
2.19
|2Bna2n |2 ≤ A − B2 ,
n1
which implies the inequality 2.12.
Lemma 2.4 see 2, Ruscheweyh and Sheil-Small. Let ϕ and ψ be convex in U and suppose
f ≺ ψ. Then ϕ ∗ f ≺ ϕ ∗ ψ.
Theorem 2.5. If f ∈ ΣA, B −1 ≤ B < A ≤ 1, then there exists p ≺ 1 Az/1 Bz such that
for all s and t with |s| ≤ 1 and |t| ≤ 1,
t2 f tzpsz
≺
s2 f szptz
1 − tz
1 − sz
2
.
2.20
Proof. By definition, for f ∈ ΣA, B, there exist gz and pz such that g ∈ Σ∗ 1/2,
pz ≺
1 Az
,
1 Bz
f z
pz.
gzg−z
2.21
Put Gz −zgzg−z, then
zf z
pz.
Gz
2.22
zf z zp z
zG z
−2−
− 1.
f z
pz
Gz
2.23
−
And 2.22 implies that
−
8
International Journal of Mathematics and Mathematical Sciences
Since Gz ∈ Σ∗ ,
zG z 1 z
≺
,
Gz
1−z
2.24
2z
zG z
−1≺
.
Gz
1−z
2.25
−
hence
−
For s and t such that |s| ≤ 1, |t| ≤ 1, the function
hz z
0
t
s
−
du
1 − su 1 − tu
2.26
is convex in U. Applying Lemma 2.4 with this h, we have
zf z zp z
2z
− 2 ∗ hz ≺
∗ hz.
− f z
pz
1−z
2.27
Given any function lz analytic in U with l0 0, we have
l ∗ hz sz
lu
tz
du
z ∈ U,
u
2.28
so 2.27 reduces to
log
t2 f tzpsz
s2 f szptz
≺ log
1 − tz
1 − sz
2
.
2.29
Exponentiating both sides of 2.29 leads to 2.20.
3. Argument and Convolution Properties of Functions in ΣA, B
In this section, we will solve some problems related to argument and convolution properties
of functions in the class ΣA, B. To solve these problems, we need the following Lemmas.
Lemma 3.1 see 17, Goluzin, 3, page 62. Let f be an analytic function in U which is normalized and starlike, then
fz
z
1/2
>
1
.
2
This inequality is sharp with extremal function fz z/1 − z2 .
3.1
International Journal of Mathematics and Mathematical Sciences
9
Lemma 3.2 see 1, Ruscheweyh. Let α ≤ 1, φ ∈ Rα and f ∈ S∗ α. Then
φ ∗ Hf
U ⊂ coHU,
φ∗f
3.2
for any analytic function H ∈ HU, where coHU denote the closed convex hull of HU.
Theorem 3.3. Let fz ∈ ΣA, B −1 ≤ B < A ≤ 1. Then
A − Br
2 arcsin r
arg −z2 f z ≤ arcsin
1 − ABr 2
|z| r.
3.3
Proof. Let fz ∈ ΣA, B. Then there exists a function gz ∈ Σ∗ 1/2 satisfying
f z
pz,
gzg−z
3.4
where
pz ≺
1 Az
.
1 Bz
3.5
And p|z| < r is contained in the disk
1 − ABr 2 A − Br
<
w −
1 − B2 r 2 1 − B2 r 2
3.6
arg pz ≤ arcsin A − Br .
1 − ABr 2
3.7
from which it follows that
Hence we can obtain the inequality:
A − Br arg zgz arg −zg−z .
arg −z2 f z ≤ arcsin
2
1 − ABr
3.8
Hence it suffices to find the upper bounds of | argzgz|. Since g ∈ Σ∗ 1/2, we have gz /
0
for z ∈ U∗ and h ≡ 1/g ∈ S∗ 1/2. If we define kz gz2 /z, then k ∈ S∗ and we can
apply Lemma 3.1 to obtain
kz
z
1/2
>
1
.
2
3.9
10
International Journal of Mathematics and Mathematical Sciences
From the relation between g, h and k we obtain
zgz ≺ 1 z z ∈ U,
3.10
which implies
zgz − 1 ≤ r
|z| r.
3.11
Hence
arg zgz ≤ arcsin r
|z| r,
3.12
for z |z| r, and so the result is proved.
Theorem 3.4. Let f ∈ ΣA, B −1 ≤ B < A ≤ 1 satisfying the condition
zg z
3 1
1
< −
< − α,
2
gz
2 2
3.13
and φ ∈ Σ with z2 φz ∈ Rα . Then φ ∗ f ∈ ΣA, B.
Proof. Let f ∈ ΣA, B such that
f z
1 Az
≺
gzg−z 1 Bz
3.14
and define the functions G and H by
Gz : −zgzg−z,
Hz : −
zf z
.
Gz
3.15
Thus for any fixed z ∈ U,
Hz ≺
1 Az
.
