Hindawi Publishing Corporation
Journal of Inequalities and Applications
Volume 2009, Article ID 147069, 12 pages
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, necho@pknu.ac.kr
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 product or 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.
Copyright q 2009 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
Let A be the class of analytic functions in the open unit disk U {z ∈ C : |z| < 1} with
the usual normalization f0 f 0 − 1 0. If f and g are analytic in U, we say that f is
subordinate to g, written f ≺ g or fz ≺ gz, if there exists an analytic function w in U with
w0 0 and |wz| < 1 for z ∈ U such that fz gwz.
Let N be the class of all functions φ which are analytic and univalent in U and for
which φU is convex with φ0 1 and Re{φz} > 0 for z ∈ U. We denote by S ∗ and K the
subclasses of A consisting of all analytic functions which are starlike and convex, respectively.
Let M denote the class of functions of the form
fz ∞
1 ak zk ,
z k0
1.1
which are analytic in the punctured open unit disk D U\{0}. For 0 ≤ η, β < 1, we denote by
MSη, MKη and MCη, β the subclasses of M consisting of all meromorphic functions
which are, respectively, starlike of order η, convex of order η and colse-to-convex of order β
and type η in U see, for details, 1, 2.
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Making use of the principle of subordination between analytic functions, we introduce
the subclasses MSη, φ, MKη, φ and MCη, β; φ, ψ of the class M for 0 ≤ η, β < 1 and
φ, ψ ∈ N, which are defined by
zf z
1
−
− η ≺ φz in U ,
1−η
fz
zf z
1
MK η; φ : f ∈ M :
− 1 − η ≺ φz in U ,
1−η
f z
MS η; φ :
MC η, β; φ, ψ :
f ∈M:
zf z
1
−
− β ≺ ψz in U .
f ∈ M : ∃g ∈ MS η; φ s.t.
1−β
gz
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 functions φ and ψ involved in these definitions, we can obtain the well-known
subclasses of M. For examples, we have
1z
MS η ,
MS η;
1−z
1z
MK η;
MK η ,
1−z
1z 1z
,
MC η, β;
MC η, β .
1−z 1−z
1.3
Now we define the function φa, c; z by
φa, c; z :
∞
ak1 k
1 z ,
z k0 ck1
z ∈ U; a ∈ R; c ∈ R \ Z−0 ; Z−0 : {−1, −2, . . .} ,
1.4
1.5
where νk is the Pochhammer symbol or the shifted factorial defined in terms of the
Gamma function by
⎧
Γν k ⎨1,
νk :
⎩νν 1 · · · ν k − 1,
Γν
if k 0, ν ∈ C \ {0},
if k ∈ N : {1, 2, . . .}, ν ∈ C.
1.6
Let f ∈ M. Denote by La, c : M → M the operator defined by
La, cfz φa, c; z ∗ fz
z ∈ D,
1.7
where the symbol ∗ stands for the Hadamard product or convolution. The operator La, c
was introduced and studied by Liu and Srivastava 6. Further, we remark in passing that this
Journal of Inequalities and Applications
3
operator La, c is closely related to the Carlson-Shaffer operator 7 defined on the space of
analytic and univalent functions in U.
Corresponding to the function φa, c; z, let φ† a, c; z be defined such that
φa, c; z ∗ φλ a, c; z 1
z1 − zλ
λ > 0.
1.8
Analogous to La, c, we now introduce a linear operator Lλ a, c on M as follows:
Lλ a, cfzc φλ a, c; z ∗ fz,
1.9
a, c ∈ R \ Z−0 ; λ > 0; z ∈ U; f ∈ M .
1.10
We note that
L1 2, 1fz fz,
L1 1, 1fz zf z 2fz.
1.11
We note that the operator Lλ a, c is 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. Noor 8 also, see 9–13.
