A Subclass of Meromorphic Close to Convex Functions of Janowski's Type

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Volume 2012, Article ID 682162,12pages 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,[email protected]

Received 9 April 2012; Accepted 19 June 2012

Academic Editor: Aloys Krieg

Copyrightq2012 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 subclassΣA, B −1≤B < A≤1of 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

LetHUbe the set of all analytic functions on the open unit diskU:{z:|z|<1}, and letA be the subclass ofHUwhich contains the functions normalized byf0 0 andf0 1. A functionfz∈ Ais said to be starlike of orderαif and only if

zfz fz

> α, 1.1

for someα0≤α <1and for allz∈U. The class of starlike functions of orderαis denoted byS∗α.

Iffandgare analytic inU, we say thatfis subordinate tog, written as follows:

f≺g inU or fz≺gz z∈U, 1.2

if there exists a Schwarz functionwz, which is analytic inUwith

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such that

fz gwz z∈U. 1.4

In particular, if the functiongis univalent inU, the above subordination is equivalent to

f0 g0, fU⊂gU. 1.5

The Hadamard productor convolutionof two series

fz ∞

n0

anzn, gz

∞

n0

bnzn 1.6

is defined by

f∗gz ∞

n0

anbnzn. 1.7

For a convex functionf, it follows from Alexander’s Theorem thatzfz fz∗z/1−z2 is a starlike function. In view of the identityfz fz∗z/1−z, it is then clear that the classes of convex and starlike functions can be unified by considering functionsf satisfying f∗gis starlike for a fixed functiong∈ 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

orderαis defined by

Rα:

f∈ A:f∗ z

1−z2−2α ∈ S ∗_α

, 1.8

whileR1consists off∈ Asatisfyingfz/z >1/2.

By using the convex hull method1,2and the method of diﬀerential subordination

3, Shanmugam 4 introduced and investigated convolution properties of various sub-classes 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.7also 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 in8,9.

LetΣdenote the class of all univalent meromorphic functionsfznormalized by

fz 1 z

∞

k1

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which are analytic in the punctured unit disk

U∗_{z_:_z_∈_C,₀_<_|z|_<_1}_U_{− {0}.} _1.10

We denote byΣ∗αthe subclass ofΣconsisting ofgzformed by

gz 1 z

∞

k1

bkzk, 1.11

which are meromorphic starlike of orderαinU∗. In particular, we denote byΣ∗, whenα0. Also, a functionfzof the form1.9is said to be meromorphic close-to-convex inU∗if there is agzinΣ∗such that

_zf_z

gz

<0 z∈U. 1.12

We denoteΣkthe set of functions close-to-convex inU∗.

In a recent paper, Gao and Zhou10 introduced an interesting subclassKs of the

analytic function classAand the univalent functionS, which contains the functionsf satis-fying the following inequality:

z2_f_z gzg−z

<0 z∈U, 1.13

for somegz ∈ S∗1/2. Here,S∗1/2is the class of starlike functions of order 1/2 inU. After that, many classes related toKsinvestigated and studied by some authors. Especially,

Wang et al.11,12, Kowalczyk and Le´s-Bomba13, Xu et al.14, and Seker15introduced the generalized class of the classKs. 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. Letf∈Σand be given by1.9and−1≤B < A≤1. Thenf ∈ΣA, B, if there

existsgz∈Σ∗1/2such that forz∈U, fz gzg−z ≺

1Az

1Bz. 1.14

Theorem A. Ifgz∈Σ∗1/2, then−zgzg−z∈Σ∗.

Proof. At first, we know that

−

_zg_z

gz

> 1

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forz∈U, sinceg∈Σ∗1/2. And lethz −zgzg−z. Then zhz

hz 1 zgz

gz −

zg−z

g−z . 1.16

So we have

−

zhz hz

1−

zgz gz

−

_−zg_−z

g−z

>−11 2

1

2. 1.17 Henceh∈Σ∗.

Therefore, we know thatΣA, Bis a subclass ofΣk, by Theorem A.

We will obtain some coeﬃcient bounds and some argument and convolution proper-ties belonging to this class.

2. On the Coefficient Estimates of Functions in

ΣA, B

Now we give the coeﬃcient estimates of functions inΣA, B. We need the following Lemma to estimate of coeﬃcient of functions inΣA, B.

