Journal of Inequalities and Applications Volume 2009, Article ID 147069,12pages doi:10.1155/2009/147069
Research Article
Inclusion Properties for Certain Classes of
Meromorphic Functions Associated with a Family
of Linear Operators
Nak Eun Cho
Department of Applied Mathematics, Pukyong National University, Pusan 608-737, South Korea
Correspondence should be addressed to Nak Eun Cho,[email protected]
Received 3 March 2009; Accepted 1 May 2009
Recommended by Ramm Mohapatra
The purpose of the present paper is to investigate some inclusion properties of certain classes of meromorphic functions associated with a family of linear operators, which are defined by means of the Hadamard productor convolution. Some invariant properties under convolution are also considered for the classes presented here. The results presented here include several previous known results as their special cases.
Copyrightq2009 Nak Eun Cho. 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.
1. Introduction
LetA be the class of analytic functions in the open unit disk U {z ∈ C : |z| < 1}with the usual normalizationf0 f0−1 0. Iff andg are analytic inU, we say thatf is subordinate tog, writtenf ≺gorfz≺gz, if there exists an analytic functionwinUwith
w0 0 and|wz|<1 forz∈Usuch thatfz gwz.
LetN be the class of all functionsφ which are analytic and univalent in Uand for whichφUis convex withφ0 1 and Re{φz}>0 forz∈U. We denote byS∗andKthe subclasses ofAconsisting of all analytic functions which are starlike and convex, respectively.
LetMdenote the class of functions of the form
fz 1 z
∞
k0
akzk, 1.1
which are analytic in the punctured open unit diskDU\{0}. For 0≤η, β <1, we denote by
Making use of the principle of subordination between analytic functions, we introduce the subclassesMSη, φ,MKη, φand MCη, β;φ, ψof the classM for 0 ≤ η, β < 1 and
φ, ψ∈ N, which are defined by
MSη;φ:
f∈ M: 1 1−η
−zfz
fz −η
≺φzinU
,
MKη;φ: f∈ M: 1 1−η
− 1zf
z
fz
−η
≺φzinU
,
MCη, β;φ, ψ:
f∈ M:∃g ∈ MSη; φ s.t. 1 1−β
−zfgzz−β
≺ψzinU
.
1.2
We note that the classes mentioned above are the familiar classes which have been used widely on the space of analytic and univalent functions in U see 3–5 and for special choices for the functionsφandψinvolved in these definitions, we can obtain the well-known subclasses ofM. For examples, we have
MS
η;1z 1−z
MSη,
MK
η;1z 1−z
MKη,
MC
η, β;1z 1−z,
1z 1−z
MCη, β.
1.3
Now we define the functionφa, c;zby
φa, c;z: 1
z ∞
k0
ak1
ck1zk, 1.4
z∈U; a∈R; c∈R\Z−0; Z−0 :{−1,−2, . . .}, 1.5
where νk is the Pochhammer symbolor the shifted factorial defined in terms of the Gamma functionby
νk: ΓΓννk
⎧ ⎨ ⎩
1, ifk0, ν∈C\ {0},
νν1· · ·νk−1, ifk∈N:{1,2, . . .}, ν∈ C. 1.6
Letf∈ M. Denote byLa, c:M → Mthe operator defined by
La, cfz φa, c;z∗fz z∈D, 1.7
operatorLa, cis closely related to the Carlson-Shaffer operator7defined on the space of analytic and univalent functions inU.
Corresponding to the functionφa, c;z, letφ†a, c;zbe defined such that
φa, c;z∗φλa, c;z 1
z1−zλ λ >0. 1.8
Analogous toLa, c, we now introduce a linear operatorLλa, conMas follows:
Lλa, cfzcφλa, c;z∗fz, 1.9
a, c∈R\Z−
0; λ >0; z∈U; f∈ M
. 1.10
We note that
L12,1fz fz, L11,1fz zfz 2fz. 1.11
We note that the operatorLλa, cis motivated essentially to the integral operator for analytic
functions defined by Choi et al. 3, which extends the Noor integral operator studied by K. I. Noor and M. A. Noor8 also, see9–13.
