Properties of multivalent functions associated with the integral operator defined by the hypergeometric function

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R E S E A R C H

Open Access

Properties of multivalent functions

associated with the integral operator deﬁned

by the hypergeometric function

Mohsan Raza

1*

_{and Sarfraz Nawaz Malik}

2

*_{Correspondence:}

mohsan976@yahoo.com

1_{Department of Mathematics, GC}

University, Faisalabad, Pakistan Full list of author information is available at the end of the article

Abstract

In this paper, we introduce a new class of multivalent functions by using a generalized integral operator deﬁned by the hypergeometric function. Some properties such as inclusion, radius problem and integral preserving are considered.

MSC: 30C45; 30C50

Keywords: Bazilevic functions; multivalent functions; integral operator; hypergeometric functions

1 Introduction and preliminaries

LetApdenote the class of functionsf(z) of the form

f(z) =zp+

∞

n=p+

anzn p∈N={, , , . . .}, (.)

which are analytic in the open unit discE. AlsoA=A, the usual class of analytic

func-tions deﬁned in the open unit discE={z:|z|< }. A functionf ∈Apis ap-valent starlike function of orderρif and only if

Rezf _(z)

f(z) >ρ, ≤ρ<p,z∈E.

This class of functions is denoted byS∗_p(ρ). It is noted thatS_p∗() =S∗_p. Letf(z) andg(z) be analytic inE, we sayf(z) is subordinate tog(z), writtenf≺gorf(z)≺g(z) if there exists a Schwarz functionw(z),w() =  and|w(z)|<  inE, thenf(z) =g(w(z)). In particular, ifg is univalent inE, then we have the following equivalence

f(z)≺g(z) ⇐⇒ f() =g() and f(E)⊂g(E).

For any two analytic functionsf(z) andg(z) with

f(z) =

∞

n=

bnzn+ and g(z) =

∞

n=

cnzn+, z∈E,

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the convolution (the Hadamard product) is given by

(f ∗g)(z) =

∞

n=

bncnzn+, z∈E.

A functionf ∈Ais said to be in the class, denoted bySD(k,δ) (≤δ< ), if and only if

Re

zf(z) f(z)

>kzf

_(z)

f(z) – 

+δ, k≥,z∈E. (.)

Similarly, a functionf ∈Ais said to be in the class, denoted byCD(k,δ) ofk-uniformly convex of orderδ(≤δ< ), if

Re

 +zf

_(z)

f(z)

>kzf

_(z)

f(z)

+δ, k≥,z∈E. (.)

Geometric interpretation The functionsf∈SD(k,δ) andf ∈CD(k,δ) if and only if zf_f₍_z(z₎) andzf_f₍_z(z₎)+ , respectively, take all the values in the conic domaink,δdeﬁned by

k,δ=

u+iv:u>k (u– )₊_v₊_δ

withp(z) =zf_f₍(_zz₎) orp(z) =zf_f₍(_zz₎)+  and considering the functions which mapEonto the

conic domaink,δ such that ∈k,δ. One may rewrite the conditions (.) or (.) in the form

p(z)≺qk,δ(z).

The functionqk,δ(z) plays the role of extremal for these classes and is given by

qk,δ(z) =

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

+(–δ)z

–z , k= ,

 +_πδγ(log +√z

–√z)

_, _k_{= ,}

 + δ

–ksinh[(_πarccosk)arctanh

√

z],  <k< ,  +_kδ_–sin(

π

R(t)

u_√(z)

t

 √_–_x√_–(_tx₎dx) +

δ

k_–, k> .

(.)

Forf(z) inAp, the operatorDμ+p–_:_Ap_−→_Ap_{is deﬁned by}

Dμ+p–f(z) = z p

( –z)μ+p∗f(z) (μ> –p), or equivalently

Dμ+p–_f_{(z) =} zp(zμ–f(z))μ+p–

(μ+p– )! , (.)

whereμis any integer greater than –p. Iff(z) is given by (.), then it follows that

Dμ+p–f(z) =zp+

∞

n=p+

(μ+n– )! (n–p)!(μ+p– )!anz

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The symbolDμ+p–_when_p_{= , was introduced by Ruscheweyh [] and}_Dμ+p–_{is called the}

