Some Properties of Certain Multivalent Analytic Functions Involving the Cătas Operator

(1)

Volume 2011, Article ID 752341,25pages doi:10.1155/2011/752341

Research Article

Some Properties of Certain Multivalent Analytic

Functions Involving the C ˘atas Operator

E. A. Elrifai, H. E. Darwish, and A. R. Ahmed

Department of Mathematics, Faculty of Science, University of Mansoura, Mansoura 35516, Egypt

Correspondence should be addressed to H. E. Darwish,[email protected]

Received 2 March 2011; Accepted 8 May 2011

Academic Editor: Marco Squassina

Copyrightq2011 E. A. Elrifai et al. 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.

We introduce a certain subclass of multivalent analytic functions by making use of the principle of subordination between these functions and C˘atas operator. Such results as subordination and superordination properties, convolution properties, inclusion relationships, distortion theorems, inequality properties, and suﬃcient conditions for multivalent starlikeness are provide. The results presented here would provide extensions of those given in earlier works. Several other new results are also obtained.

1. Introduction

LetApndenote the class of functions of the following form:

fz zp

∞

kn

apkzpk

p, n∈N:{1,2,3, . . .}, 1.1

which are analytic in the open unit diskU:{z:z∈Cand |z|<1}. For simplicity, we write

Ap1:Ap, A11 A. 1.2

A functionfz∈Apnis said to be in the classS∗p,nγofp-valent starlike functions of order

(2)

Re

_zf_z

fz

> γ 0≤γ < p;z∈U. 1.3

LetHa, nbe the class of analytic functions of the following form:

Fz aanznan1zn1· · · z∈U. 1.4

Letf, g∈Apn, wherefzis given by1.1andgzis defined by

gz zp

∞

kn

bpkzpk. 1.5

Then the Hadanard productor convolutionf∗gof the functionsfzandgzis defined by

f∗gz:zp

∞

kn

apkbpkzpk:

g∗fz. 1.6

We consider the following multiplier transformations.

Definition 1.1see1. Letfz∈Apn. Forp, n ∈N, δ, λ ≥ 0, ≥ 0, define the multiplier

transformationsIpδ, λ, onApnby the following infinite series:

Ipδ, λ, fz:zp

∞

kn

_p_λk

p

δ

akpzkp. 1.7

It is easily verified from1.7, that

pIpδ1, λ, fz p1−λ

Ipδ, λ, fz λz

Ipδ, λ, fz

. 1.8

It should be remarked that the class of multiplier transforms Ipδ, λ, is a generalization of several other linear operators considered, in earlier investigationssee2–

12.

Iffzis given by1.1, then we have

Ipδ, λ, fz

f∗ϕδ_p,λ,z, 1.9

where

ϕδ

p,λ,z zp

∞

kn

pλk

p

δ

(3)

In particular, we set

I1δ, λ, fz:Iδ, λ, fz. 1.11

For two functions fz and gz, analytic in U, we say that the function fz is subordinate togzin U, and writefz ≺ gz z ∈ Uif there exists a Schwarz function

wz, which is analytic in U with

w0 0, |wz|<1 z∈U 1.12

such that

fz gwz z∈U. 1.13

Indeed, it is known that

fz≺gz, z∈U ⇒f0 g0, fU⊂gU. 1.14

Furthermore, if the function g is univalent in U, then we have the following equivalence:

fz≺gz, z∈U⇐⇒f0 g0, fU⊂gU. 1.15

By making use of the linear operator Ipδ, λ, and the above-mentioned principle of subordination between analytic functions, we introduce and investigate the following subclass of the classApnofp-valent analytic functions.

Definition 1.2. A functionfz∈Apnis said to be in the class ßα,βp δ, λ, , n;A, Bif it satisfies

the following subordination condition:

1−α

Ipδ, λ, fz

zp

β

α

Ipδ1, λ, fz

Ipδ, λ, fz

zp

β

≺ 1Az

1Bz z∈U,

1.16

whereand throughout this paper unless otherwise mentionedthe parametersα, β, p, n, λ, δ, , A, andBare constrained as follows:

α∈C,Reβ>0, λ, ≥0, λ, ∈R, δ≥0, −1≤B≤1, A /B∈R, p, n∈N. 1.17

For simplicity, we write

ß1₁,β1,1,1; 1; 1,−1 ßβ. 1.18

(4)

If we setδ 0, λ p 1 in the class ßα,βp δ, λ, , n;A, B, which was studied by Liu13. In particular, Zhu14determined the suﬃcient conditions such that ßnα, β, A,0⊂

S∗_p,nρ.

C˘atas 1, 5, 15, Cho and Srivastava 6, Cho and Kim 7, and Kumar et al. 10

obtained many interesting results associated with the multiplier operator.

In the present paper, we aim at proving such results as subordination and super-ordination properties, convolution properties, inclusion relationships, distortion theorems, inequality properties, and suﬃcient conditions for multivalent starlikeness of the class

ßα,βp δ, λ, , n;A, B. The results presented here would provide extensions of those given in

earlier works. Several other new results are also obtained.

2. Preliminary Results

In order to establish our main results, we need the following definition and lemmas.