Next, by using the operator Lλa, c, we introduce the following classes of
meromprphic functions forφ, ψ∈ N,a, c∈R\Z−0,λ >0 and 0≤η, β <1:
MSλ
a,cη;φ:f ∈ M:Lλa, cf∈ MSη;φ,
MKλ a,c
η;φ:f ∈ M:Lλa, cf∈ MKη;φ,
MCλ
a,cη, β;φ, ψ:f ∈ M:Lλa, cf∈ MCη, β;φ, ψ.
1.12
We also note that
fz∈ MKλ a,c
η;φ⇐⇒ −zfz∈ MSλa,cη;φ. 1.13
In particular, we set
MSλ a,c
η;1Az
1Bz
MSλ a,c
η;A, B −1< B < A≤1,
MKλ a,c
η;1Az
1Bz
MKλ a,c
η;A, B −1< B < A≤1.
1.14
In this paper, we investigate several inclusion properties of the classesMSλa,cη;φ,
MKλ
invariant properties under convolution are also considered for the classes mentioned above. Furthermore, relevant connections of the results presented here with those obtained in earlier works are pointed out.
2. Inclusion Properties Involving the Operator
L
λa, cThe following lemmas will be required in our investigation.
Lemma 2.1. Let φλia, c;z,φλai, c;z, andφλa, ci;z i 1,2 be defined by1.9. Then for
λi>0,ai, ci∈R\Z−0 i1,2,
φλ1a, c;z φλ2a, c;z∗fλ1,λ2z, 2.1
φλa2, c;z φλa1, c;z∗fa1,a2z, 2.2
φλa, c1;z φλa, c2;z∗fc1,c2z, 2.3
where
fs,tz 1 z
∞
k0
sk1
tk1
zk z∈D. 2.4
Proof. From1.8, we know that
φλa, c;z 1
z ∞
k0
ck1λk1 ak11k1z
k z∈D. 2.5
Therefore2.1,2.2and2.3follow from2.5immediately.
Lemma 2.2see14, pages 60–61. Lett≥s >0. Ift≥2orst≥3, then the functionz2f s,tz
belongs to the classK, wherefs,tis defined by2.4.
Lemma 2.3see15. Letf∈ Kandg∈ S∗. Then for every analytic functionhinU,
f∗hgU
f∗gU ⊂cohU, 2.6
wherecohUdenote the closed convex hull ofhU.
At first, the inclusion relationship involving the classMSλa,cη;φis contained inTheorem 2.4.
Theorem 2.4. Letλ2≥λ1>0,a, c∈R\Z−0 andφ∈ NwithRe{φz}<2−η/1−η 0≤η <
1. Ifλ2≥2orλ1λ2≥3, then
MSλ2
a,c
η;φ⊂ MSλ1
a,c
Proof. Letf∈ MSλ2
a,cη;φ. From the definition ofMSa,cλ2η;φ, we have
1 1−η
−z
Lλ2a, cfz
Lλ2a, cfz
−η
φwz z∈U, 2.8
zz2L
λ2a, cfz
z2Lλ
2a, cfz
2−1−ηφwz−η≺ 1z
1−z z∈U, 2.9
wherewis analytic inUwith|wz|<1z∈Uandw0 0φ0−1. By using1.9,2.1 and2.8, we get
−z
Lλ1a, cfz
Lλ1a, cfz
−z
φλ1a, c;z∗fz
φλ1a, c;z∗fz
−z
φλ2a, c;z∗fλ1,λ2z∗fz
φλ2a, c;z∗fλ1,λ2z∗fz
fλ1,λ2z∗
−zLλ2a, cfz
fλ1,λ2z∗ Lλ2a, cfz
fλ1,λ2z∗
1−ηφwz ηLλ2a, cfz
fλ1,λ2z∗ Lλ2a, cfz
.
2.10
Therefore by using2.8, we obtain
1 1−η
−zLλ1a, cfz
Lλ1a, cfz
−η
1
1−η
fλ1,λ2z∗
1−ηφwz ηLλ2a, cfz
fλ1,λ2z∗ Lλ2a, cfz
−η
1
1−η
z2f
λ1,λ2z∗
1−ηφwz ηz2L
λ2a, cfz
z2fλ
1,λ2z∗z2Lλ2a, cfz
−η
.
