R E S E A R C H
Open Access
On certain subclasses of multivalent
functions defined by multiplier
transformations
Jagannath Patel
1, Nak Eun Cho
2*and Ashis Kumar Palit
3Dedicated to Professor Hari M Srivastava
*Correspondence:
2Department of Applied
Mathematics, Pukyong National University, Busan, Korea Full list of author information is available at the end of the article
Abstract
The purpose of the present paper is to introduce and investigate various properties of a certain class of multivalent functions in the open unit disk defined by a multiplier transformation. In particular, we obtain some inclusion relationships and integral preserving properties of this class of functions. Relevant connections of the results presented in this paper with various known results are also pointed out.
MSC: 30C45
Keywords: analytic function; multivalent function; multiplier transformation; subordination; integral operator
1 Introduction
LetApdenote the class of functions of the form
f(z) =zp+
∞
k=
ap+kzp+k (.)
which are analytic in the open unit diskU={z∈C:|z|< }. We writeA=A.
Suppose thatf andgare analytic inU. We say that the functionf is subordinate togin
U, and we writef ≺gorf(z)≺g(z) (z∈U), if there exists an analytic functionωinUwith ω() = and|ω(z)|< for allz∈Usuch thatf(z) =g(ω(z)) inU. Ifgis univalent inU, then the following equivalence relationship holds true:
f(z)≺g(z) ⇐⇒ f() =g() and f(U)⊂g(U).
For functionsf given by (.) andg(z) =zp+∞k=bp+kzp+k, the Hadamard product (or
convolution) off andgis defined by
(f ∗g)(z) =f(z)∗g(z) =zp+
∞
k=
ap+kbp+kzp+k.
For fixed parametersA,B(–B<A), letP(A,B) be the class of functions of the form
ϕ(z) = +cz+cz+· · · (.)
which are analytic inUand satisfy the condition
ϕ(z)≺ +Az
+Bz (z∈U).
The classP(A,B) was investigated in []. We denote bySp∗(A,B) the class of functions f ∈Ap such thatzf /pf ∈P(A,B). Analogously,K(A,B) is the class of functionsf ∈Ap
such that (zf )/pf ∈P(A,B). It is easily seen that
S∗ p
–ρ
p , –
=Sp∗(ρ) and Kp
–ρ
p , –
=Kp(ρ) (ρ<p),
the subclasses ofAp, which are respectively,p-valently starlike of orderρandp-valently
convex of orderρinU. We also note that S∗
p(ρ)⊆Sp∗() =Sp∗ and Kp(ρ)⊆Kp() =Kp (ρ<p),
whereSp∗andKpare the subclasses ofApconsisting of functions that arep-valently
star-like andp-valently convex inU, respectively.
In the present investigation, we shall make use of theGauss hypergeometric functionF
defined inUby
F(a,b;c;z) = ∞
k=
(a)k(b)k
(c)k
zk
k!
a,b,c∈C;c∈/Z–={, –, –, . . .}, (.)
where (x)ndenotes the Pochhammer symbol (or shifted factorial) given by
(x)n=
⎧ ⎨ ⎩
x(x+ )(x+ )· · ·(x+n– ) (n∈N),
(n= ).
We note that the series defined by (.) converges absolutely for allz∈Uand hence rep-resents an analytic function inU[, Chapter ].
Motivated by the multiplier transformation introduced in [] on A, we introduce an operatorφp(n,λ) onApby
φp(n,λ)(z) =zp+ ∞
k=
p+k+λ p+λ
zp+k λ> –p,n∈Z={,±,±, . . .};z∈U.
The operatorφp(n,λ) is related to the multiplier transformation studied in [].
Corresponding to the functionφp(n,λ), we define a new functionφ(
†)
p (n,λ) in terms of
the Hadamard product by
φp(n,λ)(z)∗φp(†)(n,λ)(z) =
zp
( –z)p+μ (μ> –p;z∈U). (.)
We now introduce the operatorIpn(λ,μ) :Ap−→Apby
If the functionf is given by (.), then from (.) and (.) we deduce that
Ipn(λ,μ)f(z) =zp+
∞
k=
(p+μ)k
()k
p+λ p+k+λ
n
ap+kzp+k (z∈U).
In view of (.), it follows that
zIpn(λ,μ)f(z) = (p+λ)Ipn–(λ,μ)f(z) –λIpn(λ,μ)f(z) (f ∈Ap;z∈U). (.)
