R E S E A R C H
Open Access
Properties for certain subclasses of
meromorphic functions defined by a
multiplier transformation
Nak Eun Cho
1and Min Yoon
2**Correspondence:
2Department of Statistics, Pukyong
National University, 45 Yongso-ro, Busan, 608-737, Korea
Full list of author information is available at the end of the article
Abstract
Some inclusion and convolution properties of certain subclasses of meromorphic functions associated with a family of multiplier transformations, which are defined by means of the Hadamard product (or convolution), are investigated. We also obtain closure properties for certain integral operators.
MSC: 30C45; 30C80
Keywords: meromorphic function; subordination; starlike of order
α
; convex of orderα
; prestarlike of orderα
; convolution; integral operator1 Introduction
LetAdenote the class of analytic functionsfin the open unit diskU={z∈C:|z|< }with the usual normalizationf() =f() – = . LetS*(α) andK(α) denote the subclasses of
Aconsisting of starlike and convex functions of orderα(≤α< ) and letS*() =S*and
K() =K. Iff andgare analytic inU, we say thatf is subordinate tog inU, written as f≺gorf(z)≺g(z), if there exists a Schwarz functionwsuch thatf(z) =g(w(z)) (z∈U).
A functionf ∈Ais said to be prestarlike of orderαinUif z
( –z)(–α) ∗f(z)∈S
*(α) (≤α< ),
wheref∗gdenotes the familiar Hadamard product (or convolution) of two analytic func-tionsf andginU. We denote this class byR(α) (see, for details, []). We note thatR() =K andR(/) =S*(/).
LetNbe the class of all functionshwhich are analytic and univalent inUand for which h(U) is convex withh() = .
LetMdenote the class of functions of the form
f(z) = z+
∞
k= akzk
which are analytic in the punctured open unit diskD=U\ {}.
For anyn∈N={, , , . . .}, we denote the multiplier transformationsDnλof functions
f ∈Mby
Dnλf(z) = z+
∞
k=
k+ +l
λ n
akzk (λ> ;z∈D).
Obviously, we have
Dsλ
Dtλf(z)
=Dsλ+tf(z)
for all nonnegative integerssandt. The operatorsDnλandDnare the multiplier transforma-tions introduced and studied by Sarangi and Uraligaddi [] and Uralegaddi and Somanatha [, ], respectively. Analogous toDn
λ, we here define a new multiplier transformationIλn,μ
as follows. Letfn(z) = /z+
∞
k=((k+ +λ)/λ)nzk,n∈N, and letfn†,μbe such that
fn(z)∗fn†,μ(z) =
z+
∞
k= (μ)k+
()k+
zk (μ> ;z∈D),
where (ν)k is the Pochhammer symbol (or the shifted factorial) defined (in terms of the gamma function) by
(ν)k:=
(ν+k)
(ν) =
⎧ ⎨ ⎩
ifk= andν∈C\ {},
ν(ν+ )· · ·(ν+k– ) ifk∈N:={, , . . .}andν∈C.
Then
Iλn,μf(z) =f
†
n,μ(z)∗f(z). (.)
We note thatI,f(z) =zf(z) + f(z) andI, f(z) =f(z). It is easily verified from (.) that
zIλn,+μf(z)
=λIλn,μf(z) – (λ+ )Iλn,+μf(z) (.)
and
zIλn,μf(z)=μIλn,μ+f(z) – (μ+ )Iλn,μf(z). (.)
The definition (.) of the multiplier transformationIλn,μ is motivated essentially by the
Choi-Saigo-Srivastava operator [] for analytic functions, which includes the Noor inte-gral operator studied by Liu [] (also, see [–]).
We also define the functionφ(a,c;z) by
φ(a,c;z) := z+
∞
k= (a)k+ (c)k+
zk z∈U;a∈R;c∈R\Z–;Z–:={–, –, . . .}. (.)
By using the operatorIλn,μ, we introduce the following class of analytic functions for
γ > ,λ> ,s∈R,μ> andh∈N:
Mn
λ,μ(γ;h) :=
f ∈M: ( –γ)zIλn,μf(z) +γz
Iλn,μf(z)
≺
In the present paper, we derive some inclusion relations, convolution properties and integral preserving properties for the classMn
λ,μ(γ;h).
The following lemmas will be required in our investigation.
