• No results found

Classes of functions associated with bounded Mocanu variation

N/A
N/A
Protected

Academic year: 2020

Share "Classes of functions associated with bounded Mocanu variation"

Copied!
20
0
0

Loading.... (view fulltext now)

Full text

(1)

R E S E A R C H

Open Access

Classes of functions associated with bounded

Mocanu variation

Jacek Dziok

*

*Correspondence: jdziok@ur.edu.pl

Institute of Mathematics, University of Rzeszów, Rzeszów, 35-310, Poland

Abstract

In the paper we introduce the class of linear combinations of functions which are subordinated to a convex function. Some relationships between this class and the class of real-valued functions with bounded variation on [0, 2

π

] are obtained. Next, we define classes of functions associated with bounded Mocanu variation. By using the properties of multivalent prestarlike functions, we obtain various inclusion relationships between defined classes of functions. Some applications of the main results are also considered.

MSC: 30C45; 30C50; 30C55

Keywords: analytic functions; bounded variation; prestarlike function; Mocanu functions, linear operator; Hadamard product; subordination

1 Introduction

LetAdenote the class of functions which areanalyticinU:={z∈C:|z|< },A:={f

A:f() = }, and letAp(p∈N:={, , . . .}) denote the class of functionsfAof the form

f(z) =zp+ ∞

n=p+

anzn (zU). ()

Mocanu [] introduced the classM(α) of functionsfAsuch thatf(z)f

(z)

z =  (zU) and

Re

( –α)zf (z)

f(z) +α

 +zf (z)

f(z)

>  (zU). ()

In particular,Sc:=M(),S:=M() are the well-known classes ofconvex functionsand

starlike functions, respectively.

It is clear thatfScif and only iff is univalent inUandf(U) is a convex domain. Also, byScwe denote the class of functionsfAwhich are univalent inUandf(U) is a convex domain.

We say that a functionfAisclose-to-convexif there existsgScsuch that

Ref (z)

g(z)>  (zU).

We denote byCCthe class of all close-to-convex functions.

(2)

We say thatfAissubordinatetoFA, and we writef(z)≺F(z) (or simplyfF), if there exists a function

ω:=ωA:ω(z)≤ |z|(zU) ,

such thatf(z) =F(ω(z)) (zU). In particular, ifFis univalent inU, we have the following equivalence

f(z)≺F(z) ⇐⇒ f() =F()∧f(U)⊂F(U).

LethSc,μ≥, and let us define

(h) :=

μq+ ( –μ)q:q,q≺h .

We note that the classP:=K+z

–z

is the well-known class of Caratheodory functions. Now we define generalizations of the classesM(α) andCCassociated with functions of bounded variation.

Letp∈N,δ∈R,φ,ϕ,ξAp,= (φ,ϕ). We denote byMδμ(,ξ,h) the class of functions

fApsuch that

( –δ)(ξφ)∗f (ξϕ)∗f +δ

φf

ϕf(h),

where∗denotes the Hadamard product (or convolution). Moreover, let us define

μ(,h) :=M δ

μ(,ξ,h), Mδμ(ϕ,h) :=M δ μ

(ϕ,ϕ),h

,

(,h) :=Mμ(,z,h), (ϕ,h) :=

(z)/p,ϕ,h,

S

p(ϕ,α) :=W

ϕ, + ( – α/p)z/( –z),

where

ξ(z) =zp+ ∞

n=p+

p nz

n, ϕ

(z) =

z

(z), ϕ (z) =

z

(z) (zU).

Letμ= (μ,μ),μ,μ≥,h= (h,h)∈Sc×Sc. We say that a functionfApbelongs to the classCMδ

μ(,ξ,h) if there existsg(ϕ,h) such that

( –δ)(ξφ)∗f (ξϕ)∗g +δ

φf

ϕg(h).

Moreover, let us defineCWμ(,h) :=CMμ(,zp,h).

These general classes of functions reduce to well-known classes by judicious choices of the parameters; see, for example, [–]. In particular, the class

μ(ϕ,h) contains the functionsfApsuch that

(f) :=

δ p

 +z(ϕf) (z)

(ϕf)(z)

+ ( –δ)z(ϕf) (z)

(3)

It is related to the class of functions with bounded Mocanu variation defined by Coonce and Ziegler [] and intensively investigated by Nooret al.[–]. The classes

S

p(α) :=Sp

zp  –z,

α p

, Spc:=Sp

zp[p+ ( –p)z]

p( –z) , 

are the classes ofmultivalent starlikefunctions of orderα andmultivalent convex func-tions, respectively. Choosing parameters

φ(z) = z

 –z, φ(z) = z

( –z), h(z) =  +z

 –z (zU)

we obtain the well-known class (φ,h) of functions of bounded boundary rotation (see, for example, [, , , , ]). Moreover, it is clear that

S=S

(), Sc=Sc(), CC=CW(,)

(φ,φ), (h,h)

.

The main object of this paper is to investigate convolution properties related to the prestarlike functions and various inclusion relationships between defined classes of func-tions. Some characterizations of the class(h) are also given.