In particular, we note that forz∈U, Ip(, –p)f(z) =f(z),
Ip(δ, –p)f(z) =
(p+δ) z
tδ–f(t)dt
zδ (δ> –p)cf.Eqn. (.),
Ip–(λ, –p)f(z) =zf (z) +λf(z)/(p+λ),
Ip–(, –p)f(z) =zf (z) +zf (z)/p and
Ip–(, –p)f(z) =zf (z) + zf (z) +zf (z)/p.
The operatorIn
p(λ, –p) (n∈Z–) is closely related to the Sˇalˇagean derivative operator
[]. The operatorIλn=In(λ, ) was recently studied in [, , ]. For anyn∈Z, the operator
In=In(, ) was studied in [].
By using the operatorIn
p(λ,μ), we introduce the subclass ofApas follows.
Definition For fixed parametersA,B(–B<A),n∈Z,λ,μ> –pandα, we say that a functionf∈Apis in the classSpn,λ,μ(α;A,B) if
( –α) I
n
p(λ,μ)f(z)
In+
p (λ,μ)f(z)
+αI
n–
p (λ,μ)f(z)
In
p(λ,μ)f(z)
≺ +Az
+Bz (z∈U). It is readily seen that
S
p,,–p(;A,B) =Sp∗(A,B) and Sp–,,–p(;A,B) =Kp(A,B).
For the sake of convenience, we write
Sn
p,λ,μ(A,B) =Spn,λ,μ(;A,B) =
f ∈Ap:
In
p(λ,μ)f(z)
In+
p (λ,μ)f(z)
≺ +Az
+Bz,z∈U
.
The object of the present paper is to investigate some inclusion properties of the class Sn
p,λ,μ(α;A,B). Integral-preserving and convolution properties in connection with the
op-eratorIn
p(λ,μ) are also considered. Relevant connections of the results presented here with
those obtained in the earlier works are pointed out.
2 Preliminaries
We denote byHthe class of all analytic functions inUand byBthe class of functions
∈Hsuch thatω() = and| (z)|< forz∈U.
Lemma ([], see also [, p.]) Let h be analytic and convex (univalent)in Uwith h() = .Suppose also that the functionϕdefined by(.)is analytic inU.If
ϕ(z) +zϕ(z) κ ≺h(z)
κ= ,(κ);z∈U,
then
ϕ(z)≺ψ(z) = κ zκ
z
tκ–h(t)dt≺h(z) (z∈U) (.)
andψis the best dominant of(.).
Lemma [, p.] Suppose that the function:C×U−→Csatisfies the condition
(ix,y;z)ε
forε> ,real x,y–( +x)/and all z∈U.If the functionϕ,given by(.)is analytic in
Uand
Reϕ(z),zϕ(z);z>ε,
thenRe(ϕ(z)) > inU.
Lemma [] Let <γ<γ < and∈Hsatisfy
(z)≺ +γz, () = .
(i)Ifϕ∈H,ϕ() = and satisfies
(z)β+ ( –β)ϕ(z)≺ +γz (z∈U), where
β= ⎧ ⎨ ⎩
–γ
+γ ( <γ+γ ),
–(γ+γ) (–γ) (γ
+γγ+γ),
(.)
thenRe(ϕ(z)) > inU.
(ii)Ifω∈Hwithω() = satisfies
(z) +ω(z)≺ +γz (z∈U),
then,for < γ+γ,we have
ω(z)γ+γ –γ
(z∈U). (.)
Lemma [] Ifω∈Band
ϕ(z) = +γ ω(z) +γ δ tδ–ω(tz)dt
<γ < ,δ > ,z∈U,
thenRe(ϕ(z)) >β(β< )inU,where
β= ⎧ ⎨ ⎩
–γ
+γ δ ( <γ
+δ),
–γ(+δ) (–γδ
) (+δ
γ
√
+δ)
(.)
andδ=δ/( +δ).Further,for <γ/( + δ),we have
ϕ(z) – γ( +δ) –γ δ
(z∈U). (.)
The value ofβin(.)and the bound in(.)are best possible.
3 Inclusion relationships for the classSn
p,λ,μ(
α
;A,B)Unless otherwise mentioned, we assumethroughoutthe sequel that
n∈Z, λ,μ> –p, α> and – B<A.
Theorem We have
Sn
p,λ,μ(α;A,B)⊂Spn,λ,μ(A,B).