2.11
It follows from2.9andLemma 2.2thatz2L
λ2a, cfz∈ S∗andz
2f
λ1,λ2 ∈ K, respectively.
Let us putswz: 1−ηφwz η. Then by applyingLemma 2.3to2.10, we obtain
z2fλ
1,λ2∗swzz2Lλ2a, cf
z2fλ
1,λ2∗z2Lλ2a, cf
since sis convex univalent. Therefore from the definition of subordination and 2.12, we have
1 1−η
−z
Lλ1a, cfz
Lλ1a, cfz
−η
≺φz z∈U, 2.13
or, equivalently,f∈ MSλ1
a,cφ, which completes the proof ofTheorem 2.4.
By using1.13,2.2and2.3, we have the followingTheorem 2.5andTheorem 2.6.
Theorem 2.5. Letλ >0,a2≥a1>0,c∈R\Z−0 andφ∈ NwithRe{φz}<2−η/1−η 0≤ η <1. Ifa2≥2ora1a2≥3, then
MSλ a1,c
η;φ⊂ MSλa2,c
η;φ. 2.14
Theorem 2.6. Letλ >0,a∈R\Z0−,c2≥c1>0andφ∈ NwithRe{φz}<2−η/1−η 0≤ η <1. Ifc2≥2orc1c2≥3, then
MSλ a,c2
η;φ⊂ MSλa,c1η;φ. 2.15
Next, we prove the inclusion theorem involving the classMKλa,cη;φ.
Theorem 2.7. Letλ2≥λ1>0,a, c∈R\Z−0 andφ∈ NwithRe{φz}<2−η/1−η 0≤η <
1. Ifλ2≥2orλ1λ2≥3, then
MKλ2
a,cη;φ⊂ MKa,cλ1η;φ. 2.16
Proof. Applying1.13andTheorem 2.4, we observe that
fz∈ MKλ2
a,cη;φ⇐⇒ Lλ2a, cfz∈ MK
η;φ
⇐⇒ −zLλ2a, cfz
∈ MSη;φ
⇐⇒ Lλ2a, c
−zfz∈ MSη;φ
⇐⇒ −zfz∈ MSλ2
a,cη;φ
⇒ −zfz∈ MSλ1
a,c
η;φ
⇐⇒ Lλ1a, c
−zfz∈ MSη;φ
⇐⇒ −zLλ1a, cfz
∈ MSη;φ
⇐⇒ Lλ1a, cfz∈ MK
η;φ
⇐⇒fz∈ MKλ1
a,c
η;φ,
2.17
By using a similar method as in the proof ofTheorem 2.7, we obtain the following two theorems.
Theorem 2.8. Letλ >0,a2≥a1>0,c∈R\Z−0 andφ∈ NwithRe{φz}<2−η/1−η 0≤ η <1. Ifa2≥2ora1a2≥3, then
MKλ a1,c
η;φ⊂ MKλa2,cη;φ. 2.18
Theorem 2.9. Letλ >0,a∈R\Z0−,c2≥c1>0andφ∈ NwithRe{φz}<2−η/1−η 0≤ η <1. Ifc2≥2orc1c2≥3, then
MKλ a,c2
η;φ⊂ MKλa,c1η;φ. 2.19
Takingφz 1Az/1Bz −1 < B < A≤1; z∈Uin Theorems2.4–2.9, we have the following corollaries below.
Corollary 2.10. Let1A1−η<2−η1B −1< B < A≤1; 0≤η <1andc∈R\Z−0. Ifλ2≥λ1>0andλ2≥min{2,3−λ1}, anda2≥a1>0anda2 ≥min{2,3−a1}, then
MSλ2
a1,c
η;A, B⊂ MSλ1
a1,c
η;A, B⊂ MSλ1
a2,c
η;A, B,
MKλ2
a1,c
η;A, B⊂ MKλ1
a1,c
η;A, B⊂ MKλ1
a2,c
η;A, B.