Definition 2.1 see 16. Denote by Q the set of all functions fz that are analytic and

injective onU−Ef, where

Ef

ε∈∂U: lim

z→εfz ∞

, 2.1

and such thatfε/0 forε∈∂U−Ef.

Lemma 2.2see17. Let the functionhbe analytic and univalent (convex) inUwithh0 1.

Suppose also that the functionkgiven by

kz 1cnzncn1zn1· · · 2.2

is analytic inU. If

kz zkz

ζ ≺hz Reζ>0; ζ /0; z∈U, 2.3

then

kz≺χz ζ

nz

−ζ/n

_z

0

tζ/n−1htdt≺hz z∈U, 2.4

andχzis the best dominant of2.3.

Lemma 2.3see18. Letqzbe a convex univalent function inUand letσ, η∈Cwith

Re

1zq

_z

qz

>max

0,−Re

σ η

(5)

If the functionpis analytic inUand

σpz ηzpz≺σqz ηzqz, 2.6

thenpz≺qzandqzis the best dominant.

Lemma 2.4see16. Letqbe convex univalent inUandk∈C. Further assume that Rek>0 if

pz∈H q0,1∩Q, 2.7

andpz kzpzis univalent inU, then

qz kzqz≺pz kzpz 2.8

implies thatqz≺pzandqzis the best subdominant.

Lemma 2.5Jach’s Lemma19. Letwzbe a noncostant analytic function inUwithw0 0.

If|w|attains its maximum value on the circle|z|r <1 atz0, then

z0wz0 kwz0, 2.9

wherek≥1 is a real number.

Lemma 2.6see20. LetFbe analytic and convex inU. Iffz, gz∈Aandfz, gz≺Fz;

then

λfz 1−λgz≺Fz 0≤λ≤1. 2.10

Lemma 2.7see21,22. Letk, v∈C. Suppose also that m is convex and univalent inUwith

m0 1, Rekmz v>0 z∈U. 2.11

Ifuis analytic inUwithu0 1, then the following subordination:

uz zu

_z

kuz v≺mz, z∈U 2.12

implies that

uz≺mz z∈U. 2.13

Lemma 2.8see23. Letfz 1∞_k₁akzkanalytic inUandgz 1∞k1bkzkbe analytic

(6)

Lemma 2.9see24. Letδ /0, δ∈R, v/δ >0,0≤ρ <1, p∈H1, n, andpz≺1kzk:

vM/nδv, where

M:Mn

δ, ν, ρ

1−ρ|δ|1nδ/v

₁₋_δ_ρδ₁ _nδ/v2. 2.14

Ifqz∈H1, nsatisfies the following subordination condition:

pz 1−δδ1−ρqz ρ≺1Mz, 2.15

then

Reqz>0 z∈U. 2.16

3. Main Results

We begin by presenting our first subordination property given by Theorem3.1below.

Theorem 3.1. Letfz∈ßα,βp δ, λ, ;n;A, Bwith Reα>0. Then

Ipδ, λ, fz

zp

β

≺

pβ

λnα

1

0

1Azu

1Bzuu

pβ/λnα−1_du_≺ 1Az

1Bz z∈U. 3.1

Proof. Define the functionPzby

Pz:

Ipδ, λ, fz

zp

β

z∈U. 3.2

ThenPzis analytic inUwithP0 1. By taking the derivatives in the both sides in equality

3.2and using1.8, we get

1−α

Ipδ, λ, fz

zp

β

α

Ipδ1, λ, fz

Ipδ, λ, fz

zp

β

Pz λαzPz

βp ≺

1Az

1Bz z∈U.

(7)

An application of Lemma2.2to3.3yields

Ipδ, λ, fz

zp

β

≺

pβ

λnα z

−pβ/λnα

z

0

tpβ/λnα−11At

1Btdt

ζ

n

₁

0

uζ/n−11Azu

1Bzudu≺

1Az

1Bz z∈U,

3.4

where

ζ

pβ

λα . 3.5

The proof of Theorem3.1is thus completed.

Theorem 3.2. Letqzbe univalent inU, 0/α∈C. Suppose also thatqzsatisfies

Re

1zq

_z

qz

>max

0,−Re

pβ

λα

. 3.6

Iffz∈Apnsatisfying the following subordination:

1−α

Ipδ, λ, fz

zp

β

α

Ipδ1, λ, fz

Ipδ, λ, fz

zp

β

≺qz αλzq

_z

pβ,

3.7

then

Ipδ, λ, fz

zp

β

≺qz, 3.8

andqzis the best dominant.

Proof. Let the functionPzbe defined by3.2. We know that3.3holds true. Combining

3.3and3.7, we find that

Pz λαzP

_z

pβ ≺qz

λαzqz

pβ. 3.9

(8)

Takingqz 1Az/1Bzin Theorem3.2, we get the following result.

Corollary 3.3. Letα∈Cand−1 ≤B < A ≤1. Suppose also that1Az/1Bzsatisfies the

condition3.6. Iffz∈Apnsatisfies the following subordination:

1−α

Ipδ, λ, fz

zp

β

α

Ipδ1, λ, fz

Ipδ, λ, fz

zp

β

≺ 1Az

1Bz

λαA−Bz

pβ1Bz2,

3.10

then

Ipδ, λ, fz

zp

β

≺ 1Az

1Bz, 3.11

and1Az/1Bzis the best dominant.