2 Functions with bounded variation

ByMk (k≥) we denote the class of real-valued functionsmof bounded variation on [, π] which satisfy the conditions

π

dm(t) = ,

π

dm(t)≤k, ()

in terms of the Riemann-Stieltjes integral. It is clear thatMis the class of nondecreasing functions on [, π] satisfying () or, equivalently,πdm(t) = .

The classMkis related to the class(h) withμ=k+. Therefore, for the simplicity of notation, we define

Pk(h) :=(h) (μ=k/ + /,k≥).

From the result of Hallenbeck and MacGregor ([], p.), we have the following lemma.

Lemma  Let|c| ≤,c= –,h(z) = +cz–z (zU).Then qh if and only if there exists mMsuch that

q(z) =  

π

hze–itdm(t) (zU).

Theorem  The classKμ(h)is convex.

Proof Letq,r(h),α∈[, ]. Then there existqj,rjh(j= , ) such that

(4)

Since

αq+ ( –α)r=μαq+ ( –α)r

+ ( –μ)αq+ ( –α)r

andαqj+ ( –α)rjh(j= , ), we conclude thatαq+ ( –α)r(h). Hence, the class

(h) is convex.

Theorem  If≤μ<λ,thenKμ(h)⊂(h).

Proof Letq(h). Then there existq,q≺hsuch thatq=μq+ ( –μ)q. Thus, we have

q=λq+ ( –λ)q

q=

λμ λ– q+

μ– 

λ– q

.

Hence, we getq≺hand, in consequence,q(h).

Theorem  Let|c| ≤,c= –,h(z) = +cz–z (zU).Then qPk(h)if and only if there exists mMksuch that

q(z) =  

π

hze–itdm(t) (zU). ()

Proof LetqPk(h). Then there existq,q≺hsuch thatq= (k+)q– (k–)q. Thus, by Lemma  there existm,m∈Msuch that

q(z) =  

π

hze–itdm(t) m= k +  

m– k –   m . Since π

dm(t) = k +   π

dm(t) – k –   π

dm(t) = 

and π

dm(t)≤ k +   π

dm(t) + k –   π

dm(t) =k,

we havemMkand consequently (). Conversely, letqAsatisfy () for somemMk. IfmM, then by Lemma  and Theorem , we haveqP(h)⊂Pk(h). Let nowmMk\M. Sincemis the function with bounded variation, by the Jordan theorem there exist real-valued functionsμ,μwhich are nondecreasing and nonconstant on [, π] such that

m=μ–μ,

π

dm(t)= π

dμ(t) + π

dμ(t). ()

Thus, putting

αj=

μj(π) –μj()

 , mj:=

αj

(5)

we getm,m∈Mand

m=αm–αm. ()

Combining () and (), we obtain

α– α= π

dm(t) = , α+ α= π

dm(t)≤k,

and so

α=

λ

+  

, α=

λ

– 

λ=

π

dm(t), ≤λk

.

Therefore, by () and () we obtain

q=

λ

+  

q–

λ

–  

q,

where

qj(z) =  

π

hze–itdmj(t) (zU,j= , ).

Thus, by Lemma  and Theorem , we haveq(h)⊂Pk(h) and the proof is complete.

Letσh denote the conformal mapping which mapsh(U) onto:={w∈C:Rew> } withσh() =  and letDdenote the set of analyticity ofσh. Moreover, let us define

A:=

qA:q(U)⊂D , Pk(h) :=

qA:σhqPk

 +z

 –z

.

Lemma  Let qA.Then qK(h) :=K(h)if and only if

π

Re(σhq)

reitdt= π ( <r< ). ()

Proof LetqA. Then, by the properties of subordination, we have

qK(h) ⇐⇒ qh ⇐⇒ σhqσhh ⇐⇒ Re(σhq)(z) >  (zU).

()

Moreover, π

Re(σhq)

reitdt=Re

|z|=r

(σhq)(z)

z idz= π(σhq)() = π ( <r< ).

Thus, condition () is equivalent toRe(σhq)(z) >  (zU) and by () we have,

(6)

Let H(D),SH(D) denote the classes of harmonic and subharmonic functions in the domainD⊂C, respectively.

Theorem  Let qA.Then qPk(h)if and only if

π

Re(σhq)

reitdt ( <r< ). ()

Proof LetqPk(h). Then there existq,q≺+z–z such thatσhq= (k+)q– (k)q. Hence, by Lemma  we have

π

Re(σhq)

reitdt

k

+  

π

 Req

reitdt

+

k

–  

π

 Req

reitdt= ( <r< ).

To obtain the converse, suppose that qAsatisfies (). By Lemma  we can assume

k> . If we put

F:=Re(σhq), F+(z) :=max

F(z),  ≥,

F–(z) :=max–F(z),  ≥ (zU),

r(z) :=

 π

π

r|z| |reitz|F

τreitdt |z| ≤r< ,τ∈ {+, –},

then the functions,Vτ

r (τ∈ {+, –}) are nonconstant and

FH(U), V+

r,Vr–∈H(Ur), F+,F–∈SH(U),

F=F+–F–, |F|=F++F–, Vrτ(z) =(z) |z|=r,τ∈ {+, –}.