Next, by using the operator Lλ a, c, we introduce the following classes of
meromprphic functions for φ, ψ ∈ N, a, c ∈ R \ Z−0 , λ > 0 and 0 ≤ η, β < 1:
MSλa,c η; φ : f ∈ M : Lλ a, cf ∈ MS η; φ ,
λ
MKa,c
η; φ : f ∈ M : Lλ a, cf ∈ MK η; φ ,
1.12
λ
MCa,c
η, β; φ, ψ : f ∈ M : Lλ a, cf ∈ MC η, β; φ, ψ .
We also note that
λ
fz ∈ MKa,c
η; φ ⇐⇒ −zf z ∈ MSλa,c η; φ .
1.13
In particular, we set
1 Az
MSλa,c η; A, B
η;
−1 < B < A ≤ 1,
1 Bz
1 Az
λ
λ
MKa,c
η;
MKa,c
η; A, B
−1 < B < A ≤ 1.
1 Bz
MSλa,c
1.14
In this paper, we investigate several inclusion properties of the classes MSλa,c η; φ,
λ
and MCa,c
η; φ associated with the operator Lλ a, c defined by 1.9.Some
λ
η; φ,
MKa,c
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Journal of Inequalities and Applications
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 Lλ a, c
The following lemmas will be required in our investigation.
Lemma 2.1. Let φλi a, c; z, φλ ai , c; z, and φλ a, ci ; z i 1, 2 be defined by 1.9. Then for
λi > 0, ai , ci ∈ R \ Z−0 i 1, 2,
φλ1 a, c; z φλ2 a, c; z∗fλ1 ,λ2 z,
2.1
φλ a2 , c; z φλ a1 , c; z∗fa1 ,a2 z,
2.2
φλ a, c1 ; z φλ a, c2 ; z∗fc1 ,c2 z,
2.3
where
fs,t z ∞
sk1 k
1 z
z k0 tk1
z ∈ D.
2.4
Proof. From 1.8, we know that
∞
ck1 λk1 k
1 z
z k0 ak1 1k1
φλ a, c; z z ∈ D.
2.5
Therefore 2.1, 2.2 and 2.3 follow from 2.5 immediately.
Lemma 2.2 see 14, pages 60–61. Let t ≥ s > 0. If t ≥ 2 or s t ≥ 3, then the function z2 fs,t z
belongs to the class K, where fs,t is defined by 2.4.
Lemma 2.3 see15. Let f ∈ K and g ∈ S∗ . Then for every analytic function h in U,
f∗hg U
⊂ cohU,
f∗g U
2.6
where cohU denote the closed convex hull of hU.
λ
η; φ is contained in Theorem 2.4.
At first, the inclusion relationship involving the class MSa,c
Theorem 2.4. Let λ2 ≥ λ1 > 0, a, c ∈ R \ Z−0 and φ ∈ N with Re{φz} < 2 − η/1 − η 0 ≤ η <
1. If λ2 ≥ 2 or λ1 λ2 ≥ 3, then
λ2
λ1
MSa,c
η; φ ⊂ MSa,c
η; φ .
2.7
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5
λ2
λ2
η; φ. From the definition of MSa,c
η; φ, we have
Proof. Let f ∈ MSa,c
1
1−η
z Lλ2 a, cfz
− η φwz
−
Lλ2 a, cfz
z ∈ U,
z z2 Lλ2 a, cfz
1z
2 − 1 − η φwz − η ≺
2
1−z
z Lλ2 a, cfz
z ∈ U,
2.8
2.9
where w is analytic in U with |wz| < 1 z ∈ U and w0 0 φ0 − 1. By using 1.9, 2.1
and 2.8, we get
z φλ1 a, c; z ∗ fz
z Lλ1 a, cfz
−
−
Lλ1 a, cfz
φλ1 a, c; z ∗ fz
z φλ2 a, c; z ∗ fλ1 ,λ2 z ∗ fz
−
φλ2 a, c; z ∗ fλ1 ,λ2 z ∗ fz
fλ1 ,λ2 z ∗ −z Lλ2 a, cfz
fλ1 ,λ2 z ∗ Lλ2 a, cfz
fλ1 ,λ2 z ∗ 1 − η φwz η Lλ2 a, cfz
.