1 Bz
3.16
From the condition 3.13, we can easily obtain that
zf z
−
Gz
< 2 − α,
3.17
International Journal of Mathematics and Mathematical Sciences
11
and this inequality implies that z2 Gz ∈ S∗ α. Put ϕz φ ∗ Gz. Since
Gz is an odd
meromorphic starlike function, so ϕz is. Define a function k by kz zϕz/z. Since ϕ
is an odd meromorphic starlike function, k ∈ Σ∗ 1/2 and ϕz −zkzk−z. Hence
φ ∗ f z −z φ ∗ f z
kzk−z
ϕz
φz ∗ −zf z
φ ∗ G z
φz ∗ HzGz
φ ∗ G z
3.18
z2 φz ∗ Hzz2 Gz
.
z2 φz ∗ z2 Gz
Since z2 φz ∈ Rα and z2 Gz ∈ S ∗ α, Lemma 3.2 yields
z2 φz ∗ Hzz2 Gz
∈ coHU,
z2 φz ∗ z2 Gz
3.19
where coHU denote the closed convex hull of HU, and because
Hz ≺
1 Az
1 Bz
z ∈ U
3.20
and 1 Az/1 Bz is convex in U, it follows that
φ ∗ f z 1 Az
≺
kzk−z
1 Bz
3.21
in U. Thus φ ∗ f ∈ ΣA, B and the proof of Theorem 3.4 is completed.
References
1 S. Ruscheweyh, Convolutions in Geometric Function Theory, vol. 83 of Séminaire de Mathématiques
Supérieures, Presses de l’Université de Montréal, Montreal, Canada, 1982.
2 S. Ruscheweyh and T. Sheil-Small, “Hadamard products of Schlicht functions and the PólyaSchoenberg conjecture,” Commentarii Mathematici Helvetici, vol. 48, pp. 119–135, 1973.
3 S. S. Miller and P. T. Mocanu, Differential Subordinations. Theory and Applications, vol. 225, Marcel
Dekker, New York, NY, USA, 2000.
4 T. N. Shanmugam, “Convolution and differential subordination,” International Journal of Mathematics
and Mathematical Sciences, vol. 12, no. 2, pp. 333–340, 1989.
5 R. M. Ali, V. Ravichandran, and S. K. Lee, “Subclasses of multivalent starlike and convex functions,”
Bulletin of the Belgian Mathematical Society, vol. 16, no. 3, pp. 385–394, 2009.
6 S. Supramaniam, R. M. Ali, S. K. Lee, and V. Ravichandran, “Convolution and differential
subordination for multivalent functions,” Bulletin of the Malaysian Mathematical Sciences Society, vol.
32, no. 3, pp. 351–360, 2009.
12
International Journal of Mathematics and Mathematical Sciences
7 R. Chandrashekar, R. M. Ali, S. K. Lee, and V. Ravichandran, “Convolutions of meromorphic
multivalent functions with respect to n-ply points and symmetric conjugate points,” Applied
Mathematics and Computation, vol. 218, no. 3, pp. 723–728, 2011.
8 K. S. Padmanabhan and R. Manjini, “Certain applications of differential subordination,” Publications
de l’Institut Mathématique, vol. 39, no. 53, pp. 107–118, 1986.
9 K. S. Padmanabhan and R. Parvatham, “Some applications of differential subordination,” Bulletin of
the Australian Mathematical Society, vol. 32, no. 3, pp. 321–330, 1985.
10 C. Gao and S. Zhou, “On a class of analytic functions related to the starlike functions,” Kyungpook
Mathematical Journal, vol. 45, no. 1, pp. 123–130, 2005.
11 Z.-G. Wang and D.-Z. Chen, “On a subclass of close-to-convex functions,” Hacettepe Journal of
Mathematics and Statistics, vol. 38, no. 2, pp. 95–101, 2009.
12 Z.-G. Wang, C.-Y. Gao, and S.-M. Yuan, “On certain new subclass of close-to-convex functions,”
Matematichki Vesnik, vol. 58, no. 3-4, pp. 119–124, 2006.
13 J. Kowalczyk and E. Leś-Bomba, “On a subclass of close-to-convex functions,” Applied Mathematics
Letters, vol. 23, no. 10, pp. 1147–1151, 2010.
14 Q.-H. Xu, H. M. Srivastava, and Z. Li, “A certain subclass of analytic and close-to-convex functions,”
Applied Mathematics Letters, vol. 24, no. 3, pp. 396–401, 2011.
15 B. Seker, “On certain new subclass of close-to-convex functions,” Applied Mathematics and Computation,
vol. 218, no. 3, pp. 1041–1045, 2011.
16 A. W. Goodman, Univalent Ffunctions. Vol. I, Mariner Publishing Co., Tampa, Fla, 1983.
17 G. M. Goluzin, “Some estimates for coefficients of univalent functions,” Matematicheskii Sbornik, vol.
3, no. 45, pp. 321–330, 1938 Russian.
Advances in
Operations Research
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Advances in
Decision Sciences
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Mathematical Problems
in Engineering
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Journal of
Algebra
Hindawi Publishing Corporation
http://www.hindawi.com
Probability and Statistics
Volume 2014
The Scientific
World Journal
Hindawi Publishing Corporation
http://www.hindawi.com
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
International Journal of
Differential Equations
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Volume 2014
Submit your manuscripts at
http://www.hindawi.com
International Journal of
Advances in
Combinatorics
Hindawi Publishing Corporation
http://www.hindawi.com
Mathematical Physics
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Journal of
Complex Analysis
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
International
Journal of
Mathematics and
Mathematical
Sciences
Journal of
Hindawi Publishing Corporation
http://www.hindawi.com
Stochastic Analysis
Abstract and
Applied Analysis
Hindawi Publishing Corporation
http://www.hindawi.com
Hindawi Publishing Corporation
http://www.hindawi.com
International Journal of
Mathematics
Volume 2014
Volume 2014
Discrete Dynamics in
Nature and Society
Volume 2014
Volume 2014
Journal of
Journal of
Discrete Mathematics
Journal of
Volume 2014
Hindawi Publishing Corporation
http://www.hindawi.com
Applied Mathematics
Journal of
Function Spaces
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Optimization
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014