R E S E A R C H
Open Access
Subordination properties for a general class
of integral operators involving
meromorphically multivalent functions
Nak Eun Cho
1*and Rekha Srivastava
2*Correspondence:
necho@pknu.ac.kr 1Department of Applied
Mathematics, Pukyong National University, Busan, 608-737, Korea Full list of author information is available at the end of the article
Abstract
The purpose of the present paper is to investigate some subordination- and
superordination-preserving properties of certain nonlinear integral operators defined on the space of meromorphicallyp-valent functions in the punctured open unit disk. The sandwich-type theorems associated with these integral operators are established. Relevant connections of the various results presented here with those involving relatively simpler nonlinear integral operators are also indicated.
MSC: Primary 30C45; secondary 30C80
Keywords: differential subordination; differential superordination; meromorphic functions; integral operators; convex functions; close-to-convex functions; subordination (or Löwner) chain
1 Introduction, definitions and preliminaries
LetH=H(U) denote the class of analytic functions in the open unit disk
U=z:z∈Cand|z|< .
Fora∈Candn∈N={, , , . . .}, let
H[a,n] =f∈H:f(z) =a+anzn+an+zn++· · ·
.
Letf andFbe members ofH. The functionf is said to be subordinate toF, orFis said to be superordinate tof, if there exists a functionwanalytic inU, with
w() = and w(z)< (z∈U), such that
f(z) =Fw(z) (z∈U). In such a case, we write
f ≺F or f(z)≺F(z) (z∈U).
Furthermore, if the functionFis univalent inU, then we have (cf.[, ] and [])
f ≺F ⇐⇒ f() =F() and f(U)⊂F(U).
Definition (Miller and Mocanu []) Let
φ:C→C,
and lethbe univalent inU. Ifpis analytic inUand satisfies the differential subordination
φp(z),zp(z)≺h(z) (z∈U), (.)
thenpis called a solution of the differential subordination. The univalent functionqis called a dominant of the solutions of the differential subordination or, more simply, a dom-inant if
p≺q (z∈U)
for allpsatisfying (.). A dominantq˜that satisfies the following condition:
˜
q≺q (z∈U)
for all dominantsqof (.) is said to be the best dominant.
Definition (Miller and Mocanu []) Let
ϕ:C→C,
and lethbe analytic inU. Ifpandϕ(p(z),zp(z)) are univalent inUand satisfy the differ-ential superordination
h(z)≺ϕp(z),zp(z) (z∈U), (.)
thenpis called a solution of the differential superordination. An analytic functionqis called a subordinant of the solutions of the differential superordination or, more simply, a subordinant if
q≺p (z∈U)
for allpsatisfying (.). A univalent subordinantq˜that satisfies the following condition:
q≺ ˜q (z∈U)
Definition (Miller and Mocanu []) We denote byQthe class of functionsf that are analytic and injective onU\E(f), where
E(f) =
ζ:ζ∈∂Uand lim
z→ζf(z) =∞
,
and are such that
f(ζ)= ζ∈∂U\E(f). We also denote the classDby
D:=ϕ∈H[, ] :ϕ() = andϕ(z)= (z∈U).
Letpdenote the class of functions of the form
f(z) =
zp + a
zp–+
a
zp– +· · ·+an+p–z
n+· · · (p∈N)
which are analytic in the punctured open unit diskD=U\ {}. Let*andkbe the
sub-classes ofconsisting of all functions which are, respectively, meromorphically starlike and meromorphically convex inD(see, for details, [, ]).
For a functionf ∈p, we introduce the following general integral operatorIαφ,,βϕ,γ,δ de-fined by
Iαφ,,βϕ,γ,δ(f)(z) := γ –pβ
zγφ(z) z
tδ–fα(t)ϕ(t)dt /β
f ∈p;α,β,γ,δ∈C;β∈C\ {};δ–pα=γ –pβ;R{γ –pβ}> ;φ,ϕ∈D
. (.)
Several members of the family of integral operatorsIαφ,,βϕ,γ,δ(f) defined by () have been extensively studied by many authors (see, for example, [–]; see also [] and []) with suitable restrictions on the parametersα,β,γ andδ, and forf belonging to some favored subclasses of meromorphic functions. In particular, Bajpai [] showed that the integral operatorI,,,, (f) belongs to the classes*and
k, wheneverf belongs to the classes*
andk, respectively.
Making use of the principle of subordination between analytic functions, Milleret al.