fλ1 ,λ2 z ∗ Lλ2 a, cfz
2.10
Therefore by using 2.8, we obtain
zLλ1 a, cfz
−
−η
Lλ1 a, cfz
fλ1 ,λ2 z ∗ 1 − η φwz η Lλ2 a, cfz
1
−η
1−η
fλ1 ,λ2 z ∗ Lλ2 a, cfz
2
z fλ1 ,λ2 z ∗ 1 − η φwz η z2 Lλ2 a, cfz
1
−η .
1−η
z2 fλ1 ,λ2 z ∗ z2 Lλ2 a, cfz
1
1−η
2.11
It follows from 2.9 and Lemma 2.2 that z2 Lλ2 a, cfz ∈ S∗ and z2 fλ1 ,λ2 ∈ K, respectively.
Let us put swz : 1 − ηφwz η. Then by applying Lemma 2.3 to 2.10, we obtain
2
z fλ1 ,λ2 ∗ swzz2 Lλ2 a, cf
2
U ⊂ coswU ⊂ sU,
z fλ1 ,λ2 ∗ z2 Lλ2 a, cf
2.12
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Journal of Inequalities and Applications
since s is convex univalent. Therefore from the definition of subordination and 2.12, we
have
1
1−η
z Lλ1 a, cfz
−
− η ≺ φz
Lλ1 a, cfz
z ∈ U,
2.13
λ1
or, equivalently, f ∈ MSa,c
φ, which completes the proof of Theorem 2.4.
By using 1.13, 2.2 and 2.3, we have the following Theorem 2.5 and Theorem 2.6.
Theorem 2.5. Let λ > 0, a2 ≥ a1 > 0, c ∈ R \ Z−0 and φ ∈ N with Re{φz} < 2 − η/1 − η 0 ≤
η < 1. If a2 ≥ 2 or a1 a2 ≥ 3, then
MSaλ1 ,c η; φ ⊂ MSaλ2 ,c η; φ .
2.14
Theorem 2.6. Let λ > 0, a ∈ R \ Z−0 , c2 ≥ c1 > 0 and φ ∈ N with Re{φz} < 2 − η/1 − η 0 ≤
η < 1. If c2 ≥ 2 or c1 c2 ≥ 3, then
λ
λ
MSa,c
η; φ ⊂ MSa,c
η; φ .
2
1
2.15
Next, we prove the inclusion theorem involving the class MKλa,c η; φ.
Theorem 2.7. Let λ2 ≥ λ1 > 0, a, c ∈ R \ Z−0 and φ ∈ N with Re{φz} < 2 − η/1 − η 0 ≤ η <
1. If λ2 ≥ 2 or λ1 λ2 ≥ 3, then
2
1
MKλa,c
η; φ ⊂ MKλa,c
η; φ .
2.16
Proof. Applying 1.13 and Theorem 2.4, we observe that
2
fz ∈ MKλa,c
η; φ ⇐⇒ Lλ2 a, cfz ∈ MK η; φ
⇐⇒ −z Lλ2 a, cfz ∈ MS η; φ
⇐⇒ Lλ2 a, c −zf z ∈ MS η; φ
λ2
⇐⇒ −zf z ∈ MSa,c
η; φ
λ1
⇒ −zf z ∈ MSa,c
η; φ
⇐⇒ Lλ1 a, c −zf z ∈ MS η; φ
⇐⇒ −z Lλ1 a, cfz ∈ MS η; φ
⇐⇒ Lλ1 a, cfz ∈ MK η; φ
1
⇐⇒ fz ∈ MKλa,c
η; φ ,
which evidently proves Theorem 2.7.