Further,for f ∈Sn
p,λ,μ(α;A,B),we also have
Ipn(λ,μ)f(z) In+
p (λ,μ)f(z)
≺q(z) = α
(p+λ)Q(z) (z∈U), (.) where
Q(z) = ⎧ ⎨ ⎩
t
p+λ α –(+tBz
+Bz)
(p+λ)(B–A)
αB dt (B= ),
t
p+λ
α –exp((p+λ)(t–)A
α z)dt (B= )
(.)
and q is the best dominant of(.).Moreover,if A–αB
p+λ(–B< ),then
Sn
p,λ,μ(α;A,B)⊂Spn,λ,μ( – ρ, –), (.)
whereρ= [F(,(p+λα)(BB–A), p+λ
α + ;
B B–)]
–.The boundρis best possible.
Proof Letf ∈Spn,λ,μ(α;A,B). Suppose that
g(z) =z In+
p (λ,μ)f(z)
zp
/(p+λ)
andr=sup{r:g(z)= , <|z|<r< }. Choosing the principal branch ofg, we note thatg
is single-valued and analytic inUr={z:|z|<r}. Taking the logarithmic differentiation in (.) and using identity (.) in the resulting equation, we get that
ϕ(z) = zg(z) g(z) =
In
p(λ,μ)f(z)
In+
p (λ,μ)f(z)
(.)
is of the form (.) and is analytic inUr. Again, carrying out logarithmic differentiation in (.) and using (.), we deduce that
( –α) I
n
p(λ,μ)f(z)
In+
p (λ,μ)f(z)
+αI
n–
p (λ,μ)f(z)
In
p(λ,μ)f(z)
=ϕ(z) + α p+λ
zϕ(z) ϕ(z) ≺
+Az
+Bz (z∈Ur). (.) Hence, by applying the result [, Corollary .], we obtain
ϕ(z)≺q(z) = α (p+λ)Q(z)≺
+Az
+Bz (z∈Ur),
whereqis the best dominant of (.) andQis given by (.).
The proof of the remaining part can now be deduced along the same lines as in [, Theorem ]. The boundρin (.) is best possible asqis the best dominant of (.). This evidently completes the proof of the theorem.
Settingn= –,λ= andμ= –pin Theorem , we get the following corollary.
Corollary If A–αB/p(–B< )and f ∈Apsatisfies
( –α)zf (z) f(z) +α
+zf
(z)
f (z)
≺p( +Az)
+Bz (z∈U), then
Re
zf (z) f(z)
>p
F
,p(B–A) αB ,
p α+ ;
B B–
–
(z∈U).
The result is best possible.
In the special case whenn= ,μ= –p,A= – ((η+λα)/(p+λ)) (η< ) andB= –, Theorem gives the following.
Corollary Ifmax{–λα, (p+λ– λα–α)/}<η<p+λ–λαand f ∈Apsatisfies
Re
( –α) z
λf(z)
z
tλ–f(t)dt
+αzf (z) f(z)
>η (z∈U),
then
Re
zλf(z)
z
tλ–f(t)dt
> (p+λ)
F
,(p+λ–λα–η)
α ,
p+λ α + ;
–
(z∈U).
Remarks . PuttingA= andB= – in Corollary , we find that forαpandz∈U,
Re
( –α)zf (z) f(z) +α
+zf
(z)
f (z)
> ⇒ f ∈Sp∗
p( + (p/α))
√
π ( + (p/α))
,
which in turn implies that
f ∈Kp
p(α– )( + (p/α))
√
π α( + (p/α))
.
Forp= , this result is contained in [].
. Settingα= ,A= – (η/p) (η<p) andB= – in Corollary andα= in Corol-lary , we get the corresponding result obtained in [].
Theorem Forη<p,we have
f ∈Spn,λ,μ
; –η p , –
⇒ f ∈Spn,λ,μ
α; –η
p, – |z|<R , where R= ⎧ ⎨ ⎩
pα+(p+λ)(p–η)–√(pα+(p+λ)(p–η))–p(p+λ)(p–η)
(p+λ)(p–η) (η= p ), p+λ
p+λ+α (η=
p ).
(.)
The bound R is best possible.
Proof Setting
In
p(λ,μ)f(z)
In+
p (λ,μ)f(z)
=η p+
–η
p
u(z) (z∈U), (.)
we see thatuis of the form (.), analytic and has a positive real part inU. Taking the logarithmic differentiation in (.) and using identity (.), we deduce that
Re
( –α) I
n
p(λ,μ)f(z)
In+
p (λ,μ)f(z)
+αI
n–
p (λ,μ)f(z)
In
p(λ,μ)f(z)
–η
p
–η p
Reu(z)– pα|zu(z)| (p+λ)(|η+ (p–η)u(z)|)
. (.)
