On Certain Subclasses of Analytic Functions Defined by Differential Subordination

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Volume 2011, Article ID 103521,10pages doi:10.1155/2011/103521

Research Article

On Certain Subclasses of Analytic Functions

Defined by Differential Subordination

Hesam Mahzoon

Department of Mathematics, Firoozkooh Branch, Islamic Azad University, Firoozkooh, Iran

Correspondence should be addressed to Hesam Mahzoon,mahzoon [email protected]

Received 3 June 2011; Accepted 25 August 2011

Academic Editor: Stanisława R. Kanas

Copyrightq2011 Hesam Mahzoon. 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 and study certain subclasses of analytic functions which are defined by diﬀerential subordination. Coeﬃcient inequalities, some properties of neighborhoods, distortion and covering theorems, radius of starlikeness, and convexity for these subclasses are given.

1. Introduction

LetTjbe the class of analytic functionsfof the form

fz z− ∞

kj1

akzk, ak≥0, j ∈N{1,2, . . .}, 1.1

defined in the open unit discU{z∈C:|z|<1}.

LetΩbe the class of functionsωanalytic inUsuch thatω0 0,|ωz|<1.

For any two functionsfandginTj,fis said to be subordinate togthat is denoted

f≺g, if there exists an analytic functionω∈Ωsuch thatfz gωz 1.

Definition 1.1see 2. Forn ∈ Nand λ ≥ 0, the Al-Oboudi operator Dn_λ : Tj → Tj

is defined as D0

λfz fz, D1λfz 1 −λfz λzfz Dλfz, and Dλnfz

DλDn−1

λ fz.

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Further, iffz z−_k∞_j₁akzk, then

Dn

λfz z−

∞

kj1

1 k−1λnakzk ak≥0. 1.2

For any function f ∈ Tjandδ≥0, thej, δ-neighborhood offis defined as

Nj,δf

⎧ ⎨

⎩gz z−

∞

kj1

bkzk∈ Tj:

∞

kj1

k|ak−bk| ≤δ

⎫ ⎬

⎭. 1.3

In particular, for the identity functionez z, we see that

Nj,δe

⎧ ⎨

⎩gz z−

∞

kj1

bkzk∈ Tj:

∞

kj1

k|bk| ≤δ

⎫ ⎬

⎭. 1.4

The concept of neighborhoods was first introduced by Goodman 4and then generalized by

Ruscheweyh 5.

Definition 1.2. A functionf ∈ Tjis said to be in the classTjn, m, A, B, λif

Dnm λ fz

Dn

λfz ≺

1Az

1Bz, z∈ U, 1.5

wheren∈N0,m∈N,λ≥1, and−1≤B < A≤1.

We observe that Tjn, m,1 − 2α,−1,1 ≡ Tjn, m, α 6,Tj0,1,1 − 2α,−1,1 ≡

S

jα 7, the class of starlike functions of orderαandTj1,1,1−2α,−1,1≡ Cjα 7, the

class of convex functions of orderα.

2. Neighborhoods for the Class

Tj

n, m, A, B, λ

Theorem 2.1. A functionf∈ Tjbelongs to the classTjn, m, A, B, λif and only if

∞

kj1

1 k−1λn1−B 1 k−1λm−1−Aak≤A−B 2.1

forn∈N0,m∈N,λ≥1, and−1≤B < A≤1.

Proof. Letf∈ Tjn, m, A, B, λ. Then,

Dnm λ fz

Dn

λfz ≺

1Az

1Bz, z∈ U.

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Therefore,

ωz D

n

λfz−Dnλmfz

BDnm

λ fz−ADλnfz

. 2.3

Hence,

|ωz| Dnλfz−Dnλmfz

BDnm

λ fz−ADnλfz

_∞

kj1 1 k−1λn 1 k−1λm−1akzk A−Bz∞

kj1 1 k−1λn

B 1 k−1λm−Aakzk

<1.

2.4

Thus,

R

_∞

kj1 1 k−1λn

1 k−1λm−1akzk

A−Bz∞kj1 1 k−1λn

B 1 k−1λm−Aakzk

<1. 2.5

Taking|z|r, for suﬃciently smallrwith 0< r <1, the denominator of2.5is positive and

so it is positive for allrwith 0< r <1, sinceωzis analytic for|z|<1. Then, inequality2.5

yields

∞

kj1

1 k−1λn 1 k−1λm−1akrk

<A−BrB∞

kj1

1 k−1λnmakrk−A

∞

kj1

1 k−1λnakrk.

2.6

Equivalently,

∞

kj1

1 k−1λn1−B 1 k−1λm−1−Aakrk≤A−Br, 2.7

and2.1follows upon lettingr → 1.

Conversely, for|z|r, 0< r <1, we haverk< r. That is,

∞

kj1

1 k−1λn1−B 1 k−1λm−1−Aakrk

≤ ∞

kj1

1 k−1λn1−B 1 k−1λm−1−Aakr≤A−Br.

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From2.1, we have

∞

kj1

1 k−1λn1 k−1λm−1akzk

≤ ∞

kj1

1 k−1λn 1 k−1λm−1akrk

<A−Br ∞ kj1

B 1 k−1λm−A 1 k−1λnakrk

<

A−Bz

∞

kj1

B 1 k−1λm−A1 k−1λnakzk

.