()

Thus, we have

max(z),Vrτ(z) =Vrτ(z) |z| ≤r,r∈(, ),τ∈ {+, –},

maxVrτ(z) :|z| ≤r =max(z) :|z|=r

≤max(z) :|z| ≤R rR< ,τ∈ {+, –}.

Therefore, the functions

r(z) :=

⎧ ⎨ ⎩

r(z), |z|<r,

(z), r≤ |z|< 

r∈(, ),τ∈ {+, –},

are continuous subharmonic functions inU and the families{G+

r :r∈(, )}, {Gr :r∈ (, )}are locally uniformly bounded. Hence, if we define

(z) :=supGrτ(z) :r∈(, ) = lim

n→∞G

τ –n(z)

(7)

then

SH(U), Gτ r,G

τH(U r)

r∈(, ),τ∈ {+, –}

and, in consequence,G+,GH(U),G+,G> . Moreover, by () we get

F(z) =G+r(z) –Gr(z) |z| ≤r,r∈(, ), F(z)=G+

r(z) +Gr(z)

|z|=r, r∈(, ).

()

Therefore, we have

F(z) =αG(z) –αG(z) (zU), ()

where

G:= 

α

G+, G:= 

α

Gα=G+(),α=G–()

are positive harmonic functions inU. Moreover, by () we obtain

lim r→

π

Freitdt=α lim r→

π

G

reitdt+α lim r→

π

G

reitdt. ()

Now, we consider functionsq,q∈Asuch that

Reqj(z) =Gj(z) >  (zU,j= , ).

Thenq,q≺+z–z and by () we obtain

σhq=αq–αq. ()

Hence, we getα–α= . Moreover, by () and Lemma , we have α+ α=λ, where

λ=:πlimr→–π

 |F(reit)|dtand ≤λk, by (). Thus, we get

α=

λ

+ 

, α=

λ

– 

, ≤λk.

Therefore, by () and Theorem , we have q(h)⊂Pk(h), which proves the

theo-rem.

Ifh(z) =+(–a)z–z ,σh(z) =z–a–a (zU,a= ), thenPk(h) =Pk(h). Thus, by Theorem  we get the following result.

Corollary [] Let qA,h(z) =+(–a)z–z ,a= .Then qPk(h)if and only if()holds.

Remark  Theorem  and Theorem  give relationships between the classPk(h) and

(8)

The first-order differential subordination

q(z) + zq (z)

βq(z) +γh(z) ()

is called the Briot-Bouquet differential subordination. This particular differential subor-dination has a surprising number of important applications in the theory of analytic func-tions (for details, see []). In particular, Eenigenburg, Miller, Mocanu and Reade [] proved the following result.

Lemma  Let hSc

,β,γ∈C,andRe(βh(z) +γ) >  (zU).If qAsatisfies the

Briot-Bouquet differential subordination(),then qh.

Forβ=  we can extend this result.

Theorem  Let qA,Reγ> .If

q(z) +γzq(z)∈(h), ()

then q(h).

Proof From () there existq,q≺hsuch that

q(z) +γzq(z) =μq(z) + ( –μ)q(z) (zU). ()

Letq,qbe the solutions of the Cauchy problems

q(z) +γzq(z) =q(z), q() = , q(z) +γzq(z) =q(z), q() = ,

respectively. Then, by () we haveq=μq+ ( –μ)q. Moreover, by Lemma  we get q,q≺h. Therefore,q(h) and the proof is completed.

A more general problem can be formulated as the following problem.

Problem  LetRe(βh(z) +γ) >  (zU). To verify the following result: ifqAsatisfies

q(z) + zq (z)

βq(z) +γ(h),

thenq(h).

Remark  The result from the problem was used in some papers (see, for example, [, ,

] and []). It is clear, by Theorem , that the result is true forβ= , but forβ=  it is

(9)

3 Properties of multivalent prestarlike functions

Letα<p. We say that a functionfApbelongs to the classRp(α) ofmultivalent

prestar-likefunctions of orderαif

f(z)∗ z p

( –z)(p–α) ∈Sp(α).

It is easy to verify that

Rp(α) =W

zp ( –z)(p–α),

 + ( – α/p)z

 –z

.

The classR(α) :=R(α) is the well-known class ofprestarlikefunctions of orderα intro-duced by Ruscheweyh [].

We denote thedual setofVAbyV∗:={qA: (rq)(z)=  (rV,zU)}. More-over, let us define

T(β) :=

( +xz)

( +yz)β :|x|=|y|= 

(β≥),

H:=T()∗, Hk:=qk:qH (k≥),

F(,λ) :=

qA:Re

zq(z)

q(z)

> –λ  (zU)

,

F(k,β) :=rq:rHkandqF(,βk) (≤kβ).

It is clear that

V⊂V ⇒ V∗⊂V∗ ()

and

≤kβ≤β ⇒ F(k,β)⊂F(k,β). ()

In proving our main results, we need the following lemmas.