Lemma .[, Lemma , p.] Let g be analytic inUand h be analytic and convex univalent in U with h() =g().If
g(z) +
γzg
(z)≺h(z) Re{γ} ≥;γ = , (.)
then
g(z)≺h(z) =γz–γ
z
tγ–h(t)dt≺h(z)
andh is the best dominant of(.).
Lemma . [, Theorem ., p.] Let f ∈S*(α)and g∈R(α).Then for any analytic function F inU,
g∗(fF)
g∗f (U)⊂co
F(U),
where co(F(U))denotes the convex hull of F(U).
Lemma .[, Lemma , p.] Let <a≤c.Then
Rezφ(a,c;z)>
(z∈U),
whereφis given by(.).
2 Inclusion relations
Theorem . If≤γ<γ,then
Mn
λ,μ(γ;h)⊂Mnλ,μ(γ;h). Proof Let
g(z) =zIλn,μf(z)
f ∈Mnλ,μ(γ;h) :z∈U
. (.)
Then the functiong is analytic inUwithg() = . Differentiating both sides of (.), we have
( +γ)zIλn,μf(z) +γz
Iλn,μf(z)=g(z) +γzg(z)≺h(z). (.)
Hence an application of Lemma . withμ= /γyields
Since ≤γ/γ< andhis convex univalent inU, it follows from (.), (.) and (.) that
( +γ)zIλn,μf(z) +γz
Iλn,μf(z)
=γ
γ
( –γ)zIλn,μf(z) +γz
Iλn,μf(z)+
–γ
γ
g(z)
≺h(z).
Thereforef∈Mnλ,μ(γ;h), and so we complete the proof of Theorem ..
Theorem . If <μ≤μ,then
Mn
λ,μ(γ;h)⊂M
n
λ,μ(γ;h).
Proof Letf ∈Mnλ,μ(γ;h). Then
( +γ)zIλn,μf(z) +γz
In
λ,μf(z)
=zφ(μ,μ;z)∗
( +γ)zIλn,μf(z) +γzIλn,μf(z). (.)
In view of Lemma ., we see that the functionzφ(μ,μ;z) has the Herglotz representa-tion
zφ(μ,μ;z) =
|x|= dμ(x)
–xz (z∈U), (.)
whereμ(x) is a probability measure defined on the unit circle|x|< and
|x|=
dμ(x) = .
Sincehis convex univalent inU, it follows from (.) and (.) that
( +γ)zIλn,μf(z) +γz
In
λ,μf(z)
=
|x|=
h(xz)dμ(x)≺h(z),
which completes the proof of Theorem ..
Theorem . Ifμ> ,then
Mn
λ,μ+(γ;h)⊂Mnλ,μ(γ;h),
where
h(z) =μz–μ
z
tμ–h(t)dt≺h(z).
Proof Let
g(z) = ( +γ)zIλn,μf(z) +γz
Iλn,μf(z)
Then from (.) and (.), we have
z–g(z) =γ μIλn,μ+f(z) + ( –γ μ)Iλn,μf(z). (.)
Differentiating both sides of (.) and using (.), we obtain
z–zg(z) +g(z)
=γ μzIλn,μ+f(z)
+ ( –γ μ)μIλn,μ+f(z) – (μ+ )Iλn,μf(z)
. (.)
By a simple calculation with (.) and (.), we get
g(z) +zg
(z)
μ = ( +γ)
In
λ,μ+f(z)
z +γ
Iλn,μ+f(z)
. (.)
Iff ∈Mn
λ,μ+(γ;h), then it follows from (.) that
g(z) +zg
(z)
μ ≺h(z) (μ> ).
Hence an application of Lemma . yields
g(z)≺h(z) =μz–μ
z
tμ–h(t)dt≺h(z),
which shows that
f ∈Mnλ,μ+(γ;h)⊂Mnλ,μ(γ;h).
Theorem . If s∈Randλ> ,then
Mn
λ,μ(γ;h)⊂M
n+
λ,μ(γ;h),
where
h(z) =λz–λ
z
tλ–h(t)dt≺h(z).
Proof By using the same techniques as in the proof of Theorem . and (.), we have
Theorem . and so we omit the detailed proof involved.
Theorem . Letγ > ,β> and f ∈Mnλ,μ(γ;βh+ –β).Ifβ≤β,where
β=
–
γ
uγ–
+udu
–
, (.)
then f ∈Mnλ,μ(;h).The boundβis sharp for the function
h(z) =
Proof Let
g(z) =zIλn,μf(z)
f ∈Mnλ,μ(γ;βh+ –β);γ > ;β>
. (.)