Now, by using the well-known [] estimates
zu(z) r –rRe
u(z) and Reu(z) –r +r
|z|=r<
in (.), we obtain
Re
( –α) I
n
p(λ,μ)f(z)
In+
p (λ,μ)f(z)
+αI
n–
p (λ,μ)f(z)
In
p(λ,μ)f(z)
–η
p
–η p
Reu(z) – pαr
(p+λ)(η( –r) + (p–η)( –r))
,
To show that the boundRis best possible, we consider the functionf∈Apdefined by
In
p(λ,μ)f(z)
In+
p (λ,μ)f(z)
=η p+
–η
p
+z
–z (z∈U). Noting that
( –α) I
n
p(λ,μ)f(z)
In+
p (λ,μ)f(z)
+αI
n–
p (λ,μ)f(z)
In
p(λ,μ)f(z)
–η p
=
–η p
+z –z
+ pαz
(p+λ)(η( –z) + (p–η)( –z))
=
forz= –R, we complete the proof of Theorem .
Remark Forn= –,λ= ,μ= –pandα= , Theorem yields the corresponding result
contained in [].
For a function f ∈Ap, the generalized Bernardi-Libera-Livingston integral operator
Fδ,p:Ap−→Apis defined by (cf.,e.g., [])
Fδ,p(f)(z) =
δ+p zδ
z
tδ–f(t)dt=
zp+
∞
k=
δ+p δ+p+kz
p+k
∗f(z)
=zpF(,δ+p;δ+p+ ;z)∗f(z) (δ> –p;z∈U). (.)
For convenience, we writeFδ,p(f)(z) =Fδ,p(z),z∈U. It readily follows from (.) that
f ∈Ap⇒Fδ,p∈Ap.
Theorem Letδbe a real number satisfying
(δ–λ)( –B) + (p+λ)( –A). (.)
(i)If f ∈Sn
p,λ,μ(A,B),then
In
p(λ,μ)Fδ,p(z)
In+
p (λ,μ)Fδ,p(z)≺
q(z) = p+λ
Q(z)–δ+λ
≺ +Az
+Bz (z∈U), (.) where
Q(z) = ⎧ ⎨ ⎩
t
δ+p–(+Btz +Bz)(p+
λ)(B–A)/Bdt (B= ),
t
δ+p–exp{(p+λ)A(t– )z}dt (B= )
and q is the best dominant of (.). Consequently, the operator Fδ,p maps the class
Sn
p,λ,μ(A,B)into itself.
(ii)If–B< andδmax{–(p+λ–)(–BA)+λ,(p+λ)(BB–A)–p– },then
where
τ= p+λ
(δ+p)
F
,(p+λ)(B–A)
B ;δ+p+ ; B B–
–
–δ+λ
.
The boundτ is best possible.
Proof From (.) and (.), it follows that
zIpn+(λ,μ)Fδ,p
(z) = (δ+p)Ipn+(λ,μ)f(z) –δIpn+(λ,μ)Fδ,p(z) (z∈U). (.)
We put
g(z) =z In+
p (λ,μ)Fδ,p(z)
zp
/(p+λ)
(.)
andr=sup{r:g(z)= , <|z|<r< }. Choosing the principal branch ofg, it follows that
gis a single-valued and is analytic inUr. Taking the logarithmic differentiation in (.) and using identity (.) forFδ,p, we deduce that the function
ϕ(z) = zg(z) g(z) =
In
p(λ,μ)Fδ,p(z)
In+
p (λ,μ)Fδ,p(z)
(.)
is analytic inUrandϕ() = . Using identity (.) in (.), we obtain
(δ+p) I
n+
p (λ,μ)f(z)
In+
p (λ,μ)Fδ,p(z)
= (p+λ) I
n
p(λ,μ)Fδ,p(z)
In+
p (λ,μ)Fδ,p(z)
+ (δ–p) (z∈Ur). (.)
Sincef∈Spn,λ,μ(A,B), it is clear thatIpn+(λ,μ)f(z)= in <|z|< . So, by (.), we get
Ipn+(λ,μ)Fδ,p(z)
In+
p (λ,μ)f(z)
= δ+p
(p+λ)ϕ(z) + (δ–λ) (z∈Ur). (.) Again, by taking the logarithmic differentiation in (.) followed by the use of identity (.) in the resulting equation, we get
In
p(λ,μ)f(z)
In+
p (λ,μ)f(z)
=ϕ(z) + zϕ(z)
(p+λ)ϕ(z) + (δ–λ)≺ +Az
+Bz (z∈Ur).