2.20
Corollary 2.11. Let1A1−η<2−η1B −1< B < A≤1; 0 ≤η <1andλ >0. If
a2≥a1 >0anda2≥min{2,3−a1}, andc2 ≥c1>0andc2≥min{2,3−c1}, then
MSλ a1,c2
η;A, B⊂ MSλa1,c1η;A, B⊂ MSλa2,c1η;A, B,
MKλ a1,c2
η;A, B⊂ MKλa1,c1η;A, B⊂ MKλa2,c1η;A, B.
2.21
Corollary 2.12. Let1A1−η<2−η1B −1< B < A≤1; 0≤η <1anda∈R\Z−0. Ifλ2≥λ1>0andλ2≥min{2,3−λ1}, andc2≥c1>0andc2≥min{2,3−c1}, then
MSλ2
a,c2
η;A, B⊂ MSλ2
a,c1
η;A, B⊂ MSλ1
a,c1
η;A, B,
MKλ2
a,c2
η;A, B⊂ MKλ2
a,c1
η;A, B⊂ MKλ1
a,c1
η;A, B.
2.22
To prove theorems below, we need the following lemma.
Lemma 2.13. Letφ∈ NwithRe{φz}<2−η/1−η 0≤η <1. Iff∈ Mwithz2fz∈ K
Proof. Letq∈ MSη;φ. Then
−zqz 1−ηφwz ηqz z∈U, 2.23
wherewis an analytic function inUwith|wz|<1 z∈Uandw0 0. Thus we have
1 1−η
−z
fz∗qz
fz∗qz −η
1
1−η
fz∗−zqz fz∗qz −η
1
1−η
fz∗1−ηφωz ηqz
fz∗qz −η
z∈D.
2.24
By using the similar arguments to those used in the proof ofTheorem 2.4, we conclude that
2.24is subordinated toφinUand sof∗q∈ MSη;φ.
Finally, we give the inclusion properties involving the classMCλa,cη, β;φ, ψ.
Theorem 2.14. Letc ∈R\Z−0 andφ, ψ ∈ NwithRe{φz} < 2−η/1−η 0 ≤ η < 1. If
λ2≥λ1>0andλ2≥min{2,3−λ1}, anda2 ≥a1>0anda2≥min{2,3−a1}, then
MCλ2
a1,c
η, β;φ, ψ⊂ MCλ1
a1,c
η, β;φ, ψ⊂ MCλ1
a2,c
η, β;φ, ψ. 2.25
Proof. We begin by proving that
MCλ2
a1,c
η, β;φ, ψ⊂ MCλ1
a1,c
η, β;φ, ψ. 2.26
Letf∈ MCλ2
a1,cη, β;φ, ψ. Then there exists a functionq2∈ MSη;φsuch that
1 1−β
−z
Lλ2a1, cfz
q2z − β
≺ψz 0≤β <1; z∈U. 2.27
From2.27, we obtain
−zLλ2a1, cfz
wherewis an analytic function inUwith|wz|<1 z∈Uandw0 0. By virtue of2.3, Lemmas2.2and2.13, we see thatfλ1,λ2z∗q2z≡q1zbelongs toMSη;φ. Then, making
use of2.1, we have
1 1−β
−z
Lλ1a1, cfz
q1z − β
1
1−β ⎛ ⎜
⎝fλ1,λ2z∗
−zLλ2a1, cfz
fλ1,λ2z∗q2z
−β
⎞ ⎟ ⎠
1
1−β
fλ1,λ2z∗
1−βψwz βq2z fλ1,λ2z∗q2z
−β
1
1−β
z2fλ
1,λ2z∗
1−βψwz βz2q 2z z2fλ
1,λ2z∗z2q2z
−β
≺ψz z∈U.
2.29
Therefore we prove thatf∈ MCλ1
a1,cη, β;φ, ψ.
For the second part, by using arguments similar to those detailed above with2.2, we obtain
MCλ1
a1,c
η, β;φ, ψ⊂ MCλ1
a2,c
η, β;φ, ψ 2.30
Thus the proof ofTheorem 2.14is completed.
The following results can be obtained by using the same techniques as in the proof of
Theorem 2.14and so we omit the detailed proofs involved.
