New Classes of Analytic Functions Involving Generalized Noor Integral Operator

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Volume 2008, Article ID 390435,14pages doi:10.1155/2008/390435

Research Article

New Classes of Analytic Functions Involving

Generalized Noor Integral Operator

Rabha W. Ibrahim and Maslina Darus

School of Mathematical Sciences, Faculty of Science and Technology, Universiti Kebangsaan Malaysia, Selangor Darul Ehsan, Bangi 43600, Malaysia

Correspondence should be addressed to Maslina Darus,[email protected]

Received 22 March 2008; Accepted 25 April 2008

Recommended by Jozsef Szabados

The present article investigates new classes of functions involving generalized Noor integral operator. Some properties of these functions are studied including characterization and distortion theorems. Moreover, we illustrate suﬃcient conditions for subordination and superordination for analytic functions.

Copyrightq2008 R. W. Ibrahim and M. Darus. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

1. Introduction and preliminaries

LetHbe the class of functions analytic inUand letHa, nbe the subclass ofHconsisting of functions of the formfz aanznan1zn1· · ·.LetAbe the subclass ofHconsisting of functions of the formfz za2z2· · · .

Denote byDα_:_{A → A}_{the operator defined by}

Dα: z

1−zα1∗fz, α >−1, 1.1

where∗refers to the Hadamard product or convolution. Then implies that

Dnfz z

zn−1_f_zn

n! , n∈N0N∪ {0}. 1.2

We note that D0_f_z _f_z_and _D_f_z _zf_z_. _{The operator}_Dn_f _{is called Ruscheweyh}

derivative ofnth order off.Noor1,2defined and studied an integral operatorIn : A → A

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Letfnz z/1−zn1, n∈N0, and letfn−1be defined such that

fnz∗fn−1z ₁₋z_z. 1.3

Then

Infz fn−1z∗fz

z

1−zn1

−1

∗fz. 1.4

Note thatI0fz zfzandI1fz fz.The operatorInis called the Noor Integral ofnth

order off.Using1.3,1.4, and a well-known identity forDn_f,_{we have}

n1Infz−nIn1fz z

In1fz

. 1.5

Using hypergeometric functions2F1,1.4becomes

Infz

z2F11,1;n1, z

∗fz, 1.6

where2F1a, b;c, zis defined by

2F1a, b;c, z 1 ab c

z

1!

aa1bb1

cc1

z2

2! · · · . 1.7 For complex parameters

α1, . . . , αq

αj

Aj /0,−1,−2, . . .;j1, . . . , q ,

β1, . . . , βp

_β

j

Bj /0,−1,−2, . . .;j1, . . . , p ,

1.8

the Fox-Wright generalizationqΨpzof the hypergeometricqFp function bysee3–5

qΨp

⎡ ⎢ ⎢ ⎣

α1, A1

, . . . ,αq, Aq

;

z

β1, B1

, . . . ,βp, Bp

;

⎤ ⎥ ⎥ ⎦qΨp

αj, Aj

1,q;

βj, Bj

1,p;z

: ∞

n0

Γα1nA1

· · ·ΓαqnAq

Γβ1nB1

· · ·ΓβpnBp

zn

n!

∞

n0

q j1Γ

αjnAj

p

j1

βjnBj

zn

n!,

1.9

where Aj > 0 for allj 1, . . . , q, Bj > 0 for all j 1, . . . , p, and 1p_j1Bj −

q

j1Aj ≥ 0 for suitable values|z|.For special case, whenAj 1 for allj1, . . . , q,andBj 1 for allj1, . . . , p,

we have the following relationship:

qFp

α1, . . . , αq;β1, . . . , βp;z

ΩqΨp

αj,1

1,q;

βj,1

1,p;z

,

q≤p1; q, p∈N0N∪ {0}, z∈U,

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where

Ω: Γ

β1

· · ·Γβp

Γα1

· · ·Γαq

. 1.11

We introduce a functionzqΨpαj, Aj1,q;βj, Bj1,p;z−1given by

zqΨp

αj, Aj

1,q;

βj, Bj

1,p;z

∗zqΨp

αj, Aj

1,q;

βj, Bj

1,p;z

−1

z

1−zλ1 z

∞

n2

λ1_n₋₁

n−1! z

n_, _{λ >}₋₁_, 1.12

and obtain the following linear operator:

Iλ

αj, Aj

1,q;

βj, Bj

1,p

fz zqΨp

αj, Aj

1,q;

βj, Bj

1,p;z

−1_∗

fz, 1.13

wheref ∈ A, z∈U, and

zqΨp

αj, Aj

1,q;

βj, Bj

1,p;z

−1 z

∞

n2

p j1Γ

βj n−1Bj

q

j1Γ

αj n−1Aj

λ1_n₋₁zn. 1.14

For some computation, we have

Iλ

αj, Aj

1,q;

βj, Bj

1,p

fz z

∞

n2

p j1Γ

βj n−1Bj

q

j1Γ

αj n−1Aj

λ1_n₋₁anzn, 1.15

whereanis the Pochhammer symbol defined by

a_n Γ_Γan

a

⎧ ⎨ ⎩

1, n0

aa1· · ·an−1, n{1,2, . . .}. 1.16

From1.15we have the following result.

Lemma 1.1. Letfz∈ Afor allz∈Uthen

iI01,11,1;1,1/n−11,pfz fz.

iiI11,11,1;1,1/n−11,pfz zfz.

iiizIλαj, Aj1,q;βj, Bj1,pfz λ1Iλ1αj, Aj1,q;βj, Bj1,pfz−λIλαj, Aj1,q;

βj, Bj1,pfz.

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Definition 1.2. Letfz∈ Athenfz∈Sμ_λαj, Aj1,q;βj, Bj1,pif and only if

R

zIλ

αj, Aj

1,q;

βj, Bj

1,pfz

Iλ

αj, Aj

1,q;

βj, Bj

1,p

fz

> μ, 0≤μ <1, z∈U. 1.17

Definition 1.3. Letfz∈ Athenfz∈Cμ_λαj, Aj1,q;βj, Bj1,pif and only if

R

zIλ

αj, Aj

1,q;

βj, Bj

1,p

fz

Iλ

αj, Aj

1,q;

βj, Bj

1,p

fz

> μ, 0≤μ <1, z∈U. 1.18

LetFandGbe analytic functions in the unit diskU.The functionFissubordinatetoG,

writtenF ≺ G,ifGis univalent,F0 G0andFU ⊂GU.Or given two functionsFz

andGz,which are analytic inU,the functionFzis said to be subordination toGzinUif there exists a functionhz,analytic inUwith

h0 0, |hz|<1 ∀z∈U, 1.19

such that

Fz Ghz ∀z∈U. 1.20

Let φ : C2→C and let h be univalent in U. If p is analytic in U and satisfies the differential subordinationφpz, zpz ≺ hzthenpis called a solution of the differential subordination. The univalent functionqis called a dominant of the solutions of the differential subordination,p ≺ q. Ifp andφpz, zpzare univalent in Uand satisfy the differential superordination hz ≺ φpz, zpz then p is called a solution of the differential superordination. An analytic functionqis called subordinant of the solution of the differential superordination ifq ≺p.LetΦbe an analytic function in a domain containingfU, Φ0 0 andΦ0>0.

The functionf∈ Ais calledΦ—like if

R

zfz

Φfz

>0, z∈U. 1.21

This concept was introduced by Brickman 6 and established that a function f ∈ A is univalent if and only iffisΦ—like for someΦ.

Definition 1.4. LetΦ be analytic function in a domain containingfU, Φ0 0, Φ0 1,

andΦω/0 forω∈fU−0.Letqzbe a fixed analytic function inU, q0 1.The function

f ∈ Ais calledΦ—like with respect toqif

zfz

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In the present paper, we apply a method based on the diﬀerential subordination in order to obtain subordination results involving generalized Noor integral operator for a normalized analytic functionfzz∈U

q1z≺ zIλ

αj, Aj

1,q;

βj, Bj

1,p

fz

ΦIλ

αj, Aj1,q;

βj, Bj

1,p

fz ≺q2z. 1.23

In order to prove our subordination and superordination results, we need to the following lemmas in the sequel.

Definition 1.5see7. Denote byQthe set of all functionsfzthat are analytic and injective onU−Ef, where Ef : {ζ ∈ ∂U : lim_z→ζfz ∞} and are such thatfζ/0 forζ ∈ ∂U−Ef.

