Hindawi Publishing Corporation
Journal of Inequalities and Applications
Volume 2009, Article ID 190291, 9 pages
doi:10.1155/2009/190291
Research Article
Subclasses of Meromorphic Functions Associated
with Convolution
Maisarah Haji Mohd,1 Rosihan M. Ali,1 Lee See Keong,1
and V. Ravichandran2
1
2
School of Mathematical Sciences, Universiti Sains Malaysia, 11800 USM, Penang, Malaysia
Department of Mathematics, University of Delhi, Delhi 110 007, India
Correspondence should be addressed to Lee See Keong, sklee@cs.usm.my
Received 12 December 2008; Accepted 11 March 2009
Recommended by Narendra Kumar Govil
Several subclasses of meromorphic functions in the unit disk are introduced by means of
convolution with a given fixed meromorphic function. Subjecting each convoluted-derived
function in the class to be subordinated to a given normalized convex function with positive real
part, these subclasses extend the classical subclasses of meromorphic starlikeness, convexity, closeto-convexity, and quasi-convexity. Class relations, as well as inclusion and convolution properties
of these subclasses, are investigated.
Copyright q 2009 Maisarah Haji Mohd 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.
1. Introduction
Let H be the set of all analytic functions defined in the unit disk U {z : |z| < 1}. We denote
n
by A the class of normalized analytic functions fz z ∞
n2 an z defined in U. For two
functions f and g analytic in U, the function f is subordinate to g, written as
fz ≺ gz,
1.1
if there exists a Schwarz function w, analytic in U with w0 0 and |wz| < 1 such that
fz gwz. In particular, if the function g is univalent in U, then fz ≺ gz is equivalent
to f0 g0 and fU ⊂ gU.
A function f ∈ A is starlike if zf z/fz is subordinate to 1z/1−z and convex if
1zf z/f z is subordinate to 1z/1−z. Ma and Minda 1 gave a unified presentation
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Journal of Inequalities and Applications
of these classes and introduced the classes
zf z
≺ hz ,
f ∈ A fz
zf z
≺ hz ,
Ch f ∈ A 1 f z
S∗ h 1.2
where h is an analytic function with positive real part, h0 1, and h maps the unit disk U
onto a region starlike with respect to 1.
The convolution or the Hadamard product of two analytic functions fz z ∞
n
a
zn and gz z ∞
n
n2
n2 bn z is given by
f ∗ gz ∞
an bn zn .
1.3
n1
In term of convolution, a function f is starlike if f ∗ z/1 − z is starlike, and convex if
f ∗ z/1 − z2 is starlike. These ideas led to the study of the class of all functions f such that
f ∗ g is starlike for some fixed function g in A. In this direction, Shanmugam 2 introduced
and investigated various subclasses of analytic functions by using the convex hull method 3–
5 and the method of differential subordination. Ravichandran 6 introduced certain classes
of analytic functions with respect to n-ply symmetric points, conjugate points, and symmetric
conjugate points, and also discussed their convolution properties. Some other related studies
were also made in 7–9, and more recently by Shamani et al. 10.
Let M denote the class of meromorphic functions f of the form
fz ∞
1 an zn ,
z n0
1.4
which are analytic and univalent in the punctured unit disk U∗ {z : 0 < |z| < 1}.
For 0 ≤ α < 1, we recall that the classes of meromorphic starlike, meromorphic convex,
meromorphic close-to-convex, meromorphic γ-convex Mocanu sense, and meromorphic
quasi-convex functions of order α, denoted by Ms , Mk , Mc , Mkγ , and Mq , respectively, are
defined by
zf z
>α ,
M f ∈ M − R
fz
zf z
Mk f ∈ M − R 1 >α ,
f z
zf z
> α, g ∈ Ms ,
Mc f ∈ M − R
gz
zf z
zf z
γ 1 >α ,
Mkγ f ∈ M − R 1 − γ
fz
f z
zf z
Mq f ∈ M − R
> α, g ∈ Mk .
g z
s
1.5
Journal of Inequalities and Applications
3
The convolution of two meromorphic functions f and g, where f is given by 1.4 and gz n
1/z ∞
n0 bn z , is given by
f ∗ gz ∞
1 an bn zn .
z n0
1.6
Motivated by the investigation of Shanmugam 2, Ravichandran 6, and Ali et al.