The proof of the remaining part is the same as that of [, Theorem ], and we choose to omit the details. The result is best possible asqis the best dominant of (.).
Remark Lettingn= –,λ= ,μ= –p,A= – (η/p) (η<p) andB= – in Theo-rem , we have the following implications [, Corollary . and Remark .]:
Fδ,p
S∗
p(η)
⊂S∗
p(σ) and Fδ,p
Kp(η)
⊂Kp(σ),
whereδmax{–η,p– η– }andσ = (δ+p)(F(, (p–η);δ+p+ ; /))––δ. The
4 Properties involving the operatorIn p(
λ
,μ
)Theorem If f ∈Apsatisfies
( –α)I
n+
p (λ,μ)f(z)
zp +α
In
p(λ,μ)f(z)
zp ≺
+Az
+Bz (z∈U), (.) then
Re In+
p (λ,μ)f(z)
zp
> (z∈U), (.)
where
= ⎧ ⎨ ⎩
A B+ ( –
A
B)( –B)–F(, ; p+λ
α + ;
B
B–) (B= ),
–(pp++λλ+)Aα (B= ).
The result is best possible.
Proof Setting
ϕ(z) = I
n+
p (λ,μ)f(z)
zp (z∈U), (.)
we note thatϕis of the form (.) and is analytic inU. On differentiating (.) and using identity (.) in the resulting equation, we deduce that
( –α)I
n+
p (λ,μ)f(z)
zp +α
In
p(λ,μ)f(z)
zp =ϕ(z) +
α
p+λzϕ(z)≺ +Az
+Bz (z∈U). (.) The proof of the remaining part of the theorem follows by using Lemma and the tech-niques that proved Theorem in [].
With a view to stating a well-known result, we denote byP(γ) the class of functionsϕ of the form (.) which are analytic inUand satisfy the inequality
Reϕ(z)>γ (γ < ;z∈U).
It is known [] that ifϕj∈P(γj) (γj< ;j= , ), then
(ϕ∗ϕ)∈P(γ), (.)
whereγ= – ( –γ)( –γ). The boundγis best possible.
Theorem If the functions In
p(λ,μ)fj/zp∈P(Aj,Bj) (–Bj<Aj,fj∈Ap;j= , ),then
the functionhdefined inUby
h(z) =Ipn+(λ,μ)(f∗f)(z) (.)
satisfies
Re In
p(λ,μ)h(z)
In+
p (λ,μ)h(z)
provided
(A–B)(A–B)
( –B)( –B)
< (p+λ) +
[(F(, ;p+λ+ ; /) – )+ (p+λ)]
. (.)
Proof We have
In p(λ,μ)fj
zp ∈P(γj)
γj=
–Aj
–Bj
;j= ,
.
Hence, by using (.), we deduce that
Re In
p(λ,μ)h(z)
zp +
z p+λ
In
p(λ,μ)h(z)
zp
=Re In
p(λ,μ)f(z)
zp ∗
Ipn(λ,μ)f(z)
zp
> – (A–B)(A–B) ( –B)( –B)
(z∈U), (.)
which, in view of Lemma for
κ=p+λ, A= – + (A–B)(A–B) ( –B)( –B)
and B= –,
yields
Re In
p(λ,μ)h(z)
zp
> +(A–B)(A–B) ( –B)( –B)
F
, ;p+λ+ ;
–
(z∈U). (.)
From (.), by using Theorem for
α= , A= – – (A–B)(A–B) ( –B)( –B)
F
, ;p+λ+ ;
–
and B= –,
we deduce that
Reθ(z)> – (A–B)(A–B) ( –B)( –B)
F
, ;p+λ+ ;
–
(z∈U), (.)
whereθ(z) =In
p(λ,μ)h(z)/zp. If we put
ϕ(z) = I
n
p(λ,μ)h(z)
In+
p (λ,μ)h(z)
(z∈U),
thenϕis of the form (.) analytic inU, and a simple computation shows that In
p(λ,μ)h(z)
zp +
z p+λ
In
p(λ,μ)h(z)
zp
=θ(z)ϕ(z)+ p+λzϕ(z)
=ϕ(z),zϕ(z);z, (.)
where(u,v;z) =θ(z)(u+ (v/(p+λ))). Thus, by using (.) in (.), we conclude that
Reϕ(z),zϕ(z);z> – (A–B)(A–B) ( –B)( –B)
Now, for all realx,y–( +x)/, we have
Re(ix,y;z)=
y p+λ–x
Reθ(z)
–
(p+λ)
+(p+λ) + xReθ(z)
(p+λ)Re
θ(z) – (A–B)(A–B) ( –B)( –B)
(z∈U)
by (.) and (.). Hence, by making use of Lemma , we getRe(ϕ(z)) > inU. This completes the proof of Theorem .