(μ+p– )th order Ruscheweyh derivative. We now introduce a function (zp_F(a,b,c;z))–

given by

zpF(a,b,c;z)

∗zpF(a,b,c;z)

–

= z

p

( –z)μ+p (μ> –p),

and the following linear operator

Iμ,p(a,b,c)f(z) =

zpF(a,b,c;z)

–_∗

f(z), (.)

where a, b, care real or complex numbers other than , –, –, . . . , μ> –p, z∈E and f(z)∈Ap. This operator was recently introduced in []. In particular, forp= , this op-erator is studied by Noor []. Forb= , this operator reduces to the well-known Cho-Kwon-Srivastava operator Iμ,p(a,c), which was studied by Choet al.[], and forμ= ,

b=c,a=n+p, see []. Fora=n+p,b=c= , this operator was investigated by Liu [] and Liu and Noor [].

Simple computations yield

Iμ,p(a,b,c)f(z) =zp+

∞

n=p+

(c)n(μ+p)n (a)n(b)n

anzn.

From (.), we note that

Iμ,(a,b,c)f(z) =Iμ(a,b,c)f(z) (see []),

I,p(a,p,a)f(z) =f(z),I,p(a,p,a)f(z) =

zf(z) p .

Also, it can be easily seen that

zIμ,p(a,b,c)f(z)

= (μ+p)Iμ+,p(a,b,c)f(z) –μIμ,p(a,b,c)f(z), (.)

and

zIμ,p(a+ ,b,c)f(z)

=aIμ,p(a,b,c)f(z) – (a–p)Iμ,p(a,b,c)f(z).

We deﬁne the following class of multivalent analytic functions by using the operator Iμ,p(a,b,c)f(z) above.

Deﬁnition . Let f ∈Ap for p∈N. Thenf ∈UBα

μ,p(a,b,c;γ,k,δ) for a,b,c∈R\Z–, μ> –p,α> ,k≥, ≤δ<  andγ>  if and only if

( –γ)

Iμ,p(a,b,c)f(z)

Iμ,p(a,b,c)g(z)

α

+γIμ+,p(a,b,c)f(z)

Iμ+,p(a,b,c)g(z)

Iμ,p(a,b,c)f(z) Iμ,p(a,b,c)g(z)

α–

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whereg∈Apis such that

q(z) =Iμ+,p(a,b,c)g(z) Iμ,p(a,b,c)g(z) ∈

P(ρ), ρ=k+δ

k+ ,z∈E. (.)

Furthermore, for diﬀerent choices of parameters being involved, we obtain many other well-known subclasses of the classApandAas special cases.

(i) a=c,b= ,k= ,μ=m∈N, we haveBαm,p(γ,δ)studied in [].

(ii) a=c=b=p=γ = ,k=μ= ,g(z) =z, the classUBα

μ,p(a,b,c;γ,k,δ)reduces to the

class

Bα(δ) =

f ∈A() :zf

_(z)

f(z)

f(z) z

α ∈P(δ)

studied in [].

(iii) a=c=b=p=γ = ,k=μ= ,g(z) =z, theUBα

μ,p(a,b,c;γ,k,δ)reduces toB(α)is

the class of Bazilevich functions investigated by Singh []. (iv) a=c=b=p=α= ,γ = ,k=μ= ,g(z) =z, the classUBα

μ,p(a,b,c;γ,k,δ)reduces

to the class

Bδ=

f ∈A() :f(z) z ∈P(δ)

,

the class studied by Chen [].

Letf ∈Ap·andFη,p:Ap·→Ap·be deﬁned by

Fη,p(z) =

(η+p) zη

z



tη–f(t)dt, η> –p. (.)

We need the following lemmas which will be used in our main results.

Lemma .[] Let u=u+iuand v=v+iv,and letψ:D⊂C→Cbe a

complex-valued function satisfying the conditions:

(i) ψ(u,v)is continuous in a domainD⊂C_,

(ii) (, )∈Dandψ(, ) > ,

(iii) Reψ(iu,v)≤,whenever(iu,v)∈Dandv≤–_( +u).

If h(z) =  +cz+cz+· · ·is analytic in E such that(h,zh)∈D andReψ(h(z),zh(z)) > 

for z∈E,thenReh(z) > .