Iffzis subordinate toFz, thenFzis superordinate tofz. We now derive the following

superordination result for the class ßα,βp δ, λ, ;n;A, B.

Theorem 3.4. Letqzbe convex univalent inU, α∈C, with Reα>0. Also let

Ipδ, λ, fz

zp

β

∈H q0,1∩Q,

1−α

Ipδ, λ, fz

zp

β

α

Ipδ1, λ, fz

Ipδ, λ, fz

zp

β 3.12

be univalent inU. If

qz λαzq

_z

pβ ≺1−α

Ipδ, λ, fz

zp

β

α

Ipδ1, λ, fz

Ipδ, λ, fz

zp

β

,

3.13

then

qz≺

Ipδ, λ, fz

zp

β

, 3.14

(9)

Proof. Let the functionPzbe defined by3.2. Then

qz λαzq

_z

pβ ≺1−α

Ipδ, λ, fz

zp

β

α

Ipδ1, λ, fz

Ipδ, λ, fz

zp

β

Pz λαzP z

pβ.

3.15

An application of Lemma2.4yields the assertion of Theorem3.4.

Takingqz 1Az/1Bzin Theorem3.4, we get the following corollary.

Corollary 3.5. Letqzbe convex univalent inUand−1≤B < A≤1, α∈Cwith Reα>0. Also

let

Ipδ, λ, fz

zp

β

∈H q0,1∩Q,

1−α

Ipδ, λ, fz

zp

β

α

Ipδ1, λ, fz

Ipδ, λ, fz

zp

β 3.16

be univalent inU. If

1Az

1Bz

λαA−Bz

pβ1Bz2

≺1−α

Ipδ, λ, fz

zp

β

α

Ipδ1, λ, fz

Ipδ, λ, fz

zp

β

,

3.17

then

1Az

1Bz ≺

Ipδ, λ, fz

zp

β

, 3.18

and1Az/1Bzis best subdominant.

(10)

Corollary 3.6. Letq1zbe convex univalent and letq2zbe univalent inU,α∈C, Reα>0. Let

q2zsatisfies3.6. If

0/

Ipδ, λ, fz

zp

β

∈H q10,1

∩Q,

1−α

Ipδ, λ, fz

zp

β

α

Ipδ1, λ, fz

Ipδ, λ, fz

zp

β 3.19

is univalent inU, also

q1z

λαzq₁z

pβ ≺1−α

Ipδ, λ, fz

zp

β

α

Ipδ1, λ, fz

Ipδ, λ, fz

zp

β

≺q2z

λαzq₂z

pβ,

3.20

then

q1z≺

Ipδ, λ, fz

zp

β

≺q2z, 3.21

andq1z, q2zare, respectively, the best subordinate, and dominant.

Theorem 3.7. Letfz∈Apn, ξ∈C\ {0}, and 0≤γ <1. Also let the functionϕbe defined by

ϕz

zIpδ1, λ, fz/Ipδ, λ, fz

Ipδ, λ, fz/zp

β₋ 1

Ipδ1, λ, fz/Ipδ, λ, fz

Ipδ, λ, fz/zp

β₋

1 z∈U.

3.22

Ifϕsatisfies one of the following conditions:

Reϕz

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

< 1

|ξ|2Reξ Reξ>0, /

0 Reξ 0,

> 1

|ξ|2Reξ Reξ<0,

(11)

or

Iϕz

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

>− 1

|ξ|2Iξ Iξ>0, /

0 Iξ 0,

<− 1

|ξ|2Iξ Iξ<0,

3.24

then

⎛ ⎝

Ipδ1, λ, fz

Ipδ, λ, fz

zp

β

−1

⎞ ⎠

ξ

<1−γ. 3.25

Proof. We define the functionφzby

⎛ ⎝

Ipδ1, λ, fz

Ipδ, λ, fz

zp

β

−1

⎞ ⎠

ξ

1−γφz 0≤γ <1;ξ∈C\ {0};z∈U.

3.26

It is easy to see that the functionφis analytic inUwithφ0 0.

Diﬀerentiating both sides of3.26with respect tozlogarithmically, we get

zφz

φz

ξ

zIpδ1, λ, fz/Ipδ, λ, fz

Ipδ, λ, fz/zp

β₋ 1

Ipδ1, λ, fz/Ipδ, λ, fz

Ipδ, λ, fz/zp

β₋

1 z∈U;ξ∈C\ {0}.

3.27

We now consider the functionϕdefined by

ϕ: ξ

|ξ|2

zφz

φz z∈U, ξ∈C\ {0}. 3.28

Assume that there exists a pointz0∈Usuch that

max

|z|≤|z0|

_φ_z_φ_z0₁_, _3.29

by Lemma2.5, we know that

(12)

If follows from3.28and3.30that

Reϕz0

Re

ξ

|ξ|2

zφz0

φz0

k

|ξ|2Re

ξ

k |ξ|2Reξ

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

≥ 1

|ξ|2 Reξ Reξ>0,

0 Reξ 0

≤ 1

|ξ|2 Reξ Reξ<0,

3.31

Iϕz0

I

ξ

|ξ|2

zφz0

φz0

k

|ξ|2I

ξ

−k |ξ|2Iξ

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

≤ − 1

|ξ|2Iξ Iξ>0,

0 Iξ 0,

≥ − 1

|ξ|2Iξ Iξ<0.