2.17
Journal of Inequalities and Applications
7
By using a similar method as in the proof of Theorem 2.7, we obtain the following two
theorems.
Theorem 2.8. Let λ > 0, a2 ≥ a1 > 0, c ∈ R \ Z−0 and φ ∈ N with Re{φz} < 2 − η/1 − η 0 ≤
η < 1. If a2 ≥ 2 or a1 a2 ≥ 3, then
MKλa1 ,c η; φ ⊂ MKλa2 ,c η; φ .
2.18
Theorem 2.9. Let λ > 0, a ∈ R \ Z−0 , c2 ≥ c1 > 0 and φ ∈ N with Re{φz} < 2 − η/1 − η 0 ≤
η < 1. If c2 ≥ 2 or c1 c2 ≥ 3, then
MKλa,c2 η; φ ⊂ MKλa,c1 η; φ .
2.19
Taking φz 1 Az/1 Bz −1 < B < A ≤ 1; z ∈ U in Theorems 2.4–2.9, we have
the following corollaries below.
Corollary 2.10. Let 1 A1 − η < 2 − η1 B −1 < B < A ≤ 1; 0 ≤ η < 1 and c ∈ R \ Z−0 .
If λ2 ≥ λ1 > 0 and λ2 ≥ min{2, 3 − λ1 }, and a2 ≥ a1 > 0 and a2 ≥ min{2, 3 − a1 }, then
MSaλ12,c η; A, B ⊂ MSaλ11,c η; A, B ⊂ MSaλ21,c η; A, B ,
MKλa21 ,c η; A, B ⊂ MKλa11 ,c η; A, B ⊂ MKλa12 ,c η; A, B .
2.20
Corollary 2.11. Let 1 A1 − η < 2 − η1 B −1 < B < A ≤ 1; 0 ≤ η < 1 and λ > 0. If
a2 ≥ a1 > 0 and a2 ≥ min{2, 3 − a1 }, and c2 ≥ c1 > 0 and c2 ≥ min{2, 3 − c1 }, then
MSaλ1 ,c2 η; A, B ⊂ MSaλ1 ,c1 η; A, B ⊂ MSaλ2 ,c1 η; A, B ,
MKλa1 ,c2
η; A, B ⊂ MKλa1 ,c1 η; A, B ⊂ MKλa2 ,c1 η; A, B .
2.21
Corollary 2.12. Let 1 A1 − η < 2 − η1 B −1 < B < A ≤ 1; 0 ≤ η < 1 and a ∈ R \ Z−0 .
If λ2 ≥ λ1 > 0 and λ2 ≥ min{2, 3 − λ1 }, and c2 ≥ c1 > 0 and c2 ≥ min{2, 3 − c1 }, then
λ2
λ2
λ1
MSa,c
η; A, B ⊂ MSa,c
η; A, B ⊂ MSa,c
η; A, B ,
2
1
1
2
2
1
MKλa,c
η; A, B ⊂ MKλa,c
η; A, B ⊂ MKλa,c
η; A, B .
2
1
1
2.22
To prove theorems below, we need the following lemma.
Lemma 2.13. Let φ ∈ N with Re{φz} < 2 − η/1 − η 0 ≤ η < 1. If f ∈ M with z2 fz ∈ K
and q ∈ MSη; φ, then f ∗ q ∈ MSη; φ.
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Journal of Inequalities and Applications
Proof. Let q ∈ MSη; φ. Then
−zq z 1 − η φwz η qz
z ∈ U,
2.23
where w is an analytic function in U with |wz| < 1 z ∈ U and w0 0. Thus we have
1
1−η
z fz ∗ qz
−
−η
fz ∗ qz
1
1−η
1
1−η
fz ∗ −zq z
−η
fz ∗ qz
fz ∗
2.24
1 − η φωz η qz
−η
z ∈ D.
fz ∗ qz
By using the similar arguments to those used in the proof of Theorem 2.4, we conclude that
2.24 is subordinated to φ in U and so f ∗ q ∈ MSη; φ.