[] and, more recently, Owa and Srivastava [] obtained some interesting subordination-preserving properties for certain integral operators. Moreover, Miller and Mocanu [] considered differential superordinations as the dual concept of differential subordinations (see also []). It should be remarked that in recent years several authors obtained many interesting results involving various integral operators associated with differential subor-dination and superorsubor-dination (for example, see [, –]). In the present paper, we obtain the subordination- and superordination-preserving properties of the general integral op-eratorIαφ,,βϕ,γ,δdefined by () with the sandwich-type theorem.
The following lemmas will be required in our present investigation.
Lemma (Miller and Mocanu []) Suppose that the function
satisfies the following condition:
RH(is,t)
for all real s and for all t with
t–n( +s )
(n∈N).
If the function
p(z) = +pnzn+· · ·
is analytic inUand
RHp(z),zp(z)> (z∈U),
then
Rp(z)> (z∈U).
Lemma (Miller and Mocanu []) Let
β,γ ∈C (β= ) and h∈H(U) h() =c.
If
Rβh(z) +γ> (z∈U),
then the solution of the differential equation
q(z) + zq
(z)
βq(z) +γ =h(z)
z∈U;q() =c
is analytic inUand satisfies the following inequality: Rβq(z) +γ> (z∈U).
Lemma (Miller and Mocanu []) Let
p∈Q p() =a,
and let
q(z) =a+anzn+· · ·
If the function q is not subordinate top,then there exist points
z=reiθ∈U and ζ∈∂U\E(f),
for which
q(Ur)⊂p(U), q(z) =p(ζ) and zq(z) =mζp(ζ) (mn).
Let
N:=N(c) =|c| √
+ R(c) +I(c) R(c)
c∈C;R(c) > .
IfRis the univalent function defined inUby
R(z) = Nz
–z (z∈U),
then theopen door function(see []) is defined by
Rc(z) :=R z+b
+bz b=R
–(c);z∈U. (.)
Remark The functionRcdefined by (.) is univalent inU, whereRc() =c, andRc(U) = R(U) is the complex plane with slits along the half-lines given by
R(w) = and I(w)N.
Lemma (Totoi []) Letα,β,γ,δ∈Cwith
β= , δ–pα=γ–pβ, R{γ –pβ}> and φ,ϕ∈D.
If f ∈αφ,,βϕ,γ,δ,where
αφ,,βϕ,γ,δ:=
f∈p:α zf(z)
f(z) +
zϕ(z)
ϕ(z) +δ≺Rδ–pα(z)
(.)
and Rδ–pα(z)is defined by(.)with c=δ–pα,then
Iαφ,,βϕ,γ,δ(f)(z)∈p, zpIαφ,,βϕ,γ,δ(f)(z)= (z∈U)
and
R
βz(I
φ,ϕ
α,β,γ,δ(f)(z))
Iαφ,,βϕ,γ,δ(f)(z) +
zφ(z)
φ(z) +γ
> (z∈U),
A functionL(z,t) defined onU×[,∞) is the subordination chain (or Löwner chain) if
L(·,t) is analytic and univalent inUfor allt∈[,∞),L(z,·) is continuously differentiable on [,∞) for allz∈Uand
L(z,s)≺L(z,t) (z∈U; s<t). (.) Lemma (Miller and Mocanu []) Let
q∈H[a, ] and μ:C→C.
Also set
μq(z),zq(z)=:h(z) (z∈U).
If
L(z,t) =μq(z),tzq(z)
is a subordination chain and
p∈H[a, ]∩Q,
then the following subordination condition:
h(z)≺μp(z),zp(z) (z∈U) (.)
implies that
q(z)≺p(z) (z∈U).
Furthermore,if
μq(z),zq(z)=h(z)
has a univalent solution q∈Q,then q is the best subordinant.
Lemma (Pommerenke []) The function
L(z,t) =a(t)z+· · ·
with
a(t)= and lim
t→∞a(t)=∞.
Suppose that L(·,t)is analytic inUfor all tand that L(z,·)is continuously differentiable on[,∞)for all z∈U.If the function L(z,t)satisfies the following inequalities:
R
z∂L(z,t) ∂z
∂L(z,t) ∂t
and
L(z,t)Ka(t) |z|<r< ; t<∞
(.)
for some positive constants Kand r,then L(z,t)is a subordination chain.