Lemma .[] Let h be convex in the unit disc E,and let A≥.Suppose that B(z)is analytic in E withReB(z)≥A.If g is analytic in E and g() =h().Then

Azg(z) +B(z)zg(z) +g(z)≺h(z) implies that g(z)≺h(z).

Lemma .[] Let F be analytic and convex in E.If f,g∈Apand f,g≺F.Then

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Lemma .[] Let h be convex in E with h() =a andβ∈Csuch thatReβ≥.If p∈ H[a,n]and

p(z) +zp

_(z)

β ≺h(z),

then p(z)≺q(z)≺h(z),where

q(z) = β nzβ/n

z



h(t)tβ/n–_dt

and q(z)is the best dominant.

2 Main results

Theorem . Let f ∈UBα

μ,p(a,b,c;γ,k,δ)for a,b,c∈R\Z–,μ> –p,p∈N,α> ,k≥,

≤δ< andγ> .Then f ∈UBα

μ,p(a,b,c; ,k,δ). Proof Consider

h(z) =

α

, (.)

wherehis analytic inEwithh() = , andg∈Ap satisﬁes condition (.). Diﬀerentiating (.) logarithmically and using (.), we have

zh(z)

h(z) =α(μ+p)

Iμ+,p(a,b,c)f(z)

Iμ,p(a,b,c)f(z)

–Iμ+,p(a,b,c)g(z) Iμ,p(a,b,c)g(z)

.

Using (.) and simplifying, we obtain

h(z) + γzh

_(z)

α(μ+p)q(z) = ( –γ)

α

+γIμ+,p(a,b,c)f(z)

Iμ+,p(a,b,c)g(z)

Iμ,p(a,b,c)f(z) Iμ,p(a,b,c)g(z)

α–

.

Sincef∈UBα

μ,p(a,b,c;γ,k,δ), therefore, we can write

h(z) + γzh

_(z)

α(μ+p)q(z)≺qk,δ(z), z∈E.

Now using Lemma . forA=  andB(z) = _α₍_μ₊γ_p₎_q₍_z₎withReq(z) > , we haveReB(z)≥, therefore,h(z)≺qk,δ(z). Hencef ∈UBαμ,p(a,b,c; ,k,δ).

Theorem . Let f ∈UBα

μ,p(a,b,c;γ, ,δ)for a,b,c∈R\Z–, μ> –p, p∈N. Then f ∈

UBα

μ,p(a,b,c; , ,δ),where

δ=

αδ(p+μ)|q(z)|₊_γρ

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Proof Consider

h(z) =  ( –δ)

α –δ

, (.)

wherehis analytic inEwithh() = , andg∈Ap satisﬁes condition (.). Diﬀerentiating (.), we have

( –δ)

α h

_{(z) =}Iμ,p(a,b,c)f(z)

Iμ,p(a,b,c)g(z)

α–

×

(Iμ,p(a,b,c)f(z)) Iμ,p(a,b,c)g(z)

–Iμ,p(a,b,c)f(z) Iμ,p(a,b,c)g(z)

(Iμ,p(a,b,c)g(z)) Iμ,p(a,b,c)g(z)

.

Using (.) and simplifying, we obtain

( –γ)

Iμ,p(a,b,c)f(z)

Iμ,p(a,b,c)g(z)

α

+γIμ+,p(a,b,c)f(z)

α–

= ( –δ)h(z) +δ+

γ( –δ)zh(z) α(p+μ)q(z) .

Sincef∈UBα

μ,p(a,b,c;γ, ,δ), therefore we have

( –δ)h(z) +δ+

γ( –δ)zh(z) α(p+μ)q(z) ≺

 + ( – δ)z

 –z , ≤δ< ,z∈E.

This implies that

  –δ

( –δ)h(z) +δ–δ+

γ( –δ)zh(z) α(p+μ)q(z)

∈q,(E) =P.

To obtain our desired result, we show thath∈P, forz∈E. Letu=u+iu,v=v+iv,

and let:D⊂C_→_C_{be a complex-valued function such that}_u₌_h(z),_v₌_zh_{(z). Then}

(u,v) = ( –δ)u+δ–δ+

γ( –δ)v α(p+μ)q(z).