7, 11, several subclasses of meromorphic functions defined by means of convolution with
a given fixed meromorphic function are introduced in Section 2. These new subclasses
extend the classical classes of meromorphic starlike, convex, close-to-convex, γ-convex,
and quasi-convex functions given in 1.5. Section 3 is devoted to the investigation of the
class relations as well as inclusion and convolution properties of these newly defined
classes.
We will need the following definition and results to prove our main results.
Let S∗ α denote the class of starlike functions of order α. The class Rα of prestarlike
functions of order α is defined by
Rα f ∈ A f ∗
z
1 − z2−2α
∈ S∗ α
1.7
for α < 1, and
R1 fz 1
>
f ∈ A R
.
z
2
1.8
Theorem 1.1 see 12, Theorem 2.4. Let α ≤ 1, f ∈ Rα , and g ∈ S∗ α. Then, for any analytic
function H ∈ HU,
f ∗ Hg
U ⊂ co HU ,
f ∗g
1.9
where coHU denotes the closed convex hull of HU.
Theorem 1.2 see 13. Let h be convex in U and β, γ ∈ C with Rβhz γ > 0. If p is analytic
in U with p0 h0, then
pz zp z
≺ hz
βpz γ
implies pz ≺ hz.
1.10
2. Definitions
In this section, various subclasses of M are defined by means of convolution and
subordination. Let g be a fixed function in M, and let h be a convex univalent function with
positive real part in U and h0 1.
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Journal of Inequalities and Applications
0 in U∗ and
Definition 2.1. The class Msg h consists of functions f ∈ M satisfying g ∗ fz /
the subordination
−
zg ∗ f z
≺ hz.
g ∗ fz
2.1
Remark 2.2. If gz 1/z 1/1 − z, then Msg h coincides with Ms h, where
s
M h zf z
≺ hz .
f ∈ M −
fz
2.2
Definition 2.3. The class Mkg h consists of functions f ∈ M satisfying g ∗ f z / 0 in U∗ and
the subordination
zg ∗ f z
− 1
≺ hz.
g ∗ f z
2.3
Definition 2.4. The class Mcg h consists of functions f ∈ M such that g ∗ ψz / 0 in U∗ for
some ψ ∈ Msg h and satisfying the subordination
z g ∗ f z
≺ hz.
−
g ∗ ψz
2.4
Definition 2.5. For γ real, the class Mkg,γ h consists of functions f ∈ M satisfying g ∗fz /
0,
∗
g ∗ f z 0
in
U
and
the
subordination
/
z g ∗ f z
z g ∗ f z
≺ hz.
− γ 1 1
−
γ
g ∗ fz
g ∗ f z
2.5
q
Definition 2.6. The class Mg h consists of functions f ∈ M such that g ∗ ϕ z /
0 in U∗ for
k
some ϕ ∈ Mg h and satisfying the subordination
− z g ∗ f z
≺ hz.
g ∗ ϕ z
2.6
3. Main Results
This section is devoted to the investigation of class relations as well as inclusion and
convolution properties of the new subclasses given in Section 2.
Theorem 3.1. Let h be a convex univalent function satisfying Rhz < 2 − α, 0 ≤ α < 1, and g ∈ M
with z2 g ∈ Rα . If f ∈ Ms h, then f ∈ Msg h. Equivalently, if f ∈ Ms h, then g ∗ f ∈ Ms h.