Lemma 1.6 see 8. Let qz be univalent in the unit disk U and θ and let φ be analytic in a

domain D containingqUwith φw/0, whenw ∈ qU. SetQz : zqzφqz, hz :

θqz Qz.Suppose that

1Qzis starlike univalent inU,

2Rzhz/Qz>0 forz∈U.

If

θpzzpzφpz≺θqzzqzφqz, 1.24

then

pz≺qz, 1.25

andqzis the best dominant.

Lemma 1.79. Let qzbe convex univalent in the unit diskUand let ϑ andϕbe analytic in a

domainDcontainingqU.Suppose that

1zqzϕqzis starlike univalent inU,

2R{ϑqz/ϕqz}>0forz∈U.

Ifpz∈ Hq0,1∩Q,withpU⊆Dandϑpz zpzϕzbeing univalent inUand

ϑqzzqzϕqz≺ϑpzzpzϕpz, 1.26

then

qz≺pz, 1.27

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2. Characterization properties and distortion theorems

In this section, we investigate the characterization properties for the function fz ∈ A

to belong to the classes Sμ_λαj, Aj1,q;βj, Bj1,p and C μ

λαj, Aj1,q;βj, Bj1,p by obtaining

the coeﬃcient bounds. Further, we prove the distortion theorems when fz ∈ Sμ_λαj, Aj1,q;βj, Bj1,pandfz∈C

μ

λαj, Aj1,q;βj, Bj1,p.

Theorem 2.1. Letfz∈ A.Thenfz∈Sμ_λαj, Aj1,q;βj, Bj1,pif and only if

∞

n2

Hn−1anμλ1n−1−

λ1_n−λ_n≤1−μ, 0≤μ <1, 2.1

where

Hn−1:

p j1Γ

βj n−1Bj

q

j1Γ

αj n−1Aj

. 2.2

Proof. Suppose that2.1holds. Then by usingLemma 1.1and forz∈U,we have

R

zIλ

αj, Aj

1,q;

βj, Bj

1,p

fz Iλ

αj, Aj

1,q;

βj, Bj

1,p

fz

≤z

Iλ

αj, Aj

1,q;

βj, Bj

1,pfz

Iλ

αj, Aj

1,q;

βj, Bj

1,p

fz

≤ 1

_∞

n2Hn−1anλ1n−λn

1∞_n₂Hn−1anλ1n−1 .

2.3

This last expression is greater than μ, if 2.1 holds this implies that fz ∈ Sμ_λαj, Aj1,q;βj, Bj1,p. On the other hand, assume thatfz∈S

μ

λαj, Aj1,q;βj, Bj1,pthen

R

zIλ

αj, Aj

1,q;

βj, Bj

1,p

fz Iλ

αj, Aj

1,q;

βj, Bj

1,p

fz

> μ, 2.4

butR{z} ≤ |z|then

zIλ

αj, Aj

1,q;

βj, Bj

1,p

fz Iλ

αj, Aj

1,q;

βj, Bj

1,p

fz

> μ. 2.5

By a computation, we obtain2.1.

Corollary 2.2. Let the functionfzbelong to the classSμ_λαj, Aj1,q;βj, Bj1,p.Then

_a_n_≤ 1−μ Hn−1μλ1_n−1−

λ1_n−λ_n, 0≤μ <1, 2.6

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Theorem 2.3. Letfz∈ A.Thenfz∈C_λμαj, Aj1,q;βj, Bj1,pif and only if

∞

n2

nHn−1anμλ1n−1−

λ1_n−λ_n≤1−μ, 0≤μ <1, 2.7

whereHn−1is defined in2.2.

Corollary 2.4. Let the functionfzbelong to the classC_λμαj, Aj1,q;βj, Bj1,p.Then

_a_n_≤ 1−μ nHn−1μλ1_n−1−

λ1_n−λ_n, 0≤μ <1, 2.8

whereHn−1is defined in2.2.

Theorem 2.5. Letfz∈Sμ_λαj, Aj1,q;βj, Bj1,p,then

_f_z_{≥ |}_z_{| −} 1−μ H1μλ11−

λ1₂−λ2 |z|2,

_f_z_{≤ |}_z_| 1−μ H1μλ11−

λ1₂−λ2 |z|2,

2.9

forz∈UwhereHn−1is defined in2.2.