The ﬁrst two conditions of Lemma . are easily veriﬁed. To verify the third condition, we consider

Re(iu,v)

=Re

( –δ)iu+δ–δ+

γ( –δ)v α(p+μ)q(z)

=δ–δ+Re

γ( –δ)v α(p+μ)q(z)

≤δ–δ–Re

γ( –δ)( +u)q(z)

α(p+μ)|q(z)|

≤δ–δ–

γ( –δ)( +u)ρ

α(p+μ)|q(z)| =

A+Bu

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where A= α(p+μ)(δ–δ)|q(z)|–γρ( –δ), B= –γρ( –δ)≤ if ≤δ <  and

C= α(p+μ)|q(z)|_{> . From the relation}_δ

=αδ(p+μ)|q(z)| ₊_γρ

α(p+μ)|q(z)|₊_γρ, we haveA≤. This

im-plies thatRe(iu,v)≤. Using Lemma ., we haveh∈Pforz∈E. This completes the

proof.

Theorem . Let a,b,c∈R\Z–

,μ> –p,p∈N,α> ,k≥and≤δ< .Then

UBαμ,p(a,b,c;γ,k,δ)⊂UBαμ,p(a,b,c;γ,k,δ), ≤γ<γ,z∈E.

Proof Sincef ∈UBα

μ,p(a,b,c;γ,k,δ), therefore, we have

( –γ)

Iμ,p(a,b,c)f(z)

Iμ,p(a,b,c)g(z)

α +γ

Iμ+,p(a,b,c)f(z) Iμ+,p(a,b,c)g(z)

Iμ,p(a,b,c)f(z)

Iμ,p(a,b,c)g(z)

α–

=h(z)≺qk,δ(z), (.)

whereg∈Apsatisﬁes condition (.). From Theorem ., we write

Iμ,p(a,b,c)f(z)

Iμ,p(a,b,c)g(z)

α

=h(z)≺qk,δ(z), z∈E. (.)

Now, forγ≥, we obtain

( –γ)

Iμ,p(a,b,c)f(z)

Iμ,p(a,b,c)g(z)

α +γ

Iμ,p(a,b,c)f(z)

Iμ,p(a,b,c)g(z)

α–

=

 –γ

γ

Iμ,p(a,b,c)f(z)

Iμ,p(a,b,c)g(z)

α

+γ

γ

( –γ)

Iμ,p(a,b,c)f(z)

Iμ,p(a,b,c)g(z)

α +γ

Iμ,p(a,b,c)f(z)

Iμ,p(a,b,c)g(z)

α–

=γ

γ

h(z) +

 –γ

γ

h(z).

Using the convexity of the class of the functionqk,δ(z) and Lemma ., we write

γ γ

h(z) +

 –γ

γ

h(z)≺qk,δ(z), z∈E,

wherehandhare given by (.) and (.), respectively. This implies thatf ∈UBαμ,p(a,b,c;

γ,k,δ). Hence the proof of the theorem is completed.

Theorem . Let f ∈UB

μ,p(a,b,c;γ,k,δ),a,b,c∈R\Z–,μ> –p,p∈N,α> ,k≥,γ≥

and≤δ< .Then f ∈UBμ+,p(a,b,c; ,k,δ).

Proof Sincef ∈UBμ,p(a,b,c;γ,k,δ), therefore, we have

( –γ)

Iμ,p(a,b,c)f(z)

Iμ,p(a,b,c)g(z)

+γIμ+,p(a,b,c)f(z)

Iμ+,p(a,b,c)g(z)≺

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Now, consider

γIμ+,p(a,b,c)f(z)

= ( –γ)

Iμ,p(a,b,c)f(z)

Iμ,p(a,b,c)g(z)

+γIμ+,p(a,b,c)f(z)

+ (γ– )

.

This implies that

= 

γ

( –γ)

Iμ,p(a,b,c)f(z)

Iμ,p(a,b,c)g(z)

+γIμ+,p(a,b,c)f(z)

+

 – 

γ

.

Using Theorem ., Lemma . and the convexity ofqk,δ(z), we have the required result.

Now, using the operatorsIμ,p(a,b,c) andFη,pdeﬁned by (.) and (.), respectively, we have

zI_pμ(a,b,c)Fη,p(f)(z)

= (p+η)I_pμ(a,b,c)f(z) –ηI_pμ(a,b,c)Fη,p(f)(z), η> –p. (.)