Journal of Inequalities and Applications
5
Proof. Define the function F by
Fz −
zf z
.
fz
3.1
For f ∈ Ms h, it follows that
−R
zf z
fz
< 2 − α,
3.2
2 z z f z
R
> α.
z2 fz
3.3
and therefore,
Hence z2 f ∈ S∗ α. A computation shows that
−
2
z g ∗ f z
g ∗ −zf z g ∗ fFz
z g ∗ z2 fF z
2
.
g ∗ fz
g ∗ fz
g ∗ fz
z g ∗ z2 f z
3.4
Theorem 1.1 yields
2
z g ∗ f z
z g ∗ z2 fF z
2
∈ co FU ,
−
2
g ∗ fz
z g ∗ z f z
3.5
and because Fz ≺ hz, it follows that
z g ∗ f z
−
≺ hz.
g ∗ fz
3.6
Theorem 3.2. The function f ∈ Mkg h if and only if −zf ∈ Msg h.
Proof. The results follow from the equivalence relations
z g ∗ f z
z g ∗ −zf z
z g ∗ f z
≺ hz.
− 1 ≺ hz ⇐⇒ − ≺ hz ⇐⇒ − g ∗ −zf z
g ∗ f z
g ∗ f z
3.7
Theorem 3.3. Let h be a convex univalent function satisfying Rhz < 2 − α, 0 ≤ α < 1, and φ ∈ M
with z2 φ ∈ Rα . If f ∈ Msg h, then φ ∗ f ∈ Msg h.
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Journal of Inequalities and Applications
Proof. Since f ∈ Msg h, it follows that
zg ∗ f z
g ∗ fz
< 2 − α,
3.8
2
z z g ∗ f z
> α.
R
z2 g ∗ fz
3.9
−R
and thus
Let
P z −
zg ∗ f z
.
g ∗ fz
3.10
A similar computation as in the proof of Theorem 3.1 yields
−
zφ ∗ g ∗ f z z2 φz ∗ z2 g ∗ fzP z
.
φ ∗ g ∗ fz
z2 φz ∗ z2 g ∗ fz
3.11
Inequality 3.9 shows that z2 g ∗ f ∈ S∗ α. Therefore Theorem 1.1 yields
−
zφ ∗ g ∗ f z
≺ hz,
φ ∗ g ∗ fz
3.12
hence φ ∗ f ∈ Msg h.
Corollary 3.4. Msg h ⊂ Msφ∗g h under the conditions of Theorem 3.3.
Proof. The proof follows from 3.12.
In particular, when gz 1/z 1/1 − z, the following corollary is obtained.
Corollary 3.5. Let h and φ satisfy the conditions of Theorem 3.3. If f ∈ Ms h, then f ∈ Msφ h.
Theorem 3.6. Let h and φ satisfy the conditions of Theorem 3.3. If f ∈ Mkg h, then φ ∗ f ∈ Mkg h.
Equivalently Mkg h ⊂ Mkφ∗g h.
Proof. If f ∈ Mkg h, it follows from Theorem 3.2 that −zf ∈ Msg h. Theorem 3.3 shows that
φ ∗ −zf −zφ ∗ f ∈ Msg h. Hence φ ∗ f ∈ Mkg h.
Theorem 3.7. Under the conditions of Theorem 3.3, if f ∈ Mcg h with respect to ψ ∈ Msg h, then
φ ∗ f ∈ Mcg h with respect to φ ∗ ψ ∈ Msg h.
Proof. Theorem 3.3 shows that φ ∗ ψ ∈ Msg h. Since ψ ∈ Msg h, 3.9 yields z2 g ∗ ψ ∈ S∗ α.
Journal of Inequalities and Applications
7
Let the function G be defined by
Gz −
zg ∗ f z
.
g ∗ ψz
3.13
A similar computation as in the proof of Theorem 3.1 yields
−
zφ ∗ g ∗ f z z2 φz ∗ z2 g ∗ ψzGz
.
φ ∗ g ∗ ψz
z2 φz ∗ z2 g ∗ ψz
3.14
Since z2 φ ∈ Rα and z2 g ∗ ψ ∈ S∗ α, it follows from Theorem 1.1 that
−
zφ ∗ g ∗ f z
≺ hz.