3.32

But the inequalities in3.31and3.32contradict, respectively, the inequalities in3.23and

3.24. Therefore, we can conclude that

_φ_z_<₁ _z_∈_U_, _3.33

which implies that

⎛ ⎝

Ipδ1, λ, fz

Ipδ, λ, fz

zp

β

−1

⎞ ⎠

ξ

1−γφz<1−γ. 3.34

We thus complete the proof of Theorem3.7.

From Theorem3.7, we easily get the following result for the class ßβ of Bazileviˇc functions of typeβ.

Corollary 3.8. Letfz∈A, δ0, pλξ1, andγ 0. Also let the functionϕbe defined

by3.22. Ifϕsatisfies one of the following conditions:

Reϕz<1 or Iϕz/0, 3.35

(13)

Theorem 3.9. Let Reα > 0, β > 0, and fz ∈ ßp0,βδ, λ, ;n; 1−2ρ,−1 0 ≤ ρ < 1. Then

fz∈ßα,βp δ, λ, ;n,1−2ρ,−1for|z|< Rα, β, , λ, p, where

Rα, β, , λ, p −λα

λ2_α2_p2_β2

ρβ . 3.36

The boundRα, β, , λ, pis the best possible.

Proof. Suppose that

Ipδ, λ, fz

zp

β

ρ1−ρhz z∈U; 0≤ρ <1, 3.37

wherehis analytic and has a positive real part inU. By taking the derivatives in the both sides in equality3.37and using1.9, we get

Re

⎛ ⎝1−α

Ipδ, λ, fz

zp

β

α

Ipδ1, λ, fz

Ipδ, λ, fz

zp

β⎞

⎠₋ρ

1−ρRe

hz λαzh

_z

pβ

≥1−ρRe

hz−λα|zhz|

pβ

.

3.38

By making use of the following well-known estimatesee25:

zhz≤ 2r

1−r2 Rehz |z|r <1 3.39

in3.38, we obtain that

Re

⎡ ⎣1−α

Ipδ, λ, fz

zp

β

α

Ipδ1, λ, fz

Ipδ, λ, fz

zp

β

−ρ

⎤ ⎦

1−ρ

1− 2λαr

pβ1−r2

Rehz>0

3.40

(14)

To show that the boundRα, β, , λ, pis the best possible, we consider the function

fz∈Apndefined by

Ipδ, λ, fz

zp

β

ρ1−ρ1z

1−z z∈U. 3.41

By noting that

Re

⎛ ⎝1−α

Ipδ, λ, fz

zp

β

α

Ipδ1, λ, fz

Ipδ, λ, fz

zp

β⎞

⎠₋ρ

1−ρRe

1z

1−z

2λα

pβ

z

1−z2

0

3.42

forz Rα, β, , λ, p, we conclude that the bound is the best possible. Theorem3.9is thus

proved.

fz

zp

1Awz

1Bwz

1/β

∗

zp

∞

kn

_p_λk

p

−δ

zkp

, 3.43

wherewzis analytic inUwithw0 0 and|wz|<1 z∈U.

Proof. Suppose thatfz∈ßα,βp δ, λ, ;n;A, B. It follows from3.1that

Ipδ, λ, fz

zp

β

1Awz

1Bwz, 3.44

wherewzis analytic inUwithw0 0 and|wz|<1z∈U. By virtue of3.44, we easily find that

Ipδ, λ, fz zp

1Awz

1Bwz

1/β

. 3.45

Combining1.10,1.16, and3.45, we have

zp

∞

kn

pλk

p

δ

zkp

∗fz zp

1Awz

1Bwz

1/β

. 3.46

(15)

1

z

#

1Biθ1/β

zp

∞

kn

_p_λk

p

−δ

zkp

∗fz−zp1Aeiθ1/β

$

/

0 z∈U; 0< θ <2π.

3.47

Proof. Suppose thatfz∈ßα,βp δ, λ, ;n;A, Bwith Reα>0. We know that3.1holds true,

which implies that

Ipδ, λ, fz

zp

β

/

1Aeiθ

1Beiθ z∈U; 0< θ <2π. 3.48

It is easy to see that the condition3.48can be written as follows:

1

z

Ipδ, λ, fz

1Beiθ1/β−zp1Aiθ1/β

/

0 z∈U; 0< θ <2π. 3.49

Combining1.9,1.10, and3.49, we easily get the convolution property3.47asserted by Theorem3.11.

Theorem 3.12. Letα2≥α1≥0 and−1≤B1≤B2 < A2< A1≤1. Then

ßα2,β

p δ, λ, ;n;A2, B2⊂ßpα1,βδ, λ, ;n;A1, B1. 3.50

Proof. Suppose thatfz∈βα2,β

p δ, λ, ;n;A2, B2. We know that

1−α2

Ipδ, λ, fz

zp

β

α2

Ipδ1, λ, fz

Ipδ, λ, fz

zp

β

≺ 1A2z

1B2z .