Theorem If the functions In
p(λ,μ)fj/zp∈P(Aj,Bj) (–Bj<Aj,fj∈Ap;j= , , ),
then the function H defined inUby
H(z) =Ipn+(λ,μ)(f∗f∗f)(z)
satisfies
Re In
p (λ,μ)H(z)
In+
p (λ,μ)H(z)
> (z∈U),
provided
(A–B)(A–B)(A–B)
( –B)( –B)( –B)
< (p+λ) +
[(F(, ;p+λ+ ; /) – )+ (p+λ)]
.
Proof From the definition of the functionH, it is easily seen that
Re In
p (λ,μ)H(z)
zp +
z p+λ
In
p (λ,μ)H(z)
zp
=Re In
p(λ,μ)f(z)
zp ∗
In
p(λ,μ)f(z)
zp ∗
In
p(λ,μ)f(z)
zp
> – (A–B)(A–B)(A–B) ( –B)( –B)( –B)
(z∈U)
and the proof of the theorem is completed similarly to Theorem .
Theorem Let fj∈Ap(j= , ).If the functionhdefined inUby(.)satisfies
Re In
p(λ,μ)h(z)
zp
> – (p+λ) +
[(F(, ;p+λ+ ; /) – )+ (p+λ)]
(z∈U),
then
Re In
p(λ,μ)G(z)
In+
p (λ,μ)G(z)
where
G(z) = p+λ zλ
z
tλ–
h(t)dt (z∈U).
Proof Using the fact that
Re In
p(λ,μ)h(z)
zp
=Re
In
p(λ,μ)G(z)
zp +
z p+λ
In
p(λ,μ)G(z)
zp
> – (p+λ) +
[(F(, ;p+λ+ ; /) – )+ (p+λ)]
(z∈U)
and by following the same lines of proof as in Theorem , we get the required result.
Remark Puttingn= –,λ= andμ= in Theorems , and , respectively, we obtain
the corresponding results contained in [].
Theorem Letδ> and <γ ( +δ(p+λ))/ + δ(p+λ) + (δ(p+λ)).If f ∈A p
satisfies
In–
p (λ,μ)f(z)
zp
In
p(λ,μ)f(z)
zp
δ–
≺ +γz (z∈U), (.)
then f ∈Sn–
p,λ,μ( – κ, –),where
κ= ⎧ ⎨ ⎩
+δ(p+λ)
+δ(p+λ)(+γ) ( <γ +δ(p+λ) +δ(p+λ)),
Mp(λ,δ,γ) (++δδ((pp++λλ)) γ √ +δ(p+λ) +δ(p+λ)+(δ(p+λ))),
(.)
and
Mp(λ,δ,γ) =
( +δ(p+λ))– [ + δ(p+λ) + (δ(p+λ))]γ
[( +δ(p+λ))– (δ(p+λ))γ] .
Further,for <γ ( +δ(p+λ))/( + δ(p+λ)),
Ipn–(λ,μ)f(z)
In
p(λ,μ)f(z)
– < ( + δ(p+λ))γ
+δ(p+λ)( –γ) (z∈U). (.) The bound given by(.)and the estimate in(.)are best possible.
Proof Setting
(z) = In
p(λ,μ)f(z)
zp
δ
(z∈U) (.)
and choosing the principal branch in (.), we note thatis analytic inUwith() = . A simple computation shows that (.) is equivalent to
(z) + z(z)
Now, by applying Lemma (withκ=δ(p+λ),A=γ andB= –), we get
(z)≺ +γz
γ=
δ(p+λ)γ +δ(p+λ);z∈U
.
We further observe that
+ δ(p+λ)
z(z) (z) =
Ipn–(λ,μ)f(z) In
p(λ,μ)f(z)
(z∈U).