Lemma [] Let w be a nonconstant function meromorphic inU with w() = .If

w(z)=maxw(z);|z| ≤ |z|

for some z∈U,then there exists a real number k(k≥)such that zw(z) =kw(z).

Lemma ([], p.) Letβ≥and fA.Then f ∈[T(β)]∗if and only if

⎧ ⎨ ⎩

Rezf

β(z)

(z) >

–β

forβ= , Ref(z) > forβ= 

(zU),

where

(10)

Lemma ([], p.) Letβ≥.If f,g∈[T(β)]∗,then fg∈[T(β)]∗.

Lemma ([], p. and p.) Ifβ≥,then[F(,β)]∗= [T(β)]∗.

Lemma ([], p.) Letβ≥,f ∈[T(β)]∗,gF(,β– ),qA.Then

f ∗(qg)

fg (U)⊆co

q(U) , ()

where co{q(U)}denotes the closed convex hull of q(U).

From Lemma  and definitions of the classesRp(α),Sp∗(α) andF(α,β) we obtain the following two results.

Lemma  A function f belongs to the classRp(α)if and only if

f(z)

zp

T(p+  – α)∗.

Lemma  A function f belongs to the classSp∗(α)if and only if

f(z)

zpF(, p– α).

Using Lemmas  and , we have the following theorem.

Theorem  If f,gRp(α),then fgRp(α).

Theorem  Ifα≤α<p,then

Rp(α)⊂Rp(α). ()

Proof By condition () we have

F(, p+  – α)⊂F(, p+  – α).

Thus, by () and Lemma , we obtain

T(p+  – α)∗=F(, p+  – α)∗⊂F(, p+  – α)∗

=T(p+  – α) ∗

,

and by Lemma  we get the inclusion relation ().

Making use of Lemmas -, we get the following theorem.

(11)

For complex parameters a, b, c (c = , –, –, . . .), the hypergeometric function

F(a,b;c;z) is defined by

F(a,b;c;z) := ∞

n=

(a)n(b)n (c)nn! z

n (zU),

where

(λ)n:= ⎧ ⎨ ⎩

 (n= ),

λ(λ+ )· · ·(λ+n– ) (n∈N) is the Pochhammer symbol. Next, we define

ϕp(a,b;c)(z) :=zpF(a,b;c;z) (zU).

In particular, the function

lp(a,c)(z) :=ϕp(a, ;c)(z) (zU) ()

will be called the multivalent incomplete hypergeometric function.

Theorem  If either

Rea≤Rec, Ima=Imc and (p+  –ac)/≤α<p ()

or

 <ac and

pc

α<p, ()

then

lp(a,c)∈Rp(α). ()

Proof Let () hold true. By Theorem  it is sufficient to prove () forα=

(p+  –ac). It is easy to show that

ϕp(a,b;c) =lp(a,c)∗ϕp(,b; )

and

ϕp(; p– α; )(z) =

zp

( –z)(p–α) (zU).

Thus, condition () is equivalent toϕp(a,b;c)∈Sp∗(α), whereb=a+c–  > . Using the notationF(z) :=F(a,b;c;z) (zU), we have to show that

Re

zF(z)

F(z)

> –b

(12)

or that the meromorphic functionω,ω() = , defined by

zF(z)

F(z) =b

ω(z)

 –ω(z) (zU) ()

is bounded by  inU. If this is false, we findz∈Usuch that|w(z)|= ,|w(z)|<  (|z|<|z|). According to Lemma , there existsk≥ such that

zw(z) =kw(z), w(z) =eiθ. ()

Taking the logarithmic derivative of (), we get

zF(z)

F(z) =

(b+ )ω(z) –   –ω(z) +

(z)

[ –ω(z)]ω(z) (zU). ()

The hypergeometric functionFsatisfies the Gaussian hypergeometric differential equa-tion

z( –z)F+c– (a+b+ )zFabF= .

Therefore, by () and (), we obtain

zA(z) =B(z) (zU), ()

where

A(z) =a+

ba+( z)

ω(z)

ω(z), B(z) =ω(z)

(z)

ω(z) +c–  + (b+  –c)ω(z)

.

Thus, by () we have

A(z) =a+ (k–  +c)eiθ, B(z) =k–  +c+aeiθ.

SinceA(z) =eiθB(z), we have|A(z)|=|B(z)|. Furthermore,A(z)=  under restrictions (). Thus () gives|z|= , a contradiction. Now let () hold true. It is clear that the function

lp(a, )(z) =

zp

( –z)a (zU)

belongs to the classSp∗(pa)⊂Sp∗(pc) andlp(a,c)∗lp(c, ) =lp(a, ). Thus, by the defi-nition of the classRp(α) and Theorem , we havelp(a,c)∈Rp(pc)⊂Rp(α).