Lemma 2.1. Letg ∈Σ∗1/2and be given by1.11, then one has

_2b

2n−1−2b0b2n−22b1b2n−3− · · · −1n−12bn−2bn −1nb2n−1 ≤

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 thatG−z −Gz, soGzis an odd meromorphic starlike function. If we let

Gz 1 z

∞

n1

C2n−1z2n−1, 2.3

it’s well-known16, Volume 2, page 232that

|C2n−1| ≤

1

n. 2.4 Substituting the series expressions ofgz,Gzin2.2and comparing the coeﬃcients of two side of this equation, using2.3we can get an equality:

C2n−12b2n−1−2b0b2n−22b1b2n−3− · · · −1n−12bn−2bn −1nb2n−1, 2.5

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Theorem 2.2. Letfzgiven by1.9and−1≤B < A≤1. If ∞

n1

1|B|n|an| 1|A|

1 n

≤A−B, 2.6

thenf ∈ΣA, B.

Proof. Let the functionsfzandgzbe given by1.9and1.11, respectively. Furthermore,

we letgz∈Σ∗1/2. Then

Gz −zgzg−z 1 z

∞

n1

C2n−1z2n−1, 2.7

whereC2n−1is given by2.5and|C2n−1| ≤1/n. Now we obtain

Δ: zfz−zgzg−z − Azfz−Bzgzg−z

∞

n1

nanzn

∞

n1

C2n−1z2n−1

− B−A1 z −

∞

n1

AC2n−1z2n−1−

∞

n1

Bnanzn

.

2.8

Thus, for|z|r 0≤r <1, we have from2.6

Δ≤∞

n1

n|an|rn

∞

n1

|C2n−1|r2n−1−

A−B1 r −

∞

n1

|AC2n−1|r2n−1−

∞

n1

n|Ban|rn

∞

n1

1|B|n|an|rn

∞

n1

1|A||C2n−1|r2n−1−A−B1

r ≤∞

n1

1|B|n|an|rn

∞

n1

1|A|1 nr

2n−1₋_A₋_B1

r ≤0.

2.9

Thus we have

zfz−zgzg−z < Azgzg−z−Bzfz , 2.10

which is equivalent to

_g_zf_gz_−z −1 < A−B f _z gzg−z

, 2.11

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Theorem 2.3. Letfz∈ΣA, B −1≤B < A≤1andgzis given by1.9and1.11,

respec-tively. Then fork≥1 one has

4k2|a2k|2

k−1

n1

41−B2n2|a2n|2

k

n1

1−B22n−12|a2n−1|221−AB2n−1

a2n−1C2n−1

1−A2|C2n−1|2

≤A−B2,

2.12

whereC2k−1is given by2.5.

Proof. Letfz∈ΣA, B. Then we have

−zfz Gz

1Awz

1Bwz, 2.13

wherewis an analytic function inU,|wz|<1 forz∈UandGz −zgzg−z. Then, −zf_z₋_G_z _A_·_G_z _Bzf_z_w_z_. _2.14

Thus, putting

wz ∞

n1

tnzn, 2.15

we obtain

−∞

n1

nanzn−

∞

n1

C2n−1z2n−1

A−B1 z

∞

n1

A·C2n−1z2n−1

∞

n1

nBanzn

_∞

n1

tnzn

.

2.16

With comparing the coeﬃcients, we can write

−2k

n1

nanzn− k

n1

C2n−1z2n−1

∞

n2k1

cnzn

A−B1 z

k

n1

A·C2n−1z2n−1

2k−1

n1

nBanzn

_∞

n1

tnzn

.

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Then, squaring the modulus of the both sides of the above equality and integrating along |z|rand using the fact the|wz|<1, we obtain

k

n1

|2n−1a2n−1C2n−1|2r4n−2 k

n1

|2na2n|2r4n

<A−B2 1 r2

k

n1

|AC2n−1B2n−1a2n−1|2r4n−2

k−1

n1

|2Bna2n|2r4n.

2.18

Lettingr → 1 on the both sides, we obtain

k

n1

|2n−1a2n−1C2n−1|2− k

n1

|AC2n−1B2n−1a2n−1|2

k

n1

|2na2n|2−

k−1

n1

|2Bna2n|2≤A−B2,

2.19

which implies the inequality2.12.

Lemma 2.4see2, Ruscheweyh and Sheil-Small. Letϕ andψ be convex inU and suppose

f≺ψ. Thenϕ∗f≺ϕ∗ψ.

Theorem 2.5. Iff∈ΣA, B −1≤B < A≤1, then there existsp≺1Az/1Bzsuch that

for allsandtwith|s| ≤1 and|t| ≤1,

t2_f_tz_p_sz

s2_f_sz_p_tz ≺

1−tz 1−sz

2

. 2.20

Proof. By definition, forf∈ΣA, B, there existgzandpzsuch thatg ∈Σ∗1/2,

pz≺ 1Az 1Bz, fz

gzg−z pz.