Proof. Iffz∈Sμ_λαj, Aj1,q;βj, Bj1,pthen in view ofTheorem 2.1, we have

H1μλ11−

λ12−λ2 ∞

n2

_a_n_≤∞

n2

Hn−1anμλ1n−1−

λ1_n−λn

≤1−μ.

2.10

This yields

∞

n2

_a_n_≤ 1−μ H1μλ11−

λ1₂−λ2

. 2.11

Now

_f_z_{≥ |}_z_{| − |}_z_|2∞

n2

_a_n

≥ |z| − 1−μ H1μλ11−

λ1₂−λ2 |z|2.

2.12

Also,

_f_z_{≤ |}_z_| 1−μ H1μλ11−

λ1₂−λ2

|z|2. 2.13

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Corollary 2.6. Under the hypothesis ofTheorem 2.5,fzis included in a disk with its center at the origin and radiusrgiven by

r1 1−μ

H1μλ11−

λ1₂−λ2

. 2.14

In the same way, we can prove the following result.

Theorem 2.7. Letfz∈Cμ_λαj, Aj1,q;βj, Bj1,pthen

_f_z_{≥ |}_z_{| −} 1−μ

2H1μλ11−

λ12−λ2 |z|2,

_f_z_{≤ |}_z_| 1−μ

2H1μλ11−

λ1₂−λ2 |z|2,

2.15

forz∈UwhereHn−1is defined in2.2.

Corollary 2.8. Under the hypothesis ofTheorem 2.7,fzis included in a disk with its center at the

origin and radiusrgiven by

r1 1−μ

2H1μλ11−

λ1₂−λ2

. 2.16

We next study some properties of the classesSμ_λαj, Aj1,q;βj, Bj1,pandCμλαj, Aj1,q;

βj, Bj1,p.

Theorem 2.9. Letλ >−1and0≤μ1< μ2<1.Then

Sμ2

λ

αj, Aj

1,q;

βj, Bj

1,p

⊂Sμ1

λ

αj, Aj

1,q;

βj, Bj

1,p

. 2.17

Proof. By usingTheorem 2.1.

Theorem 2.10. Let−1< λ1≤λ2and0≤μ <1.Then

Sμ_λ₁αj, Aj

1,q;

βj, Bj

1,p

⊇Sμ_λ₂αj, Aj

1,q;

βj, Bj

1,p

. 2.18

Theorem 2.11. Letλ >−1and0≤μ1< μ2<1.Then

Cμ2

λ

αj, Aj

1,q;

βj, Bj

1,p

⊂Cμ1

λ

αj, Aj

1,q;

βj, Bj

1,p

. 2.19

Theorem 2.12. Let−1< λ1≤λ2and0≤μ <1.Then

C_λμ₁αj, Aj

1,q;

βj, Bj

1,p

⊇Cμ_λ₂αj, Aj

1,q;

βj, Bj

1,p

. 2.20

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3. Sandwich results

By making use of Lemmas 1.6 and 1.7, we prove the following subordination and superordination results.

Theorem 3.1. Letqz/0be univalent inUsuch thatzqz/qzis starlike univalent inUand

R

1 α

γqz

zqz qz −

zqz qz

>0, α, γ ∈C, γ /0. 3.1

Iff ∈ Asatisfies the subordination

α

zIλ

αj, Aj

1,q;

βj, Bj

1,p

fz

ΦIλ

αj, Aj

1,q;

βj, Bj

1,p

fz

γ

1z

Iλ

αj, Aj

1,q;

βj, Bj

1,p

fz

Iλ

αj, Aj

1,q;

βj, Bj

1,p

fz − zΦIλ

αj, Aj

1,q;

βj, Bj

1,p

fz

ΦIλ

αj, Aj

1,q;

βj, Bj

1,p

fz

≺αqz γzq

_z

qz ,

3.2

then

zIλ

αj, Aj

1,q;

βj, Bj

1,p

fz

ΦIλ

αj, Aj

1,q;

βj, Bj

1,p

fz ≺qz, 3.3

Proof. Our aim is to applyLemma 1.6. Setting

pz: z

Iλ

αj, Aj

1,q;