Finally, we give the inclusion properties involving the class MCλa,c η, β; φ, ψ.
Theorem 2.14. Let c ∈ R \ Z−0 and φ, ψ ∈ N with Re{φz} < 2 − η/1 − η 0 ≤ η < 1. If
λ2 ≥ λ1 > 0 and λ2 ≥ min{2, 3 − λ1 }, and a2 ≥ a1 > 0 and a2 ≥ min{2, 3 − a1 }, then
MCλa21 ,c η, β; φ, ψ ⊂ MCλa11 ,c η, β; φ, ψ ⊂ MCλa12 ,c η, β; φ, ψ .
2.25
Proof. We begin by proving that
MCλa21 ,c η, β; φ, ψ ⊂ MCλa11 ,c η, β; φ, ψ .
2.26
Let f ∈ MCλa21 ,c η, β; φ, ψ. Then there exists a function q2 ∈ MSη; φ such that
1
1−β
z Lλ2 a1 , cfz
−
− β ≺ ψz
q2 z
0 ≤ β < 1; z ∈ U .
2.27
From 2.27, we obtain
−z Lλ2 a1 , cfz 1 − β ψwz β q2 z,
2.28
Journal of Inequalities and Applications
9
where w is an analytic function in U with |wz| < 1 z ∈ U and w0 0. By virtue of 2.3,
Lemmas 2.2 and 2.13, we see that fλ1 ,λ2 z ∗ q2 z ≡ q1 z belongs to MSη; φ. Then, making
use of 2.1, we have
1
1−β
z Lλ1 a1 , cfz
−
−β
q1 z
⎛
⎞
f
,
cfz
∗
−z
L
z
a
λ
,λ
λ
1
1
2
2
1 ⎜
⎟
− β⎠
⎝
1−β
fλ1 ,λ2 z ∗ q2 z
1
1−β
1
1−β
≺ ψz
fλ1 ,λ2 z ∗
1 − β ψwz β q2 z
−β
fλ1 ,λ2 z ∗ q2 z
2.29
z2 fλ1 ,λ2 z ∗ 1 − β ψwz β z2 q2 z
−β
z2 fλ1 ,λ2 z ∗ z2 q2 z
z ∈ U.
Therefore we prove that f ∈ MCλa11 ,c η, β; φ, ψ.
For the second part, by using arguments similar to those detailed above with 2.2, we
obtain
MCλa11 ,c η, β; φ, ψ ⊂ MCλa12 ,c η, β; φ, ψ
2.30
Thus the proof of Theorem 2.14 is completed.
The following results can be obtained by using the same techniques as in the proof of
Theorem 2.14 and so we omit the detailed proofs involved.
Theorem 2.15. Let λ > 0 and φ, ψ ∈ N with Re{φz} < 2−η/1−η 0 ≤ η < 1. If a2 ≥ a1 > 0
and a2 ≥ min{2, 3 − a1 }, and c2 ≥ c1 > 0 and c2 ≥ min{2, 3 − c1 }, then
MCλa1 ,c2 η, β; φ, ψ ⊂ MCλa1 ,c1 η, β; φ, ψ ⊂ MCλa2 ,c1 η, β; φ, ψ .
2.31
Theorem 2.16. Let a ∈ R \ Z−0 and φ, ψ ∈ N with Re{φz} < 2 − η/1 − η 0 ≤ η < 1. If
λ2 ≥ λ1 > 0 and λ2 ≥ min{2, 3 − λ1 }, and c2 ≥ c1 > 0 and c2 ≥ min{2, 3 − c1 }, then
2
2
1
MCλa,c
η, β; φ, ψ ⊂ MCλa,c
η, β; φ, ψ ⊂ MCλa,c
η, β; φ, ψ .