2.21

PutGz −zgzg−z, then

−zfz

Gz pz. 2.22

And2.22implies that

−zfz fz

zpz

pz −2− zGz

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SinceGz∈Σ∗,

−zGz Gz ≺

1z

1−z, 2.24

hence

−zGz Gz −1≺

2z

1−z. 2.25

Forsandtsuch that|s| ≤1,|t| ≤1, the function

hz

_z

0

s 1−su−

t 1−tu

du 2.26

is convex inU. ApplyingLemma 2.4with thish, we have

−zfz fz

zpz pz −2

∗hz≺ 2z

1−z∗hz. 2.27

Given any functionlzanalytic inUwithl0 0, we have

l∗hz

sz

tz

ludu

u z∈U, 2.28

so2.27reduces to

log

t2_f_tz_p_sz s2_f_sz_p_tz

≺log

1−tz 1−sz

2

. 2.29

Exponentiating both sides of2.29leads to2.20.

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 classΣA, B. To solve these problems, we need the following Lemmas.

Lemma 3.1see17, Goluzin,3, page 62. Letfbe an analytic function inUwhich is

normal-ized and starlike, then

_f_z

z

1/2

> 1

2. 3.1

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Lemma 3.2see1, Ruscheweyh. Letα≤1,φ∈ Rαandf∈ S∗α. Then

φ∗Hf

φ∗f U⊂coHU, 3.2

for any analytic functionH∈ HU, where coHUdenote the closed convex hull ofHU.

Theorem 3.3. Letfz∈ΣA, B −1≤B < A≤1. Then

_arg

−z2_f_z _≤_arcsinA−Br

1−ABr2 2 arcsinr |z|r. 3.3

Proof. Letfz∈ΣA, B. Then there exists a functiongz∈Σ∗1/2satisfying

fz

gzg−z pz, 3.4

where

pz≺ 1Az

1Bz. 3.5

Andp|z|< ris contained in the disk

w−1−ABr

2

1−B2_r2

< A−Br

1−B2_r2 3.6

from which it follows that

_arg _p_z _≤_arcsinA−Br

1−ABr2. 3.7

Hence we can obtain the inequality:

_arg

−z2_f_z _≤_arcsinA−Br 1−ABr2 arg

zgz arg−zg−z . 3.8

Hence it suﬃces to find the upper bounds of|argzgz|. Sinceg∈Σ∗1/2, we havegz/0 forz ∈U∗ andh≡ 1/g ∈ S∗1/2. If we definekz gz2/z, thenk ∈ S∗ and we can applyLemma 3.1to obtain

kz z

1/2

> 1

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From the relation betweeng, handkwe obtain

zgz≺1z z∈U, 3.10

which implies

_zg_z₋₁ _≤_r _|z|_r_. _3.11

Hence

_arg

zgz ≤arcsinr |z|r, 3.12

forz|z|r, and so the result is proved.

Theorem 3.4. Letf ∈ΣA, B −1≤B < A≤1satisfying the condition

1 2 <−

zgz gz

< 3 2 −

1

2α, 3.13

andφ∈Σwithz2φz∈ Rα. Thenφ∗f∈ΣA, B.

Proof. Letf∈ΣA, Bsuch that

fz gzg−z ≺

1Az

1Bz 3.14

and define the functionsGandHby

Gz:−zgzg−z, Hz:−zf _z

Gz . 3.15

Thus for any fixedz∈U,

Hz≺ 1Az

1Bz. 3.16

From the condition3.13, we can easily obtain that

−

_zf_z

Gz

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and this inequality implies thatz2_G_z_{∈ S}∗_α_{. Put}_ϕ_z_φ_∗_G_z_{. Since}_G_z_{is an odd}

meromorphic starlike function, soϕzis. Define a functionkbykz zϕz/z. Sinceϕ is an odd meromorphic starlike function,k∈Σ∗1/2andϕz −zkzk−z. Hence

φ∗fz kzk−z

−zφ∗fz ϕz

φz∗

−zf_z

φ∗Gz

φz∗HzGz

φ∗Gz

z2φz∗Hzz2Gz

z2_φ_z_∗_z2_G_z .

3.18

Sincez2_φ_z_{∈ R}

αandz2Gz∈ S∗α,Lemma 3.2yields

z2_φ_z_∗_H_z_z2_G_z

z2_φ_z_∗_z2_G_z ∈coHU, 3.19

where coHUdenote the closed convex hull ofHU, and because

Hz≺ 1Az

1Bz z∈U 3.20

and1Az/1Bzis convex inU, it follows that

φ∗fz kzk−z ≺

1Az

1Bz 3.21

inU. Thusφ∗f ∈ΣA, Band the proof ofTheorem 3.4is completed.

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A Subclass of Meromorphic Close to Convex Functions of Janowski's Type

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