Remark  The results related to multivalent prestarlike functions were obtained in the

(13)

4 The main inclusion relationships From now on we make the assumptions:

Reh(z) >α, Rehj(z) >α (zU,j= , ). ()

Moreover, letψ∗(φ,ϕ) := (ψφ,ψϕ). Then we have

(h) :=

zp/( –z),hSp∗(α) and (ϕ,h)⊂Sp∗(ϕ,α). ()

Lemma  If≤λ<δ,then

(ϕ,h)⊂(ϕ,h)⊂W(ϕ,h).

Proof Letf

(ϕ,h) and let

q(z) :=z(ϕf) (z)

p(ϕf)(z) (zU).

Then we obtain

q(z) +δ

p zq(z)

q(z) =(f)(z) (zU),

whereis defined by (). Since(f)≺h, by Lemma  we haveqh. Moreover,

(f) =

λ δLδ(f) +

δλ δ q.

Thus, by Theorem  we get(f)≺hor, equivalently,f(ϕ,h).

Theorem  IfψRp(α),then

(,h)∩(ϕ,h)⊂(ψ,h), ()

(ϕ,h)⊂(ψϕ,h). ()

Proof Letf(,h)∩(ϕ,h). Then there existω,ω∈such that

φf

ϕf =μ(hω) + ( –μ)(hω)

andF=ϕf(h)⊂Sp∗(α). Thus, applying the properties of convolution, we get

(ψφ)∗f

(ψϕ)∗f =μ

ψ∗[(hω)F]

ψF + ( –μ)

ψ∗[(hω)F]

ψF . ()

By Theorem  we conclude that

qj(z) :=ψ∗[(hωj)F]

ψF (z)∈co

(14)

Therefore,qjhand by () we havef(ψ,h), which proves the inclusion (). To prove (), we assume thatf(ϕ,h). Thenf((φ,ϕ),h), whereφ(z) =zpϕ(z) (z

U). Thus, by () we obtain thatf((ψφ,ψϕ),h). Since (ψφ)(z) =zp(ψϕ)(z) (zU), we havef(ψϕ,h), which proves () and completes the proof.

Theorem  Letψ,ξRp(α), ≤δ≤.Then

μ(,ξ,h)∩(,h)⊂Mδμ(ψ,ξ,h), ()

(ϕ,h)⊂Mδ(ψϕ,h). ()

Proof Let

μ(,ξ,h)∩(,h). Then, applying Theorem  and Theorem , we obtain

f(ψ,h) andf(ξψ,h) or, equivalently,

q:=

φψf

ϕψf(h), q:=

ξφψf

ξϕψf(h).

Since the class(h) is convex by Theorem , we conclude that

( –δ)ξ∗(ψφ)∗f

ξ∗(ψϕ)∗f +δ

(ψφ)∗f

(ψϕ)∗f(h).

Thus, we have f(ψ ,ξ,h) and, in consequence, we get (). From () and

Lemma  we have ().

Theorem  Letψ,ξRp(α), ≤δ≤.Then

CWμ(,h)⊂CWμ(ψ,h), ()

CMδ

μ(,ξ,h)∩CWμ(,h)⊂CMδμ(ψ,ξ,h). ()

Proof LetfCWμ(,h). Then there existg(ϕ,h) andω,ω∈such that

φf

ϕg =μ(h◦ω) + ( –μ)(h◦ω).

SinceF=ϕgSp∗(α), applying the properties of convolution, we obtain

(ψφ)∗f

(ψϕ)∗g =μ

ψ∗[(h◦ω)F]

ψF + ( –μ)

ψ∗[(h◦ω)F]

ψF . ()

Analysis similar to that in the proof of Theorem  gives

(ψφ)∗f

(ψϕ)∗g(h).

Moreover, by Theorem  we havegWμ(ψϕ,h) and, in consequence,fCWμ(ψ

(15)

Then, applying () and Theorem , we obtainfCWμ(ψ,h) andfCWμ((ξψ)∗

,h) or, equivalently,

q:=

ψφf

ψϕg(h), q:=

ξψφf

ξψϕg(h).

Since the class(h) is convex by Theorem , we conclude that

( –δ)ξψφf

ξψϕg+δ

ψφf

ψϕg(h).

This gives () and completes the proof.

Combining Theorems - with Theorem , we obtain the following corollary.

Corollary  If either()or(),then

(ϕ,h)⊂

lp(a,c)∗ϕ,h

,

CWμ(,h)⊂CWμ

lp(a,c)∗,h,

(,h)∩(ϕ,h)⊂

lp(a,c,z)∗,h.

Moreover,ifξRp(α)and≤δ≤,then

(ϕ,h)⊂

lp(a,c)∗ϕ,h,

μ(,ξ,h)∩(,h)⊂Mδμ

lp(a,c)∗,ξ,h,

CMδ

μ(,ξ,h)∩CWμ(,h)⊂CMδμ

lp(a,c)∗,ξ,h.

Sincelp(a,c)∗lp(c,a)∗φ=φ, by Theorem  we obtain the next result.

Corollary  If either()or(),then

lp(c,a)∗ϕ,h(ϕ,h),

CWμ

lp(c,a)∗,hCWμ(,h),

lp(c,a)∗,h

lp(c,a)∗ϕ,h(,h).