2
1
1
2.32
Remark 2.17. For a λ 1 λ > −1 and c 1, Theorems 2.4, 2.5, 2.7, 2.8, and 2.14 reduce to
the corresponding results obtained by Cho and Noor 16.
3. Inclusion Properties Involving Various Operators
λ
The next theorem shows that the classes MSa,c
η; φ, MKλa,c η; φ and MCλa,c η, β; φ, ψ are
invariant under convolution with convex functions.
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Journal of Inequalities and Applications
Theorem 3.1. Let λ > 0, a > 0, c ∈ R \ Z−0 , φ, ψ ∈ N with Re{φz} < 2 − η/1 − η 0 ≤ η < 1
and let g ∈ M with z2 gz ∈ K. Then
λ
λ
η; φ ⇒ g∗f ∈ MSa,c
η; φ,
i f ∈ MSa,c
ii f ∈ MKλa,c η; φ ⇒ g∗f ∈ MKλa,c η; φ,
iii f ∈ MCλa,c η, β; φ, ψ ⇒ g∗f ∈ MCλa,c η, β; φ, ψ.
λ
Proof. i Let f ∈ MSa,c
η; φ. Then we have
1
1−η
⎛
⎞
gz
∗
−z
L
cfz
a,
λ
z Lλ a, c g ∗ f z
1 ⎜
⎟
− η⎠.
−η −
⎝
1−η
gz ∗ Lλ a, cfz
Lλ a, c g ∗ f z
3.1
By using the same techniques as in the proof of Theorem 2.4, we obtain i.
ii Let f ∈ MKλa,c φ. Then, by 1.13, −zf z ∈ MSa,c η; φ and hence from i,
λ
η; φ. Since
gz ∗ −zf z ∈ MSa,c
gz ∗ −zf z −z g ∗ f z,
3.2
we have ii applying 1.13 once again.
iii Let f ∈ MCλa,c η, β; φ, ψ. Then there exists a function q ∈ MSη; φ such that
−z Lλ a, cfz 1 − β ψwz β qz
0 ≤ β < 1; z ∈ U ,
3.3
where w is an analytic function in U with |wz| < 1 z ∈ U and w0 0. From Lemma 2.13,
we have that g ∗ q ∈ MSη; φ. Since
1
1−β
z Lλ a, c g ∗ f z
−
−β
g ∗ q z
⎛
⎞
gz
∗
−z
L
cfz
a,
λ
1 ⎜
⎟
− β⎠
⎝
1−β
gz ∗ qz
1
1−β
≺ ψz
we obtain iii.
z2 gz ∗
z ∈ U,
1 − β ψwz β z2 qz
−β
z2 gz ∗ z2 qz
3.4
Journal of Inequalities and Applications
11
Now we consider the following operators defined by
∞
1 c k−1
1
z
Re{c} ≥ 0; z ∈ D,
z k1 k c
1 − xz
1
log
Ψ2 z 2
log 1 0; |x| ≤ 1, x /
1; z ∈ D .
1−z
z 1 − x
Ψ1 z 3.5
It is well known 17 that the operators z2 Ψ1 and z2 Ψ2 are convex univalent in U. Therefore
we have the following result, which can be obtained from Theorem 3.1 immediately.
Corollary 3.2. Let a > 0, λ > 0, c ∈ R \ Z−0 , φ, ψ ∈ N with Re{φz} < 2 − η/1 − η 0 ≤ η < 1
and let Ψi i 1, 2 be defined by 3.5, respectively. Then
λ
λ
η; φ ⇒ Ψi ∗f ∈ MSa,c
η; φ,
i f ∈ MSa,c
ii f ∈ MKλa,c η; φ ⇒ Ψi ∗f ∈ MKλa,c η; φ,
iii f ∈ MCλa,c η, β; φ, ψ ⇒ Ψi ∗f ∈ MCλa,c η, 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 Government MOEHRD, Basic Research Promotion
Fund KRF-2008-313-C00035.
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