Then we have
g(z) +γzg(z) = ( +γ)zIλn,μf(z) +γzIλn,μf(z)
≺βh(z) + –β.
Hence an application of Lemma . yields
g(z)≺β
γz
–γ
z
tγ–h(t)dt+ –β= (h∗ψ)(z), (.)
where
ψ(z) = β
γz
–γ z
tγ–
–tdt+ –β. (.)
If <β≤β, whereβis given by (.), then from (.), we have
Reψ(z)= β
γ
uγ–Re
–uzdu
+ –β
> β
γ
uγ–
+udu+ –β
≥
.
By using the Herglotz representation forψ, it follows from (.) and (.) that
zIλn,μf(z)≺(h∗ψ)(z)≺h(z)
sincehis convex univalent inU. This shows thatf∈Mn
λ,μ(;h).
Forh(z) = /( –z) andf ∈Mdefined by
zIλn,μf(z) =
β γz
–γ
z
tγ–
–tdt+ –β,
it is easy to verify that
( +γ)zIλn,μf(z) +γz
Iλn,μf(z)
=βh(z) + –β.
Thusf ∈Mnλ,μ(γ;βh+ –β). Furthermore, forβ>β, we have
RezIλn,μf(z)
→β
γ
uγ–
+udu+ –β<
(z→–),
which implies thatf ∈/Mn
λ,μ(;h). Hence the boundβcannot be increased whenh(z) =
3 Convolution properties Theorem . If f ∈Mn
λ,μ(γ;h)and
Rezg(z)>
(g∈M;z∈U), then
f ∗g∈Mnλ,μ(γ;h).
Proof Letf ∈Mn
λ,μ(γ;h) andg∈M. Then we have
( +γ)zIλn,μ(f∗g)(z) +γz
Iλn,μ(f ∗g)(z)
=zg(z)∗ψ(z),
where
ψ(z) = ( +γ)zI
n
λ,μf(z)
+ γz
In
λ,μf(z)
≺
h(z).
The remaining part of the proof of Theorem . is similar to that of Theorem ., and so
we omit the details involved.
Corollary . Let f ∈Mnλ,μ(γ;h)be given by(.).Then the function
σm(z) =
tSm(tz)dt (z∈U),
where
Sm(z) = z+
m–
n= anzn–
m∈N\ {};z∈U,
is also in the classMnλ,μ(γ;h).
Proof We have
σm(z) = z+
m–
n= an
n– z
n+= (f∗g m)(z)
m∈N\ {}, (.)
where
f(z) = z+
∞
n=
anzn–∈Mnλ,μ(γ;h)
and
gm(z) = z+
m–
n= zn
while, it is known [] that
Rezgm(z)
=Re
+ m–
n= zn
n+
>
m∈N\ {};z∈U. (.)
In view of (.) and (.), an application of Theorem . leads toσm∈Mnλ,μ(γ;h).
Theorem . If f ∈Mn
λ,μ(γ;h)and
zg(z)∈R(α) (g∈M;z∈U), then
(f ∗g)∈Mnλ,μ(γ;h).
Proof By using a similar method as in the proof of Theorem ., we have
( +γ)zIλn,μ(f∗g)(z) +γz
Iλn,μ(f ∗g)(z)
=z
g(z)∗(zψ(z))
zg(z)∗z (z∈U), (.)
where
ψ(z) = ( +γ)zInλ,μf(z) +γz
Iλn,μf(z)
≺
h(z).
Sincehis convex univalent inU, it follows from (.) and Lemma . that Theorem .
holds true.
If we takeα= andα= / in Theorem ., we have the following corollary.
Corollary . If f ∈Mn
λ,μ(γ;h)and g∈Msatisfies one of the following conditions:
(i) zg(z)is convex univalent inU or
(ii) zg(z)∈S*( ), then(f ∗g)∈Mn
λ,μ(γ;h).
4 Integral operators
Theorem . If f ∈Mnλ,μ(γ;h),then the function F defined by
F(z) = c– zc
z
tc–f(t)dt Re{c}> (.)
is in the classMn
λ,μ(γ;h),where
h(z) = (c– )z–(c–)
z
tch(t)dt≺h(z).
Proof Letf ∈Mnλ,μ(γ;h). Then from (.), we obtain
Define the functionGby
z–G(z) = ( +γ)Iλn,μF(z) +γz
Iλn,μF(z)
(z∈D). (.)