Moreover,ifξRp(α)and≤δ≤,then

lp(c,a)∗ϕ,h(ϕ,h),

μ

lp(c,a)∗,ξ,h

lp(c,a)∗,hMδμ(,ξ,h),

CMδ μ

lp(c,a)∗,ξ,hCWμ

lp(c,a)∗,hCMδμ(,ξ,h).

Let us define the linear operators:Ap−→Ap,Jλ×:Ap×Ap−→Ap×Ap,

(f)(z) :=λ

zf(z)

p + ( –λ)f(z),

×(f,g) :=

(f),(g)

(zU,Reλ> ).

(16)

Since(ϕ) =lp(pλ+ ,pλ)∗ϕ, puttinga=pλ,c=λp+  in Corollary , we have the following corollary.

Corollary  If p–Rep/λα<p,then

(ϕ),h

(ϕ,h),

CWμ

×(),h

CWμ(,h),

×(),h

(ϕ),h

(,h).

Moreover,ifξRp(α)and≤δ≤,then

(ϕ),h

(ϕ,h),

μ

Jλ×(),ξ,h

Jλ×(),h

μ(,ξ,h),

CMδ μ

×(),ξ,h

CWμ

×(),h

CMδ

μ(,ξ,h).

In particular, forλ= , we get the following corollary.

Corollary  If≤α<p,then

p–(z),h(ϕ,h),

CWμ

p–z(z),hCWμ(,h),

p–z(z),h

p–(z),h(,h).

Moreover,ifξRp(α)and≤δ≤,then

p–(z),h(ϕ,h),

μ

p–z(z),ξ,h

p–z(z),hMδμ(,ξ,h),

CMδ μ

p–z(z),ξ,hCWμ

p–z(z),hCMδμ(,ξ,h).

5 Applications to classes defined by linear operators The classes

μ(,ξ,h) and CM δ

μ(,ξ,h) generalize well-known important classes, which were investigated in earlier works. Most of these classes were defined by using linear operators and special functions.

LetA, . . . ,AqandB, . . . ,Bs(q,s∈N) be positive real numbers such that

 + s

k=

Bkq

k=

Ak≥.

For complex parametersa, . . . ,aqandb, . . . ,bs(q,s∈N) such that Aakk,Bbkk = , –, –, . . . , we define the Fox-Wright generalization of the hypergeometric function by

qs

(a,A), . . . , (aq,Aq);

(b,B), . . . , (bs,Bs); z

:= ∞

n=

(a+An)· · ·(aq+Aqn)

(b+Bn)· · ·(bs+Bsn)

zn

(17)

IfAn=  (n= , . . . ,q) andBn=  (n= , . . . ,s), then we obtain the generalized hypergeo-metric function

qFs(a, . . . ,aq;b, . . . ,bs;z) :=ωqs

(a, ), . . . , (aq, );

(b, ), . . . , (bs, );

z

(zU),

where

ω= (b)· · ·(bs)

(a)· · ·(aq)

. ()

Moreover, in terms of Fox’sH-function, we have

qs

(a,A), . . . , (aq,Aq);

(b,B), . . . , (bs,Bs);

z

=Hq,s+,q

z ( –a,A), . . . , ( –aq,Aq)

(, )( –b,B), . . . , ( –bs,Bs)

.

Other interesting and useful special cases of the Fox-Wright function defined by () in-clude (for example) the generalized Bessel function

ν defined by

Jνμ(z) := ∞

n=

(–z)n

n!( +ν+μn) (zU)

which, forμ= , corresponds essentially to the classical Bessel function, and the gener-alized Mittag-Leffler function,μdefined by

,μ(z) := ∞

n=

zn

(ν+λn) (zU).

For real numbersλ,t(λ> –p), we define the function

ζ(a,b,t)(z) :=

ωzpqs

(a,A), . . . , (aq,Aq)

(b,B), . . . , (bs,Bs) ;z

,t(z), ()

whereωis defined by () and

,t(z) = ∞

n=p

n+λ p+λ

t

zn (zU).

It is easy to verify that

(a+ ,b,t) =A(a;b,t) + (apA)ζ(a,b,t), ()

(a,b,t) =B(a,b+ ,t) + (bpB)ζ(a,b+ ,t),

(p+λ)ζ(a,b,t+ ) =(a,b,t) +λζ(a,b,t),

ζ(a,b,t) =lp(a,c)∗ζ(c,b,t) (A= ), ()

ζ(a,c,t) =lp(b,c)∗ζ(a,b,t) (B= ),

(18)

Concerning the functionζ(a,b,t), we consider the following classes of functions:

V(a,b,t) :=

ζ(a,b,t),h, CV(a,b,t) :=CWμ

ζ(a,b,t),h.

By using the linear operator

p[a,b,t]f =ζ(a,b,t)∗f (fAp) ()

we can define the classV(a,b,t) alternatively in the following way:

fV(a,b,t) ⇐⇒ a

A

p[a+ ,b,t]f(z)

p[a,b,t]f(z)

+pa

A

(h).