βj, Bj

1,p

fz

ΦIλ

αj, Aj

1,q;

βj, Bj

1,p

fz. 3.4

Computation shows that

zpz

pz 1

zIλ

αj, Aj

1,q;

βj, Bj

1,p

fz

Iλ

αj, Aj

1,q;

βj, Bj

1,p

fz −

zΦIλ

αj, Aj

1,q;

βj, Bj

1,p

fz

ΦIλ

αj, Aj

1,q;

βj, Bj

1,p

fz , 3.5

which yields the following subordination:

αpz γzp

_z

pz ≺αqz

γzqz

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By setting

θω:αω, φω: γ

ω, γ /0, 3.7

it can be easily observed that θω is analytic in C and φω is analytic in C\ {0} and that

φω/0 whenω∈C\{0}.Also, by letting

Qz zqzφqzγzq

_z

qz,

hz θqzQz αqz γzq

_z

qz,

3.8

we find thatQzis starlike univalent inUand that

R

zhz Qz

1 α

γqz

zqz qz −

zqz qz

>0. 3.9

Then the relation3.3follows by an application ofLemma 1.6.

Corollary 3.2. Let the assumptions ofTheorem 2.1hold. Then the subordination

α−γ

zIλ

αj, Aj

1,q;

βj, Bj

1,p

fz

Iλ

αj, Aj

1,q;

βj, Bj

1,p

fz

γ

1z

Iλ

αj, Aj

1,q;

βj, Bj

1,p

fz

Iλ

αj, Aj

1,q;

βj, Bj

1,p

fz

≺αqz γzq

_z

qz ,

3.10

implies

zIλ

αj, Aj

1,q;

βj, Bj

1,pfz

Iλ

αj, Aj

1,q;

βj, Bj

1,p

fz ≺qz, 3.11

Proof. By lettingΦω:ω.

Corollary 3.3. Iff ∈ Aand assume that3.1holds then

1 z

Iλ

αj, Aj

1,q;

βj, Bj

1,p

fz

Iλ

αj, Aj

1,q;

βj, Bj

1,p

fz ≺

1Az

1Bz

A−Bz

1Az1Bz 3.12

implies

zIλ

αj, Aj

1,q;

βj, Bj

1,p

fz

Iλ

αj, Aj

1,q;

βj, Bj

1,p

fz ≺

1Az

1Bz, −1≤B < A≤1, 3.13

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Proof. By settingΦω:ω, αγ 1,andqz: 1Az/1Bz,where−1≤B < A≤1.

1z

Iλ

αj, Aj

1,q;

βj, Bj

1,p

fz

Iλ

αj, Aj

1,q;

βj, Bj

1,p

fz ≺

1z

1−z

2z

1−z2 3.14

implies

zIλ

αj, Aj

1,q;

βj, Bj

1,p

fz Iλ

αj, Aj

1,q;

βj, Bj

1,p

fz ≺

1z

1−z, 3.15

and1z/1−zis the best dominant.

Proof. By settingΦω:ω, αγ 1,andqz: 1z/1−z.

1z

Iλ

αj, Aj

1,q;

βj, Bj

1,p

fz

Iλ

αj, Aj

1,q;

βj, Bj

1,p

fz ≺e

Az_Az _3.16

implies

zIλ

αj, Aj

1,q;

βj, Bj

1,p

fz Iλ

αj, Aj1,q;

βj, Bj

1,p

fz ≺e

Az_, _3.17

andeAzis the best dominant.

Proof. By settingΦω:ω, αγ 1,andqz:eAz, |A|< π.

Theorem 3.6. Letqz/0be convex univalent in the unit diskU.Suppose that

R

α γqz

>0, α, γ ∈Cfor z∈U, 3.18

and that zqz/qz is starlike univalent in U. If zIλαj, Aj1,q;βj, Bj1,pfz/ΦIλαj, Aj1,q;βj, Bj1,pfz∈ Hq0,1∩Qwheref∈ A,

α

zIλ

αj, Aj

1,q;

βj, Bj

1,p

fz

ΦIλ

αj, Aj

1,q;

βj, Bj

1,p

fz

γ

1z

Iλ

αj, Aj

1,q;

βj, Bj

1,p

fz

Iλ

αj, Aj

1,q;