3.51

Since−1≤B1≤B2< A2≤A1≤1, we easily find that

1−α2

Ipδ, λ, fz

zp

β

α2

Ipδ1, λ, fz

Ipδ, λ, fz

zp

β

≺ 1A2z

1B2z ≺

1A1z

1B1z

3.52

that is,fz∈ßα2,β

(16)

Ifα2 > α1 ≥ 0, by Theorem3.1and3.52, we know thatfz∈ß0p,βδ, λ, ;n;A1, B1,

that is,

Ipδ, λ, fz

zp

β

≺ 1A1z

1B1z

. 3.53

At the same time, we have

1−α1

Ipδ, λ, fz

zp

β

α1

Ipδ1, λ, fz

Ipδ, λ, fz

zp

β

1−α1

α2

_I

pδ, λ, fz

zp

β

α1 α2

⎡ ⎣1−α2

Ipδ, λ, fz

zp

β

α2

Ipδ1, λ, fz

Ipδ, λ, fz

zp

β⎤

⎦.

3.54

Moreover, since 0≤α1/α2<1 andh1z: 1A1z/1B1z, is analytic and convex in

U. Combining3.52–3.54and Lemma2.6, we find that

1−α1

Ipδ, λ, fz

zp

β

α1

Ipδ1, λ, fz

Ipδ, λ, fz

zp

β

≺ 1A1z

1B1z,

3.55

that is,fz∈ßα1,β

p δ, λ, ;n;A1, B1, which implies that the assertion3.50of Theorem3.12

holds.

Letρdenote the class of functions of the following form:

tz 1

∞

kn

tkzk n∈N, 3.56

which are analytic and convex inUand satisfy the following condition:

Retz>0 z∈U. 3.57

(17)

S∗_p,nμ, φ:

f∈Apn:

1

p−μ

_zf_z

fz −μ

≺φzφ∈ρ; 0≤μ < p;z∈U,

Cp,n

μ;φ

fz∈Apn:

1

p−μ

1zf

_z

fz −μ

≺φzφ∈ρ; 0≤μ < p;z∈U.

3.58

Next, by using the operator defined by 1.7, we define the following two subclasses

Sp,nδ, λ, ;μ;φandCp,nδ, λ, ;μ;φof the classApn:

Sp,n

δ, λ, ;μ;φ:%fz∈Apn:Ipδ, λ, fz∈S∗p,n

μ;φ&,

Cp,n

δ, λ, ;μ;φ:'fz∈Apn:Ipδ, λ, fz∈Cp,n

μ;φ(.

3.59

Clearly, we know that

f∈Cp,n

δ, λ, ;μ;φ⇐⇒ zf

_z

p ∈Sp,n

δ, λ, ;μ;φ. 3.60

We now derive some inclusion relationships for the classes Sp,nδ, λ, ;μ;φ and

Cp,nδ, λ, ;μ;φ, by similarly applying the method of proof of Proposition 1 obtained by Cho

et al.26and Wang et al.27.

Theorem 3.13. Let 0≤μ < p, λ >−p, andφ∈ρwith

Reφz>max

0,−δμ

p−μ,−

λμ−p

p−μ

z∈U. 3.61

Then

Sp,n

δ1, λ, ;μ;φ⊂Sp,n

δ, λ, ;μ;φ⊂Sp,n

δ, λ1, ;μ;φ. 3.62

Theorem 3.14. Let 0≤μ < p, λ >−pandφ∈ρwith3.61holds. Then

Cp,n

δ1, λ, ;n;μ;φ⊂Cp,n

δ, λ, ;μ;φ⊂Cp,n

δ, λ1, ;μ;φ. 3.63

Proof. By virtue of3.60and Theorem3.13, we observe that

fz∈Cp,n

δ1, λ, ;μ;φ⇐⇒Ipδ1, λ, fz∈Cp,n

μ;φ

⇐⇒z

Ipδ1, λ, fz

p ∈S

∗

p,n

μ;φ

⇐⇒Ipδ1, λ,

_zf_z

p

∈S∗_p,nμ;φ

⇒ zfz

p ∈Sp,n

(18)

⇐⇒zfz

p ∈Sp,n

δ, λ, ;μ;φ

⇐⇒Ipδ, λ,

_zf_z

p

∈S∗_p,nμ, φ

⇐⇒z

Ipδ, λ, fz

p ∈S

∗

p,n

μ;φ

⇐⇒Ipδ, λ, fz∈Cp,n

μ;φ

⇐⇒fz∈Cp,n

δ, λ, ;μ;φ,

fz∈Cp,n

δ, λ, ;μ;φ⇐⇒zf

_z

p ∈Sp,n

δ, λ, ;μ;φ

⇒ zfz

p ∈Sp,n

δ, λ1, ;μ;φ

⇐⇒z

Ipδ, λ1, fz

p ∈S

∗

p,n

μ;φ

⇐⇒Ipδ, λ1, fz∈Cp,n

μ;φ

⇐⇒fz∈Cp,n

δ, λ1, ;μ;φ.

3.64

From3.64, we conclude that the assertion of Theorem3.14holds true.

Takingφz 1Az/1Bzin Theorems3.13and3.14, we get the following results.