Corollary  If p–Rea/A≤α<p,m∈N,then

V(a+m,b,t)⊂V(a,b,t), CV(a+m,b,t)⊂ CV(a,b,t). ()

Proof It is sufficient to prove the corollary form= . Letandζ(a,b,t) be defined by () and (), respectively. Then by () we haveζ(a+ ,b,t) =JpA

a

(ζ(a,b,t)). Hence, by using Corollary , we conclude that

ζ(a+ ,b,t),h

ζ(a,b,t),h,

CWμ

ζ(a+ ,b,t),hCWμ

ζ(a,b,t),h.

This clearly forces the inclusion relations () form= .

Analogously to Corollary , we prove the following corollary.

Corollary  Let m∈N.If p–Reb/B≤α<p,then

V(a,b,t)⊂V(a,b+m,t), CV(a,b,t)⊂CV(a,b+m,t).

If–Reλα<p,then

V(a,b,t+m)⊂V(a,b,t), CV(a,b,t+m)⊂CV(a,b,t).

It is natural to ask about the inclusion relations in Corollaries  and  whenmis positive real. Using Theorems  and , we shall give a partial answer to this question.

Corollary  If lp(a,c)∈Rp(α),then

V(c,b,t)⊂V(a,b,t), CV(c,b,t)⊂CV(a,b,t) (A= ), ()

V(b,a,t)⊂V(b,c,t), CV(b,a,t)⊂CV(b,c,t) (B= ). ()

Proof Let us putψ=lp(a,c),ϕ=ζ(c,b,t). Then, by Theorem , Theorem  and rela-tionship (), we obtain

ζ(c,b,t),h

ζ(a,b,t),h, CWμ

ζ(c,b,t),hCWμ

ζ(a,b,t),h.

(19)

Combining Corollary  with Theorem , we obtain the following result.

Corollary  If either()or(),then the inclusion relations()and()hold true.

The linear operatorp[a,b,t] defined by () includes (as its special cases) other linear operators of geometric function theory which were considered in earlier works. In par-ticular, the operatorp[a,b, ] was introduced by Dziok and Raina []. It contains, as its further special cases, such other linear operators as the Dziok-Srivastava operator, the Carlson-Shaffer operator, the Ruscheweyh derivative operator, the generalized Bernardi-Libera-Livingston operator (for the precise relationships, see Dziok and Srivastava ([], pp.-)). Moreover, the linear operatorp[a,b,t] includes also the Sălăgean operator, the Noor operator, the Choi-Saigo-Srivastava operator (for the precise relationships, see Cho

et al.[]). By using these linear operators, we can consider some subclasses of the classes

V(a,b,t),CV(a,b,t), see for example [–, –, , , , , –]. Also, the obtained results generalize several results obtained in these classes of functions.

Competing interests

The author declares that they have no competing interests .

Received: 1 May 2013 Accepted: 10 July 2013 Published: 26 July 2013 References

1. Mocanu, PT: Une propriété de convexité g énéralisée dans la théorie de la représentation conforme. Mathematica (Cluj, 1929)11(34), 127-133 (1969) (in French)

2. Aouf, MK: Some inclusion relationships associated with Dziok-Srivastava operator. Appl. Math. Comput.216, 431-437 (2010)

3. Aouf, MK, Seoudy, TM: Some properties of certain subclasses ofp-valent Bazilevic functions associated with the generalized operator. Appl. Math. Lett.24(11), 1953-1958 (2011)

4. Cho, NE, Kwon, OS, Srivastava, HM: Inclusion relationships and argument properties for certain subclasses of multivalent functions associated with a family of linear operators. J. Math. Anal. Appl.292, 470-483 (2004) 5. Choi, JH, Saigo, M, Srivastava, HM: Some inclusion properties of a certain family of integral operators. J. Math. Anal.

Appl.276, 432-445 (2002)

6. Coonce, HB, Ziegler, MR: Functions with bounded Mocanu variation. Rev. Roum. Math. Pures Appl.19, 1093-1104 (1974)

7. Dziok, J: Applications of multivalent prestarlike functions. Appl. Math. Comput. (to appear) 8. Dziok, J: Applications of the Jack lemma. Acta Math. Hung.105, 93-102 (2004)

9. Dziok, J: Inclusion relationships between classes of functions defined by subordination. Ann. Pol. Math.100, 193-202 (2011)

10. Dziok, J: Characterizations of analytic functions associated with functions of bounded variation. Ann. Pol. Math.109, 199-207 (2013)

11. Dziok, J, Raina, RK: Families of analytic functions associated with the wright generalized hypergeometric function. Demonstr. Math.37, 533-542 (2004)

12. Dziok, J, Sokół, J: Some inclusion properties of certain class of analytic functions. Taiwan. J. Math.13, 2001-2009 (2009) 13. Dziok, J, Srivastava, HM: Classes of analytic functions associated with the generalized hypergeometric function. Appl.