2 Main results and their corollaries and consequences
We begin by proving a general subordination property involving the integral operator
Iαφ,,βϕ,γ,δdefined by (), which is contained in Theorem below.
Theorem Let f,g∈φα,,βϕ,γ,δ,whereαφ,,βϕ,γ,δis defined by(.).Suppose also that
R
+zν
g(z) νg(z)
> –δ z∈U;νg(z) :=z
zpg(z)αϕ(z), (.)
where
ρ= +|γ –pβ– |
–| – (γ–pβ– )|
R{γ–pβ– }
R{γ–pβ– }> . (.)
Then the subordination relation
νf(z)≺νg(z) (z∈U) (.)
implies that
zzpIαφ,,βϕ,γ,δ(f)(z) β
φ(z)≺zzpIαφ,,βϕ,γ,δ(g)(z) β
φ(z) (z∈U), (.)
where Iαφ,,βϕ,γ,δis the integral operator defined by().Moreover,the function
zzpIαφ,,βϕ,γ,δ(g)(z) β
φ(z)
is the best dominant.
Proof Let us define the functionsFandG, respectively, by
F(z) :=zzpIαφ,,βϕ,γ,δ(f)(z) β
φ(z) and G(z) :=zzpIαφ,,βϕ,γ,δ(g)(z) β
φ(z). (.)
We first show that if the functionqis defined by
q(z) := +zG
(z)
G(z) (z∈U), (.)
then
From the definition of (), we obtain
Now, by a simple calculation with (.), we obtain the following relationship:
+zν
and by using Lemma , we conclude that the differential equation (.) has a solution
q∈H(U) with We now proceed to show that
where
Eρ(s) :=R{γ–pβ– }– ρs– ρI{γ –pβ– }s
– ρ|γ–pβ– |+R{γ–pβ– }. (.)
Forρgiven by (.), we note that the coefficient ofsin the quadratic expressionEρ(s) given by (.) is positive or equal to zero and alsoEρ(s) is a perfect square. Hence from (.), we see that (.) holds true. Thus, by using Lemma , we conclude that
Rq(z)> (z∈U).
That is, the functionG(z) defined by (.) is convex inU.
We next prove that the subordination condition (.) implies that
F(z)≺G(z) (z∈U) (.)
for the functionsFandGdefined by (.). Without loss of generality, we can assume that
Gis analytic and univalent onUand that
F(ζ)= |ζ|= .
We now consider the functionL(z,t) defined by
L(z,t) :=γ–pβ–
γ –pβ G(z) +
+t γ –pβzG
(z) (z∈U; ≤t<∞).
We note that
∂L(z,t)
∂z
z=z
=G() + t
γ –pβ
= (z∈U; ≤t<∞)
and
R
z∂L(z,t)/∂z ∂L(z,t)/∂t
=R
γ–pβ– + ( +t) +zG
(z)
G(z)
> (z∈U).
Furthermore, sinceGis convex, by using the well-known growth and distortionsharp
inequalities for convex functions (see []), we can prove that the second condition of Lemma is satisfied. Therefore, by virtue of Lemma ,L(z,t) is a subordination chain. We observe from the definition of a subordination chain that
νg(z) =
γ –pβ–
γ–pβ G(z) +
γ–pβzG
(z) =L(z, )
and
This implies that
L(ζ,t) /∈L(U, ) =νg(U) (ζ∈∂U; ≤t<∞).
We now suppose thatFis not subordinate toG. Then, in view of Lemma , there exist pointsz∈Uandζ∈∂Usuch that
F(z) =G(ζ) and zF(z) = ( +t)ζG(ζ) (≤t<∞).
Hence we have
L(ζ,t) =
γ–pβ–
γ –pβ G(ζ) +
+t γ–pβζG
(ζ
)
=γ–pβ–
γ –pβ F(ζ) +
γ–pβzF (z
)
=νf(z)∈νg(U),
by virtue of the subordination condition (.). This contradicts the above observation that
L(ζ,t) /∈νg(U).
Therefore, the subordination condition (.) must imply the subordination given by (.). ConsideringF=G, we see that the functionGis the best dominant. This evidently
com-pletes the proof of Theorem .
We next prove a solution to a dual problem of Theorem in the sense that the subordi-nations are replaced by superordisubordi-nations.