Corollary 3.15. Let 0≤μ < p, λ >−p, and−1≤B < A≤1. Then

Sp,n

δ1, λ, ;μ;1Az

1Bz

⊂Sp,n

δ, λ, ;μ,1Az

1Bz

⊂Sp,n

δ, λ1, ;μ;1Az

1Bz

,

Cp,n

δ1, λ, ;μ;1Az

1Bz

⊂Cp,n

δ, λ, ;μ,1Az

1Bz

⊂Cp,n

δ, λ1, ;μ;1Az

1Bz

.

3.65

Theorem 3.16. Letfz∈ßα,βp δ, λ, ;n;A, Bwith Reα>0 and−1≤B < A≤1. Then

pβ

λnα

₁

0

1−Au

1−Buu

pβ/λnα−1_{du < Re}

Ipδ, λ, fz

zp

β

<

pβ

λnα

₁

0

1Au

1Buu

pβ/λnα−1_du.

(19)

The extremal function of 3.66is defined by

Ipδ, λ, Fα,β,A,Bz zp

pβ λαn

1

0

1Azu

1Bzuu

pβ/λnα−1_du

1/β

. 3.67

Proof. Letfz∈ ßα,βp δ, λ, ;n;A, Bwith Reα> 0. From Theorem3.1, we know that3.1

holds, which implies that

Re

Ipδ, λ, fz

zp

β

<sup

z∈U Re

pβ

λαn

₁

0

1Azu

1Bzuu

pβ/λnα−1_du

≤

pβ

λnα

₁

0

sup z∈U

Re

1Azu

1Bzu

upβ/λnα−1du

<

pβ

λnα

₁

0

1Au

1Buu

pβ/λnα−1_du,

Re

Ipδ, λ, fz

zp

β

> inf

z∈URe

pβ

λnα

1

0

1Azu

1Bzuu

pβ/λnα−1_du

≥

pβ

λnα

1

0

inf z∈URe

1Azu

1Bzuu

pβ/λnα−1_du

>

pβ

λnα

₁

0

1−Au

1−Buu

pβ/λnα−1_du.

3.68

Combining both equations of 3.68, we get 3.66. By noting that the function Ipδ, λ,

Fα,β,A,Bz defined by 3.67 belongs to the class ßα,βp δ, λ, ;n;A, B, we obtain that the equality3.66is sharp. The proof of Theorem3.16is evidently completed.

By similarly applying the method of proof of Theorem3.16, we easily get the following result.

Corollary 3.17. Letfz∈ßα,βp δ, λ, ;n;A, Bwith Reα>0 and−1≤A < B≤1. Then

pβ

λnα

1

0

1Au

1Buu

pβ/λnα−1_du

<Re

Ipδ, λ, fz

zp

β

<

pβ

λnα

1

0

1−Au

1−Buu

pβ/λnα−1_du.

3.69

The extremal function of 3.69is defined by3.67.

(20)

Corollary 3.18. Letfz ∈ ßα,βp δ, λ, ;n;A, Bwith Reα > 0 and−1 ≤ B < A ≤ 1. Then for

|z|r <1, we have

rp

pβ

λnα

₁

0

1−Aur

1−Buru

pβ/λnα−1_du

1/β

<Ipδ, λ, fz< rp

pβ λnα

1

0

1Aur

1Buru

pβ/λnα−1_du

1/β

.

3.70

Corollary 3.19. Letfz ∈ ßα,βp δ, λ, ;n;A, Bwith Reα > 0 and−1 ≤ A < B ≤ 1. Then for

|z|r <1, we have

rp

pβ

λnα

1

0

1Aur

1Buru

pβ/λnα−1_du

1/β

<Ipδ, λ, fz< rp

pβ λnα

₁

0

1−Aur

1−Buru

pβ/λnα−1_du

1/β

.

3.71

By noting that

Reν1/2≤Reν1/2≤ |ν|1/2 ν∈C; Reν≥0. 3.72

From Theorem3.16and Corollary3.17, we easily get the following results.

Corollary 3.20. Letfz∈ßα,βp δ, λ, ;n;A, Bwith Reα>0 and−1≤B < A≤1. Then

pβ

λnα

1

0

1−Au

1−Buu

pβ/λnα−1_du

1/2

<Re

Ipδ, λ, fz

zp

β/2 <

pβ

λnα

1

0

1−Au

1−Buu

pβ/λnα−1_du

1/2 .

3.73

Corollary 3.21. Letfz∈ßα,βp δ, λ, ;n;A, Bwith Reα>0 and−1≤A < B≤1. Then

pβ λnα

₁

0

1Au

1Buu

pβ/λnα−1_du

1/2

<Re

Ipδ, λ, fz

zp

β/2 <

pβ

λnα

1

0

1−Au

1−Buu

pβ/λnα−1_du

1/2 .

(21)

Theorem 3.22. Let

fz zp

∞

kn

apkzpk∈ßpα,βδ, λ, ;n;A, B. 3.75

Then

apn≤

pδ1

pλnδ

A−B

pβλnα

. 3.76

The inequality3.76is sharp, with extremal function defined by3.67.