Hence, assertion (.) follows by using part (i) of Lemma . If we put
In–
p (λ,μ)f(z)
In
p(λ,μ)f(z)
= +ω(z) (ω∈B;z∈U),
then we obtain (.) from part (ii) of Lemma .
To show that the estimates are best possible, we consider the functionf ∈Apdefined in
Uby In
p(λ,μ)f(z)
zp
δ
= +δ(p+λ)γ
tδ(p+λ)–ω(tz)dt (δ> ,ω∈B;z∈U).
From this, we obtain
In–
p (λ,μ)f(z)
In
p(λ,μ)f(z)
= +γ ω(z)
+δ(p+λ)γtδ(p+λ)–ω(tz)dt (z∈U),
and the sharpness follows from Lemma (forδ =δ(p+λ)).
Puttingn=λ= ,μ= –pandδ= in Theorem , we get the following.
Corollary If <γ (p+ )/ + p+ pand f ∈A
psatisfies
pzf (pz–) –
<γ (z∈U),
then
Re
zf (z) f(z)
>
⎧ ⎨ ⎩
p(p+)(–γ)
+p(+γ) ( <γ p+ p+), p(p+)–p(+p+p)γ
((p+)–(pγ)) (
p+ p+γ
p+
√
+p+p).
Further,for <γ (p+ )/(p+ ),
zff((zz))– < p(p+ )γ
Theorem Ifγ> and f ∈Apsatisfies
( –α)I
n
p(λ,μ)f(z)
zp +α
In–
p (λ,μ)f(z)
zp ≺ +γz (α> ;z∈U), (.)
then f ∈Sn–
p,λ,μ( – ϑ, –),where
ϑ= ⎧ ⎨ ⎩
α((p+λ)(+γ)+α–γ)–(p+λ)γ
α((p+λ)(+γ)+α) ( <γ p+λ+α
(p+λ)+α),
Np(λ,α,γ)–(–α)
α (
p+λ+α
(p+λ)+αγ
p+λ+α
√
(p+λ)+(p+λ+α))
(.)
and
Np(λ,α,γ) =
(p+λ+α)– ((p+λ+α)+ (p+λ))γ
((p+λ+α)– (p+λ)γ) .
For <γ (p+λ+α)/((p+λ) +α),we have
Ipn–(λ,μ)f(z)
In
p(λ,μ)f(z)
– < ((p+λ) +α)γ
(p+λ)( –γ) +α (z∈U). (.) Further,f ∈Spn,λ,μ( – κ, –),whereκis obtained fromκ(given in(.))forδ= and
upon replacingγ by(p+λ)γ/(p+λ+α).Moreover,for <γ ((p+λ+α)(p+λ+ ))/ [(p+λ)( + (p+λ))],
Ipn(λ,μ)f(z)
In+
p (λ,μ)f(z)
– < (p+λ)( + (p+λ))γ
(p+λ+α) + (p+λ)(α+ (p+λ)( –γ)) (z∈U). (.) The estimates are best possible.
Proof Sincef ∈Apsatisfies (.), by Theorem (forA=γ andB= –) we obtain
In
p(λ,μ)f(z)
zp ≺ +γz
γ=
(p+λ)γ p+λ+α;z∈U
. (.)
Again, on writing (.) in the form
In
p(λ,μ)f(z)
zp
–α+αI
n–
p (λ,μ)f(z)
In
p(λ,μ)f(z)
≺ +γz (z∈U)
and using part (ii) of Lemma , we deduce that
Re
–α+αI
n–
p (λ,μ)f(z)
In
p(λ,μ)f(z)
⎧ ⎨ ⎩
(p+λ+α)(–γ)
(p+λ)(+γ)+α ( <γ
p+λ+α
(p+λ)+α),
which implies assertion (.). By using part (ii) of Lemma with
ω(z) =α In–
p (λ,μ)f(z)
In
p(λ,μ)f(z)
–
,
we obtain (.). Thatf ∈Sn–
p,λ,μ( – κ, –) and (.) now follow from Theorem and
(.).
To show the sharpness of the estimates, we consider the functionf defined inUby In
p(λ,μ)f(z)
zp = +
(p+λ)γ α
tp+αλ–ω(tz)dt (ω∈B;z∈U).
Hence, by using identity (.), we get
–α+αI
n–
p (λ,μ)f(z)
In
p(λ,μ)f(z)
= +γ ω(z) +(p+αλ)γtp+αλ–ω(tz)dt
(z∈U),
and the sharpness follows from Lemma . The fact thatf ∈Sn
p,λ,μ( – κ, –) is sharp
fol-lows from (.) and the sharpness of Theorem .