Math. Comput.103, 1-13 (1999)

14. Eenigenburg, PJ, Miller, SS, Mocanu, PT, Reade, OM: Second order differential inequalities in the complex plane. J. Math. Anal. Appl.65, 289-305 (1978)

15. Liu, J-L, Noor, KI: On subordinations for certain analytic functions associated with Noor integral operator. Appl. Math. Comput.187, 1453-1460 (2007)

16. Liu, M-S, Zhu, Y-C, Srivastava, HM: Properties and characteristics of certain subclasses of starlike functions of orderβ. Math. Comput. Model.48, 402-419 (2008)

17. Lyzzaik, A: Multivalent functions of bounded boundary rotation and weakly close-to-convex functions. Proc. Lond. Math. Soc.51(3), 478-500 (1985)

18. Hallenbeck, DJ, MacGregor, TH: Linear Problems and Convexity Techniques in Geometric Function Theory, Pitman Advanced Publishing Program. Pitman, Boston (1984)

19. Jack, IS: Functions starlike and convex of orderα. J. Lond. Math. Soc.3, 469-474 (1971)

20. Miller, SS, Mocanu, PT: Differential Subordinations: Theory and Applications. Series on Monographs and Textbooks in Pure and Applied Mathematics, vol. 225. Dekker, New York (2000)

21. Moulis, EJ: Generalizations of the Robertson functions. Pac. J. Math.81, 167-174 (1979)

22. Noor, KI: On some subclasses of functions with bounded radius and bounded boundary rotation. Panam. Math. J.

6(1), 75-81 (1996)

(20)

24. Noor, KI, Hussain, S: On certain analytic functions associated with Ruscheweyh derivatives and bounded Mocanu variation. J. Math. Anal. Appl.340(2), 1145-1152 (2008)

25. Noor, KI, Malik, SN: On generalized bounded Mocanu variation associated with conic domain. Math. Comput. Model.

55(3-4), 844-852 (2012)

26. Noor, KI, Muhammad, A: On analytic functions with generalized bounded Mocanu variation. Appl. Math. Comput.

196(2), 802-811 (2008)

27. Noor, KI, Ul-Haq, W: On some implication type results involving generalized bounded Mocanu variations. Comput. Math. Appl.63(10), 1456-1461 (2012)

28. Özkan, Ö, Altınta¸s, O: Applications of differential subordination. Appl. Math. Lett.19, 728-734 (2006)

29. Paatero, V: Über die konforme Abbildung von Gebieten deren Ränder von beschränkter Drehung sind. Ann. Acad. Sci. Fenn., A33, 1-79 (1931)

30. Patel, J, Mishra, AK, Srivastava, HM: Classes of multivalent analytic functions involving the Dziok-Srivastava operator. Comput. Math. Appl.54, 599-616 (2007)

31. Padmanabhan, K, Parvatham, R: Properties of a class of functions with bounded boundary rotation. Ann. Pol. Math.

31, 311-323 (1975)

32. Piejko, K, Sokół, J: On the Dziok-Srivastava operator under multivalent analytic functions. Appl. Math. Comput.177, 839-843 (2006)

33. Pinchuk, B: Functions with bounded boundary rotation. Isr. J. Math.10, 7-16 (1971)

34. Ruscheweyh, S: Convolutions in Geometric Function Theory. Sem. Math. Sup., vol. 83. Presses University Montreal, Montreal (1982)

35. Ruscheweyh, S: Linear operators between classes of prestarlike functions. Comment. Math. Helv.52, 497-509 (1977) 36. Sokół, J, Trojnar-Spelina, L: Convolution properties for certain classes of multivalent functions. J. Math. Anal. Appl.337,

1190-1197 (2008)

37. Srivastava, HM, Lashin, AY: Subordination properties of certain classes of multivalently analytic functions. Math. Comput. Model.52, 596-602 (2010)

38. Wang, Z-G, Zhang, G-W, Wen, F-H: Properties and characteristics of the Srivastava-Khairnar-More integral operator. Appl. Math. Comput.218, 7747-7758 (2012)

doi:10.1186/1029-242X-2013-349

References

Related documents

This study was carried out to cover more cassava growing areas in Rivers State with a view to determining recent status of plant-parasitic nematodes in terms of occurrence and

The present study was undertaken with a view to test the potential of mutated and non-mutated endophytic actinomycetes and as biocontrol agents which were isolated from

Both the proposed model and the analysis of the EE in this study take into account (i) the path losses, fading, and shadowing that affect the received signal at the UE within the

This research did not include positive innovation outcomes (e.g., customer satisfaction, employee trust) as a mediator variable, as the present study considers negative

Recommendations designed to overcome other environ- mental barriers could include the promotion and funding of developmental screening tools suited to the time constraints of

The study is crucial in unearthing psychosocial effects faced by War Victims among Freedom Fighters (MAU MAU) Those experiences are of great interest to the

As compared to other studies conducted ac- cording to the same methodology, it was demonstrated that tansy aqueous extracts are characteristic for a high deterrent activity in

Background: Negative Pressure Wound Therapy (NPWT) or Vacuum Assisted Closure (VAC) is a method used to cover large wounds, decubitus ulcers and open fractures which cannot be