Proof. Combining1.16and3.75, we obtain

1−α

Ipδ, λ, fz

zp

β

α

Ipδ1, λ, fz

Ipδ, λ, fz

zp

β

1

1 λnα

pβ

_p_λn

p

δ

βapnzn· · · ≺ 1Az 1Bz.

3.77

1 λnα

pβ

_p_λn

p

δ

βapn

≤ |A−B|. 3.78

Thus, from3.78, we easily arrive at3.76asserted by Theorem3.22.

Theorem 3.23. Let 0/α∈R, β∈R, β/α >0, λ >−pand 0≤ρ <1. Iffz∈ßα,βp δ, λ, ;n;A,0

with

AAn

α, β, λ, ρ, p

1−ρ|α|1λnα/pβ

₁₋_α_ρα₁

1λnα/pβ2

, _3.79

then

Ipδ, λ, fz∈S∗p,n

ρp−1−ρλ. 3.80

Proof. Suppose thatfz∈ßα,βp δ, λ, ;n;A,0. By definition, we have

1−α

Ipδ, λ, fz

zp

β

α

Ipδ1, λ, fz

Ipδ, λ, fz

zp

β

≺1Az z∈U.

(22)

Let the functionPzbe defined by3.2. We then find from3.1and3.81that

Pz≺

pβ

λnα z

−pβ/λnα

_z

0

1Attpβ/λnα−1dt1

pβA

λnαpβz. 3.82

We now suppose that

Ipδ1, λ, fz

Ipδ, λ, fz

1−ρqz ρ 0≤ρ <1; z∈U. 3.83

Thenqz∈H1, n. It follows from3.81and3.83that

Pz'1−α α 1−ρqz ρ(≺1Az z∈U. 3.84

Reqz>0 z∈U. 3.85

Combining3.83and3.85, we find that

Re

Ipδ1, λ, fz

Ipδ, λ, fz

1−ρReqzρ > ρ z∈U. 3.86

The assertion of Theorem3.23can now easily be derived from1.8and3.86.

Theorem 3.24. Letfz ∈ßα,βρ δ, λ, ;n;A,0withβ >0, A >0,Reα >0, and|α|nRep

β/λα> Apβ/λ. Then

zIpδ, λ, fz

Ipδ, λ, fz −

p< A

pβ λ|α|nRepβ/λαpβ

λ|α| λ|α|nRepβ/λα−Apβ . 3.87

Proof. Let the functionPzbe defined by3.2. It follows from3.3that

pz λαzp

_z

pβ 1Awz, 3.88

where

wz

∞

kn

wkzk n∈N 3.89

(23)

pz 1A

pβ

λα

1

0

tpβ/λα−1wtzdt

1A

pβ

λα

∞

kn

1

kpβ/λαwkz

k_.

3.90

It follows from3.90that

zpz1A

pβ

λα

∞

kn

k1

kpβ/λαwkz

k

1A

pβ

λα

∞

kn

1

kpβ/λαwkz

k

A

pβ

λα

wz−

pβ

λα

₁

0

tp/λα−1wtzdt

.

3.91

We next find from3.90and3.91that

zpz A

pβ

λα

wz−

pβ

λα

1

0

tpβ/λα−1wtzdt

A

pβ

λα

∞

kn

k

kpβ/λαwkz

k_.

3.92

Now from3.88we get

zp_p_zz< nRe

pβ/λαpβ/λ|α|

pz , 3.93

and from3.90we get

_p_z 1A

pβ

λα

∞

kn

1

kpβ/λαwkz

k

> λ|α|

nRepβ/λα−Apβ

−λ|α|nRepβ/λα ,

3.94

from3.93and3.94, we get

zp_p_zz< A

pβ λ|α|nRepβ/λαpβ

λ|α| λ|α|nRepβ/λα−Apβ . 3.95

(24)

References

1 A. C˘atas, “On certain classes ofp-valent functions defined by multiplier transfor presented paper,” in Proceedings of the International Symposium on Geometry Function Theory and Applications, Istanbul, Turkey, 2007.

2 M. Acu, Y. Polatoglu, and E. Yavuz, “The radius of starlikeness of the certain classes ofp-valent functions defined by multiplier transformations,” Journal of Inequalities and Applications, vol. 2008, Article ID 280183, 5 pages, 2008.

3 M. Acu and S. Owa, “Note on a class of starlike functions,” in Proceedings of the Interational Short

Joint Work on Study on Calculus Operators in Univalent Functions Theory, pp. 1–10, RIMS, Kyoto, Japan,

August 2006.

4 F. M. AlObudi, “On univalent functions defined by a generalized S˜al˜agean opeator,” International

Journal of Mathematics, vol. 2004, no. 27, pp. 1429–1436, 2004.

5 A. C˘atas¸, “Neighborhoods of a certain class of analytic functions with negative coeﬃcients,” Banach

Journal of Mathematical Analysis, vol. 3, no. 1, pp. 111–121, 2009.

6 N. E. Cho and H. M. Srivastava, “Argument estimates of certain analytic functions defined by a class of multiplier transformations,” Mathematical and Computer Modelling, vol. 37, no. 1-2, pp. 39–49, 2003.