Theorem . Let f ∈Apand Fη,pbe given by(.).If

( –γ)Iμ,p(a,b,c)Fη,p(f)(z) zp +γ

Iμ,p(a,b,c)(f(z))

zp ≺qk,δ(z), z∈E, (.) with a,b,c∈R\Z–_,μ,η> –p,p∈N,γ> ,then

Iμ,p(a,b,c)Fη,p(f(z))

zp ≺h(z)≺qk,δ(z), z∈E, where

h(z) = p+η

γz(p+η)/γ

z



qk,δ(z)t

(p+η)/γ_–

dt.

Proof Let

Iμ,p(a,b,c)Fη,p(f(z))

zp =h(z), z∈E, wherehis analytic inEwithh() = . Then

zIμ,p(a,b,c)Fη,p

f(z)=pzph(z) +zp+h(z).

Using (.), we have

γ(p+η)Iμ,p(a,b,c)f(z) zp –γ η

Iμ,p(a,b,c)Fη,p(f)(z)

zp =pγh(z) +γzh

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Thus,

( –γ)Iμ,p(a,b,c)Fη,p(f(z)) zp +γ

Iμ,p(a,b,c)(f(z))

zp =h(z) +γ zh_(z)

p+η. (.)

From (.), it follows that

h(z) +γ

zh_(z)

p+η ≺qk,δ(z), z∈E.

Using Lemma ., forβ=p_γ+η,n=  anda= , we obtainh(z)≺h(z)≺qk,δ(z). That is,

Iμ,p(a,b,c)Fη,p(f(z))

zp ≺qk,δ(z).

Theorem . Let f ∈UBα

μ,p(a,b,c; , ,δ) for a,b,c∈R\Z–, μ> –p, p∈N, α,γ > ,

≤δ< .Then f ∈UBα

μ,p(a,b,c;γ, ,δ),for|z|<r,where

r= α(p

+ μ) +γ – γ

_{+ }_αγ_(p₊_μ₎_α_(p₊_μ_). _(.)

Proof Letf ∈UBα

μ,p(a,b,c; , ,δ). Then we have

α

= ( –δ)h(z) +δ, (.)

whereg∈Apsatisﬁes the condition

q(z) =Iμ+,p(a,b,c)g(z) Iμ,p(a,b,c)g(z) ∈

P, z∈E

andh∈P. Diﬀerentiating (.) and then using (.), we obtain

γIμ+,p(a,b,c)f(z)

Iμ,p(a,b,c)f(z)

Iμ,p(a,b,c)g(z)

α–

–γ

Iμ,p(a,b,c)f(z)

Iμ,p(a,b,c)g(z)

α

=γ(p–δ)zh

_(z)

α(p+μ)q(z).

This implies that

  –δ

( –γ)

Iμ,p(a,b,c)f(z)

Iμ,p(a,b,c)g(z)

α

+γIμ+,p(a,b,c)f(z)

Iμ,p(a,b,c)f(z)

Iμ,p(a,b,c)g(z)

α–

–δ

=h(z) + γzh

_(z)

α(p+μ)q(z), z∈E. (.)

Now, using the well-known distortion result for classP, we have

zh(z)≤rReh(z)

 –r and Reh(z)≥

 –r

(10)

Thus, due to the applications of these inequalities, we have

Re

h(z) + γzh

_(z)

α(p+μ)q(z)

≥Reh(z) – γ|zh

_(z)|

α(p+μ)|q(z)| ≥Reh(z)

 – γr

α(p+μ)( –r)

=Reh(z)

α(p+μ)( –r)_{– }_γ_r α(p+μ)( –r)

.

For|z|<r, whereris given in (.), the inequality above is positive. Sharpness of the

result follows by takingh(z) =+z

–z. Hence from (.), we have the required result.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

MR and SNM jointly discussed and presented the ideas of this article. MR made the text ﬁle and corresponded it to the journal. Both authors read and approved the ﬁnal manuscript.

Author details

1_{Department of Mathematics, GC University, Faisalabad, Pakistan.}2_{Department of Mathematics, COMSATS Institute of}

Information Technology, Defence Road, Lahore, Pakistan.

Received: 15 February 2013 Accepted: 6 September 2013 Published:07 Nov 2013

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