φ ∗ g ∗ ψz
3.15
Thus φ ∗ f ∈ Mcg h with respect to φ ∗ ψ.
Corollary 3.8. Mcg h ⊂ Mcφ∗g h under the assumptions of Theorem 3.3.
Proof. The subordination 3.15 shows that f ∈ Mcφ∗g h.
Theorem 3.9. Let Rγhz < 0. Then
i Mkg,γ h ⊂ Msg h,
ii Mkg,γ h ⊂ Mkg,β h for γ < β ≤ 0.
Proof. Define the function P by
P z −
zg ∗ f z
g ∗ fz
3.16
and the function Jg γ; f by
zg ∗ f z
zg ∗ f z
1
−
γ
Jg γ; fz − γ 1 .
g ∗ fz
g ∗ f z
3.17
For f ∈ Mkg,γ h, it follows that Jg γ; fz ≺ hz. Note also that
Jg γ; fz P z −
γzP z
.
P z
3.18
i Since Rγhz < 0 and
P z −
γzP z
≺ hz.
P z
3.19
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Journal of Inequalities and Applications
Theorem 1.2 yields P z ≺ hz. Hence f ∈ Msg h.
ii Observe that
zg ∗ f z
zg ∗ f z
Jg β; fz − β 1 1
−
β
g ∗ fz
g ∗ f z
β
β
1−
P z Jg γ; fz.
γ
γ
3.20
Furthermore Jg γ; fz ≺ hz and P z ≺ hz from i. Since 0 < β/γ < 1 and hU is
convex, we deduce that Jg β; fz ∈ hU. Therefore, Jg β; fz ≺ hz.
q
Corollary 3.10. The class Mkg h is a subset of the class Mg h.
Proof. The proof follows from the definition of the classes by taking f ϕ.
q
Theorem 3.11. The function f ∈ Mg h if and only if −zf ∈ Mcg h.
q
Proof. If f ∈ Mg h, then there exists ϕ ∈ Mkg h such that
− zg ∗ f z
g ∗ ϕ z
≺ hz.
3.21
Also,
−z g ∗ −zf z
− zg ∗ f z
≺ hz.
g ∗ −zϕ z
g ∗ ϕ z
3.22
Since ϕ ∈ Mkg h, by Theorem 3.2 , −zϕ ∈ Msg h. Hence −zf ∈ Mcg h.
Conversely, if −zf ∈ Mcg h, then
z g ∗ −zf z
≺ hz
− g ∗ ϕ1 z
3.23
for some ϕ1 ∈ Msg h. Let ϕ ∈ Mkg h be such that −zϕ ϕ1 ∈ Msg h. The proof is completed
by observing that
− zg ∗ f z
g ∗ ϕ z
z g ∗ −zf z
− ≺ hz.
g ∗ −zϕ z
3.24
q
q
Corollary 3.12. Let h and φ satisfy the conditions of Theorem 3.3. If f ∈ Mg h, then φ∗f ∈ Mg h.
q
Proof. If f ∈ Mg h, Theorem 3.11 gives −zf ∈ Mcg h. Theorem 3.7 next gives φ ∗ −zf q
−zφ ∗ f ∈ Mcg h. Thus, Theorem 3.11 yields φ ∗ f ∈ Mg h.
Journal of Inequalities and Applications
q
9
q
Corollary 3.13. Mg h ⊂ Mφ∗g h under the conditions of Theorem 3.3.
q
q
Proof. If f ∈ Mg h, it follows from Corollary 3.12 that φ ∗ f ∈ Mg h. The subordination
− zφ ∗ g ∗ f z
φ ∗ g ∗ ϕ z
q
q
≺ hz
3.25
q
gives f ∈ Mφ∗g h. Therefore Mg h ⊂ Mφ∗g h.
Open Problem
An analytic convex function in the unit disk is necessarily starlike. For the meromorphic case,
is it true that Mkg h ⊂ Msg h?
Acknowledgment
The work presented here was supported in part by the USM’s Reserach University grant and
the FRGS grant
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