Theorem Let f,g∈αφ,,βϕ,γ,δ,whereαφ,,βϕ,γ,δis defined by(.).Suppose also that
R
+zν
g(z) νg(z)
> –ρ z∈U;νg(z) :=z
zpg(z)αϕ(z),
whereρis given by(.)andνf is univalent inU,and zzpIαφ,,βϕ,γ,δ(f)(z)βφ(z)∈Q,
where Iαφ,,βϕ,γ,δis the integral operator defined by().Then the superordination relation
νg(z)≺νf(z) (z∈U) (.)
implies that
zzpIαφ,,βϕ,γ,δ(g)(z)βφ(z)≺zzpIαφ,,βϕ,γ,δ(f)(z)βφ(z) (z∈U).
Moreover,the function
zzpIαφ,,βϕ,γ,δ(g)(z) β
φ(z)
Proof Let the functionsFandGbe given by (.). We first note from (.) and (.) that
νg(z) =
γ –pβ–
γ–pβ G(z) +
γ–pβzG (z)
=:μG(z),zG(z). (.)
After a simple calculation, equation (.) yields the following relationship:
+zν
g(z)
νg(z) =q(z) +
zq(z)
q(z) +γ–pβ– ,
where the functionqis given in (.). Then, by using the same method as in the proof of Theorem , we can prove that
Rq(z)> (z∈U),
that is,Gdefined by (.) is convex (univalent) inU.
We next prove that the superordination condition (.) implies that
G(z)≺F(z) (z∈U). (.)
For this purpose, we consider the functionL(z,t) defined by
L(z,t) :=γ–pβ–
γ –pβ G(z) + t γ –pβzG
(z) (z∈U; ≤t<∞).
SinceGis convex andR{γ –pβ– }> , we can prove easily thatL(z,t) is a subordina-tion chain as in the proof of Theorem . Therefore, according to Lemma , we conclude that the superordination condition (.) must imply the superordination given by (.). Furthermore, since the differential equation (.) has the univalent solutionG, it is the best subordinant of the given differential superordination. Hence we complete the proof
of Theorem .
If we combine Theorem and Theorem , then we obtain the following sandwich-type theorem.
Theorem Let f,gk∈αφ,,βϕ,γ,δ(k= , ),where φ,ϕ
α,β,γ,δis defined by(.).Suppose also that
R
+zν
gk(z) νgk (z)
> –ρ z∈U;νgk(z) :=z
zpgk(z)
α
ϕ(z);k= , , (.)
whereρis given by(.)and the functionνf is univalent inU,and zzpIαφ,,βϕ,γ,δ(f)(z)βφ(z)∈Q,
where Iαφ,,βϕ,γ,δis the integral operator defined by().Then the subordination relation
implies that
are the best subordinant and the best dominant,respectively.
The assumption of Theorem , that is, the functions
zzpf(z)αϕ(z) and zzpIαφ,,βϕ,γ,δ(f)(z)βφ(z)
need to be univalent inU, will be replaced by another set of conditions in the following result.
Corollary Let f,gk∈αφ,,βϕ,γ,δ(k= , ),where φ,ϕ
α,β,γ,δis defined by(.).Suppose also that
the condition(.)is satisfied and
R
whereρis given by(.).Then the subordination relation
νg(z)≺νf(z)≺νg(z) (z∈U)
are the best subordinant and the best dominant,respectively.
Proof In order to prove Corollary , we have to show that the condition (.) implies the univalence ofνf(z) and
F(z) :=zzpIαφ,,βϕ,γ,δ(f)(z)βφ(z). (.)
by using the same techniques as in the proof of Theorem , we can prove the convexity (univalence) ofFand so the details may be omitted. Therefore, by applying Theorem ,
we obtain Corollary .
are the best subordinant and the best dominant,respectively.
where Iαφ,,βϕ,pβ++i,δis the integral operator defined by()withγ =pβ+ +i.Then the
subor-are the best subordinant and the best dominant,respectively.
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
Both authors jointly worked on the results and they read and approved the final manuscript.
Author details
1Department of Applied Mathematics, Pukyong National University, Busan, 608-737, Korea.2Department of Mathematics
and Statistics, University of Victoria, Victoria, BC V8W 3R4, Canada.
Acknowledgements
Dedicated to Professor Hari M Srivastava.
This research was supported by the Basic Science Research Program through the National Research Foundation of the Republic of Korea (NRF) funded by the Ministry of Education, Science and Technology (No. 2012-0002619).
Received: 14 January 2013 Accepted: 14 March 2013 Published: 5 April 2013 References
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doi:10.1186/1687-1847-2013-93