Theorem 2.15. Letλ >0andφ, ψ∈ NwithRe{φz}<2−η/1−η 0≤η <1. Ifa2≥a1>0
anda2≥min{2,3−a1}, andc2≥c1>0andc2 ≥min{2,3−c1}, then
MCλ a1,c2
η, β;φ, ψ⊂ MCλa1,c1
η, β;φ, ψ⊂ MCλa2,c1
η, β;φ, ψ. 2.31
Theorem 2.16. Leta∈ R\Z−0 andφ, ψ ∈ NwithRe{φz} < 2−η/1−η 0 ≤ η < 1. If
λ2≥λ1>0andλ2≥min{2,3−λ1}, andc2≥c1>0andc2≥min{2,3−c1}, then
MCλ2
a,c2
η, β;φ, ψ⊂ MCλ2
a,c1
η, β;φ, ψ⊂ MCλ1
a,c1
η, β;φ, ψ. 2.32
Remark 2.17. Foraλ1λ >−1andc 1, Theorems2.4,2.5,2.7,2.8, and2.14reduce to the corresponding results obtained by Cho and Noor16.
3. Inclusion Properties Involving Various Operators
Theorem 3.1. Letλ >0, a >0,c∈R\Z−0,φ, ψ∈ NwithRe{φz}<2−η/1−η 0≤η <1
and letg∈ Mwithz2gz∈ K. Then
if ∈ MSλa,cη;φ⇒g∗f ∈ MSλa,cη;φ,
iif ∈ MKλa,cη;φ⇒g∗f ∈ MKλa,cη;φ,
iiif ∈ MCλa,cη, β;φ, ψ⇒g∗f∈ MCλa,cη, β;φ, ψ.
Proof. iLetf∈ MSa,cλ η;φ. Then we have
1 1−η
−z
Lλa, cg∗fz
Lλa, cg∗fz −η
1
1−η ⎛ ⎜ ⎝gz∗
−zLλa, cfz
gz∗ Lλa, cfz −η
⎞ ⎟
⎠. 3.1
By using the same techniques as in the proof ofTheorem 2.4, we obtaini.
ii Let f ∈ MKλa,cφ. Then, by 1.13,−zfz ∈ MSa,cη;φ and hence from i, gz∗−zfz∈ MSλ
a,cη;φ. Since
gz∗−zfz−zg∗fz, 3.2
we haveiiapplying1.13once again.
iiiLetf ∈ MCλa,cη, β;φ, ψ. Then there exists a functionq∈ MSη;φsuch that
−zLλa, cfz1−βψwz βqz 0≤β <1; z∈U, 3.3
wherewis an analytic function inUwith|wz|<1 z∈Uandw0 0. FromLemma 2.13, we have thatg∗q∈ MSη;φ. Since
1 1−β
−z
Lλa, cg∗fz
g∗qz −β
1
1−β ⎛ ⎜ ⎝gz∗
−zLλa, cfz
gz∗qz −β
⎞ ⎟ ⎠
1
1−β
z2gz∗1−βψwz βz2qz z2gz∗z2qz −β
≺ψz z∈U,
3.4
Now we consider the following operators defined by
Ψ1z
1
z ∞
k1
1c
kczk−1 Re{c} ≥0;z∈D,
Ψ2z
1
z21−xlog
1−xz
1−z log 10;|x| ≤1, x /1;z∈D
.
3.5
It is well known17that the operatorsz2Ψ
1 andz2Ψ2are convex univalent inU. Therefore
we have the following result, which can be obtained fromTheorem 3.1immediately.
Corollary 3.2. Leta >0, λ >0,c∈R\Z−0,φ, ψ∈ NwithRe{φz}<2−η/1−η 0≤η <1
and letΨi i1,2be defined by3.5, respectively. Then
if ∈ MSλa,cη;φ⇒Ψi∗f∈ MSa,cλ η;φ,
iif ∈ MKλa,cη;φ⇒Ψi∗f∈ MKλa,cη;φ,
iiif ∈ MCλa,cη, β;φ, ψ⇒Ψi∗f ∈ MCλa,cη, β;φ, ψ.
Acknowledgments
The author would like to express his gratitude to the referees for many valuable advices regarding a previous version of this paper. This work was supported by the Korea Research Foundation Grant funded by the Korean GovernmentMOEHRD, Basic Research Promotion Fund KRF-2008-313-C00035.
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