Puttingn= –,λ= andμ= –pin Theorem , we have the following.
Corollary If f ∈Apsatisfies
–α+α p
f (z) pzp–+α
f (z)
pzp– ≺ +γz (γ > ,α> ;z∈U),
then
Re
+zf
(z)
f (z)
> ⎧ ⎨ ⎩
pα(p+α+(p–)γ)–pγ
α(p(+γ)+α) ( <γ p+α
p+α),
p(p+α)–p((p+α)+p)γ
α((p+α)–pγ) –p(–αα) (
p+α
p+αγ
p+α
√
p+(p+α))
and for <γ(p+α)/(p+α),
+zf
(z)
f (z) –p
< p(p+α)γ
α(p( –γ) +α) (z∈U). The result is sharp.
Remark Forp= in Corollary and Corollary , we get the corresponding results
ob-tained in [].
Theorem If f ∈Apsatisfies
Ipn(λ,μ)f(z)
In+
p (λ,μ)f(z)
–
γ
Ipn–(λ,μ)f(z)
In
p(λ,μ)f(z)
–
β
<
A–B +|B|
γ+β
+ (p+λ)( +|A|)
β
(z∈U), (.)
Proof If we set
Ipn(λ,μ)f(z) In+
p (λ,μ)f(z)
= +Aω(z)
+Bω(z) (z∈U), (.) thenωis analytic inU. Differentiating (.) logarithmically and using identity (.) in the resulting equation, we get
In–
p (λ,μ)f(z)
In
p(λ,μ)f(z)
= +Aω(z) +Bω(z)+
(A–B)zω(z)
(p+λ)( +Aω(z))( +Bω(z)) (z∈U). Now, we have
Ipn(λ,μ)f(z)
In+
p (λ,μ)f(z)
–
γ
Ipn–(λ,μ)f(z)
In
p(λ,μ)f(z)
–
β
= (A–B)γ+β ω(z) +Bω(z)
γ+β
× +
p+λ zω(z)
ω(z) +Aω(z)
β (z∈U). (.)
We claim that |ω(z)|< for z∈U. Otherwise, there exists a point z∈ Usuch that
max|z||z||ω(z)|=|ω(z)|= . By using Jack’s lemma [], we writeω(z) =eiθ, <θπ
andzω(z) =mω(z),m. Thus, from (.), it follows that
Ipn(λ,μ)f(z)
In+
p (λ,μ)f(z)
–
γ
Ipn–(λ,μ)f(z)
In
p(λ,μ)f(z)
–
β
A–B +|B|
γ+β
+ m p+λ
+Aeiθ
β
A–B +|B|
γ+β
+ (p+λ)( +|A|)
β
.
This contradicts the hypothesis (.) and hence|ω(z)|< forz∈U. This proves the
the-orem.
TakingA= –pρ,B= –,μ= –p,λ= ,n= – andγ = –βin Theorem , we get the following interesting criterion for starlikeness for multivalent functions, which improves the corresponding work in [] forp= .
Corollary Letβandρ<p.If f ∈Apsatisfies
zff((zz))–p
–β
+zf
(z)
f (z) –p
β<ξ(p,ρ,β) = ⎧ ⎨ ⎩
(p–ρ)( +(p–ρ))β (ρp ),
(p–ρ)( + ρ)
β (p
ρ<p)
for all z∈U,then f ∈Sp∗(ρ).
Corollary Letβandρ<p.If f ∈Apsatisfies
+zf
(z)
f (z)
–p
–β
+z
f (z) + zf (z)
zf (z) +f (z)
–p
β
<ξ(p,ρ,β) (z∈U),
whereξ(p,ρ,β)is defined as in Corollary,then f ∈Kp(ρ).
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 Mathematics, Utkal University, Vani Vihar, Bhubaneswar, 751004, India.2Department of Applied
Mathematics, Pukyong National University, Busan, Korea.3Department of Mathematics, Bhadrak Institute of Engineering
and Technology, Bhadrak, 756 113, India.
Acknowledgements
This work was supported by a Research Grant of Pukyong National University (2013 year) and the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (No. 2011-0007037).
Received: 3 January 2013 Accepted: 1 August 2013 Published: 21 September 2013 References
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doi:10.1186/1029-242X-2013-441
Cite this article as:Patel et al.:On certain subclasses of multivalent functions defined by multiplier transformations.