Differentiating both sides of (.) with respect toz, we get
zG(z) –G(z) = ( +γ)zIλn,μ
zF(z)+γzIλn,μ
zF(z). (.)
Furthermore, it follows from (.), (.) and (.) that
( +γ)zIλn,μf(z) +γz
Iλn,μf(z)
= ( +γ)zIλn,μ
zF(z) +cF(z) c–
+γz
Iλn,μ
zF(z) +cF(z) c–
= c
c– G(z) + c–
zG(z) –G(z)
=G(z) + c– zG
(z). (.)
Sincef∈Mnλ,μ(γ;h), from (.), we have
G(z) + c– zG
(z)≺h(z) Re{c}> ,
and so an application of Lemma . yields
G(z)≺h(z) =c– zc–
z
tch(t)dt≺h(z).
Therefore we conclude that
F∈Mnλ,μ(γ;h)⊂Mnλ,μ(γ;h).
Theorem . If f ∈Mand F are defined as in Theorem.,if
( –α)zIλn,μF(z) +αzIλn,μf(z)≺h(z) (α> ), (.)
then F∈Mn
λ,μ(;h),where
h(z) = c–
α z
–c–α z
tc–α –h(t)≺h(z) Re{c}> .
Proof Let
G(z) =zIλn,μF(z) (z∈D). (.)
ThenGis analytic inUwithG() = and
zG(z) =zInλ,μF(z)
It follows from (.), (.), (.) and (.) that
( –α)zIλn,μF(z) +αzIλn,μf(z)
= ( –α)zI
n
λ,μF(z)
+
α
c–
czIλn,μF(z) +z
Iλn,μF(z)
=G(z) + α c– zG
(z)≺h(z) Re{c}> ;α> .
Therefore, by Lemma ., we conclude that Theorem . holds true as stated.
Theorem . Let F∈Mn
λ,μ(γ;h).If the function f is defined by
F(z) = c– zc
z
tc–f(t)dt (c> ), (.)
then
σf(σz)∈Mnλ,μ(γ;h),
where
σ=σ(c) =
+ (c– )–
c– . (.)
The boundσ is sharp for the function
h(z) =β+ ( –β) +z
–z (β= ;z∈U). (.)
Proof We note that forF∈M,
F(z) =F(z)∗
z( –z) and zF
(z) =F(z)∗
z( –z) – z( –z)
.
Then from (.), we have
f(z) =cF(z) +zF
(z)
c– = (F∗g)(z) (c> ;z∈D), (.)
where
g(z) = c–
(c– ) z( –z)+
z( –z)
∈M. (.)
Next, we show that
Rezg(z)>
|z|<σ, (.)
whereσ=σ(c) is given by (.). Letting
–z=Re
we see that
cosθ= +R ( –r)
R and R≥
+r. (.)
Then for (.) and (.), we have
Rezg(z)= c–
(c– )Rcosθ+Rcosθ–
= R
c–
c –r+R –r– +
≥ R
c–
c– – r– (c– )r+ .
This evidently gives (.), which is equivalent to
Reσzg(σz)>
(z∈U). (.)
LetF∈Mnλ,μ(γ;h). Then, by using (.) and (.), an application of Theorem . yields
σf(σz) =F(z)∗σg(σz)∈Mnλ,μ(γ;h).
Forhgiven by (.), we consider the functionF∈Mdefined by
( +γ)zIλn,μF(z) +γz
Iλn,μF(z)
=β+ ( –β) +z
–z (β= ;z∈U). (.)
Then from (.), (.) and (.), we find that
( +γ)zIλn,μf(z) +γz
Iλn,μf(z)
=β+ ( –β) +z –z+
z c–
β+ ( –β) +z –z
=β+( –β)(c– + z– (c– )z )
(c– )( –z)
=β (z= –σ).
Therefore we conclude that the boundσ=σ(c) cannot be increased for eachc(c> ).
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors jointly worked on the results and they read and approved the final manuscript.
Author details
1Department of Applied Mathematics, Pukyong National University, 45 Yongso-ro, Busan, 608-737, Korea.2Department
of Statistics, Pukyong National University, 45 Yongso-ro, Busan, 608-737, Korea.
Acknowledgements
The authors would like to express their thanks to the referees for valuable advice regarding a previous version of this paper. This research was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (No. 2012-0002619).
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doi:10.1186/1029-242X-2013-46