βj, Bj

1,p

fz −

zΦIλ

αj, Aj

1,q;

βj, Bj

1,p

fz

ΦIλ

αj, Aj

1,q;

βj, Bj

1,p

fz

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is univalent inUand the subordination

αqz γzq

_z

qz ≺α

zIλ

αj, Aj

1,q;

βj, Bj

1,p

fz

ΦIλ

αj, Aj

1,q;

βj, Bj

1,p

fz

γ

1z

Iλ

αj, Aj

1,q;

βj, Bj

1,p

fz

Iλ

αj, Aj

1,q;

βj, Bj

1,p

fz

−zΦ _I

λ

αj, Aj1,q;

βj, Bj

1,p

fz

ΦIλ

αj, Aj1,q;

βj, Bj

1,p

fz

3.20

holds, then

qz≺ z

Iλ

αj, Aj

1,q;

βj, Bj

1,p

fz

ΦIλ

αj, Aj

1,q;

βj, Bj

1,p

fz 3.21

andqis the best subordinant.

Proof. Our aim is to applyLemma 1.7. Setting

pz: z

Iλ

αj, Aj

1,q;

βj, Bj

1,p

fz

ΦIλ

αj, Aj

1,q;

βj, Bj

1,p

fz. 3.22

Computation shows that

zpz pz 1

zIλ

αj, Aj

1,q;

βj, Bj

1,p

fz

Iλ

αj, Aj

1,q;

βj, Bj

1,p

fz − zΦIλ

αj, Aj

1,q;

βj, Bj

1,p

fz

ΦIλ

αj, Aj

1,q;

βj, Bj

1,p

fz , 3.23

which yields the following subordination

αqz γzq

_z

qz ≺αpz

γzpz

pz , α, γ ∈C. 3.24

By setting

ϑω:αω, ϕω: γ

ω, γ /0, 3.25

it can be easily observed that θωis analytic in Candφωis analytic inC\ {0}, and that

φω/0 whenω∈C\ {0}.Also, we obtain

R

_ϑ_q_z

ϕqz

R

α γqz

>0. 3.26

Then3.21follows by an application ofLemma 1.7.

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Theorem 3.7. Letq1z/0, q2z/0be convex univalent in the unit diskUsatisfying3.18and

3.1, respectively. Suppose thatzq_iz/qiz, i1,2, is starlike univalent inU.If

zIλ

αj, Aj

1,q;

βj, Bj

1,p

fz

ΦIλ

αj, Aj

1,q;

βj, Bj

1,p

fz ∈ H

q0,1∩Q, 3.27

wheref ∈ A,

α

zIλ

αj, Aj

1,q;

βj, Bj

1,p

fz

ΦIλ

αj, Aj

1,q;

βj, Bj

1,p

fz

γ

1z

Iλ

αj, Aj

1,q;

βj, Bj

1,p

fz

Iλ

αj, Aj

1,q;

βj, Bj

1,p

fz −

zΦIλ

αj, Aj

1,q;

βj, Bj

1,p

fz

ΦIλ

αj, Aj

1,q;

βj, Bj

1,p

fz

3.28

is univalent inU, and the subordination

αq1z

γzq₁z q1z ≺α

zIλ

αj, Aj

1,q;

βj, Bj

1,p

fz

ΦIλ

αj, Aj1,q;

βj, Bj

1,p

fz

γ

1z

Iλ

αj, Aj

1,q;

βj, Bj

1,p

fz

Iλ

αj, Aj

1,q;

βj, Bj

1,p

fz

−zΦ _I

λ

αj, Aj

1,q;

βj, Bj

1,p

fz

ΦIλ

αj, Aj

1,q;

βj, Bj

1,p

fz

≺αq2z

γzq₂z q2z

3.29

holds, then

q1z≺ zIλ

αj, Aj

1,q;

βj, Bj

1,p

fz

ΦIλ

αj, Aj

1,q;

βj, Bj

1,p

fz ≺q2z, 3.30

andq1zis the best subordinant andq2zis the best dominant.

Acknowledgment

The work presented here was supported by SAGA: STGL-012-2006, Academy of Sciences, Malaysia.

References

1 K. I. Noor, “On new classes of integral operators,”Journal of Natural Geometry, vol. 16, no. 1-2, pp. 71–80, 1999.

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