7 N. E. Cho and T. H. Kim, “Multiplier transformations and strongly close-to-convex functions,”

Bulletin of the Korean Mathematical Society, vol. 40, no. 3, pp. 399–410, 2003.

8 J. Patel, “Inclusion relations and convolution properties of certain subclasses of analytic functions defined by generalized Salagean operator,” Bulletin of the Belgian Mathematical Society. Simon Stevin, vol. 15, no. 1, pp. 33–47, 2008.

9 G. S. S˜al˜agean, “Subclasses of univalent functions,” in Proceedings of the 5th Complex Analysis

Romanin-Finnish Seminar, vol. 1013 of Lecture Notes in Math., pp. 362–372, Springer, Berlin, Germany, 1983.

10 S. S. Kumar, H. C. Taneja, and V. Ravichandran, “Classes of multivalent functions defined by Dziok-Srivastava linear operator and multiplier transformation,” Kyungpook Mathematical Journal, vol. 46, no. 1, pp. 97–109, 2006.

11 B. A. Uralegaddi and C. Somanatha, “Certain classes of univalent functions,” in Current Topics in

Analytic Function Theory, pp. 371–374, World Scientific, River Edge, NJ, USA, 1992.

12 Z. -G. Wang, N. Xu, and M. Acu, “Certain subclasses of multivalent analytic functions defined by multiplier transforms,” Applied Mathematics and Computation, vol. 216, no. 1, pp. 192–204, 2010.

13 M. -S. Liu, “On certain class of analytic functions defined by diﬀerential subordination,” Abstract and

Applied Analysis, vol. 22, pp. 388–392, 2002.

14 Y. Zhu, “Some starlikeness criterions for analytic functions,” Journal of Mathematical Analysis and

Applications, vol. 335, no. 2, pp. 1452–1459, 2007.

15 A. Cˇatas, G. I. Oros, and G. Oros, “Diﬀerential suborinations associated with multiplier transforma-tions,” Abstract and Applied Analysis, vol. 2008, Article ID 845724, 11 pages, 2008.

16 S. S. Miller and P. T. Mocanu, “Subordinants of diﬀerential superordinations,” Complex Variables, vol. 48, no. 10, pp. 815–826, 2003.

17 S. S. Miller and P. T. Mocanu, Diﬀerential Subordinations: Theory and Applications, vol. 225 of Pure and Applied Mathematics, Marcel Dekker, New York, NY, USA, 2000.

18 T. N. Shanmugam, V. Ravichandran, and S. Sivasubramanian, “Diﬀerential sandwich theorems for some subclasses of analytic functions,” The Australian Journal of Mathematical Analysis and Applications, vol. 3, pp. 1–11, 2006.

19 I. S. Jack, “Functions starlike and convex of orderα,” Journal of the London Mathematical Society, vol. 3, pp. 469–474, 1971.

20 M. -S. Liu, “A certain subclass of analytic functions,” Journal of South China Normal University, no. 4, pp. 15–20, 2002.

21 P. Eenigenburg, M. M. Miller, P. T. Mocanu, and O. Reade, “On a Briot-Bouquet diﬀerential suborolination,” in General Mathematics 3, vol. 64 of International Series of Numerical Mathematics, pp. 339–348, Birkh˜auser, Basel, Switzerland, 1983.

22 P. Eenigenburg, S. S. Miller, P. T. Mocanu, and M. O. Reade, “On a Briot- Bouquet differential subordination,” Revue Roumaine de Mathématique Pures et Appliquées , vol. 29, no. 7, pp. 567–573, 1984.

23 W. Rogosinski, “On the coeﬃcients of subordinate functions,” Proceedings of the London Mathematical

(25)

24 M. -S. Liu, “On the starlikeness for certain subclass of analytic functions and a problem of Ruscheweyh,” Advances in Mathematics, vol. 34, no. 4, pp. 416–424, 2005.

25 T. H. MacGregor, “The radius of univalence of certain analytic functions,” Proceedings of the American

Mathematical Society, vol. 14, pp. 514–520, 1963.

26 N. E. Cho, O. S. Kwon, and H. M. Srivastava, “Inclusion relationships and argument properties for certain subclasses of multivalent functions associated with a family of linear operators,” Journal of

Mathematical Analysis and Applications, vol. 292, no. 2, pp. 470–483, 2004.

27 Z. -G. Wang, R. Aghalary, M. Darus, and R. W. Ibrahim, “Some properties of certain multivalent analytic functions involving the Cho-Kwon-Srivastava operator,” Mathematical and Computer

(26)

Submit your manuscripts at

http://www.hindawi.com

Hindawi Publishing Corporation

http://www.hindawi.com Volume 2014

Mathematics

Journal of

Hindawi Publishing Corporation http://www.hindawi.com

Differential Equations

International Journal of

Volume 2014

Applied MathematicsJournal of

Mathematical PhysicsAdvances in

Complex Analysis

Journal of

Optimization

Journal of

Combinatorics

International Journal of

Journal of

Function Spaces

Abstract and Applied Analysis

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporation http://www.hindawi.com Volume 2014

The Scientific

World Journal

Discrete Dynamics in Nature and Society

Discrete Mathematics

Journal of

Some Properties of Certain Multivalent Analytic Functions Involving the Cătas Operator

Research Article