2.9

This proves that

Dnm λ fz

Dn λfz

≺ 1Az

1Bz, z∈ U, 2.10

and hencef∈ Tjn, m, A, B, λ.

Theorem 2.2. If

δ A−B

1λjn−11−B1λjm−1−A, 2.11

then Tjn, m, A, B, λ⊂Nj,δe.

Proof. It follows from2.1that iff∈ Tjn, m, A, B, λ, then

1λjn−11−B1λjm−1−A

∞

kj1

kak≤A−B, _2.12

which implies

∞

kj1

kak≤ A−B

1λjn−11−B1λjm−1−A δ. 2.13

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3. Neighborhoods for the Classes

Rj

n, A, B, λ

and

Pjn, A, B, λ

Definition 3.1. A functionf ∈ Tjis said to be in the classRjn, A, B, λif it satisfies

Dn λfz

_≺ 1Az

1Bz z∈ U, 3.1

where−1≤B < A≤1,λ≥1 andn∈ N0.

Definition 3.2. A functionf ∈ Tjis said to be in the classPjn, A, B, λif it satisfies

Dn λfz

z ≺

1Az

1Bz z∈ U, 3.2

where−1≤B < A≤1,λ≥1 andn∈ N0.

Lemma 3.3. A functionf∈ Tjbelongs to the classRjn, A, B, λif and only if

∞

kj1

1−B 1 k−1λn1ak≤A−B. 3.3

Lemma 3.4. A functionf∈ Tjbelongs to the classPjn, A, B, λif and only if

∞

kj1

1−B 1 k−1λnak≤A−B. 3.4

Theorem 3.5. Rjn, A, B, λ⊂ Nj,δe, where

δ A−B

1λjn1−B. 3.5

Proof. Iff∈ Rjn, A, B, λ, we have

1λjn

∞

kj1

1−Bkak≤A−B, 3.6

which implies

∞

kj1

kak≤ A−B

1λjn1−B δ. 3.7

Theorem 3.6. Pjn, A, B, λ⊂ Nj,δe, where

δ A−B

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Proof. Iff∈ Pjn, A, B, λ, we have

1λjn−1

∞

kj1

1−Bkak≤A−B, 3.9

which implies

∞

kj1

kak≤ A−B

1λjn−11−B δ.

3.10

Thus, in view of condition1.4, we get the required result of Theorem3.6.

4. Neighborhood of the Class

K

λ_j

n, m, A, B, C, D

Definition 4.1. A functionf ∈ Tjis said to be in the classKλ_jn, m, A, B, C, Dif it satisfies

f_g_zz−1< A−B

1−B, z∈ U, 4.1

for−1≤B < A≤1,−1≤D < C≤1,λ≥1 andg∈ Tjn, m, C, D, λ.

Theorem 4.2. Forg∈ Tjn, m, C, D, λ, one hasNj,δg⊂ Kλ_jn, m, A, B, C, Dand

1−A

1−B 1−

1λjn−11−D1λjm−1−Cδ

1λjn1−D1λjm−1−C−C−D,

4.2

where

δ≤1−D1λj−C−D1λj1−n1−D1λjm−1−C−1. 4.3

Proof. Letf∈ Nj,δgforg∈ Tjn, m, C, D, λ. Then,

∞

kj1

k|ak−bk| ≤δ,

∞

kj1

bk≤ C−D

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Consider

f_gz_z−1≤

_∞

kj1|ak−bk|

1−∞_k_j₁bk

≤ δ

1λj

1λjn1−D1λjm−1−C

1λjn1−D1λjm−1−C−C−D

1λjn−11−D1λjm−1−Cδ

1λjn1−D1λjm−1−C−C−D

A−B

1−B.

4.5

This implies thatf ∈ Kλ_jn, m, A, B, C, D.

5. Distortion and Covering Theorems

Theorem 5.1. Iff ∈ Tjn, m, A, B, λ, then

r− A−B

1jλn1−B1jλm−1−Ar

j1

≤fz≤r A−B

j1 ₀_<_|_z_|_{r <}₁_, 5.1

with equality for

fz z− A−B

j1 _z_±_r_. _5.2

Proof. In view of Theorem2.1, we have

1jλn1−B1jλm−1−A

∞

kj1

ak

≤ ∞

kj1

1 k−1λn1−B 1 k−1λm−1−Aak≤A−B.

5.3

Hence,

_f_z_≤_r ∞

kj1

akrk≤rrj1

∞

kj1

ak≤r A−B

j1_,

fz≥r− ∞

kj1

akrk≥r−rj1

∞

kj1

ak≥r− A−B

j1_.

5.4

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Theorem 5.2. Any functionf∈ Tjn, m, A, B, λmaps the disk|z|<1 onto a domain that contains the disk

|w|<1− A−B

1jλn1−B1jλm−1−A. 5.5

Proof. The proof follows upon lettingr → 1 in Theorem5.1.

Theorem 5.3. Iff ∈ Tjn, m, A, B, λ, then

1− A−B

1jλn−11−B1jλm−1−Ar

j

≤f_z_≤₁ A−B

j ₀_<_|_z_|_{r <}₁_,

5.6

with equality for

fz z− A−B

1jλn−11−B1jλm−1−Az

j1 _z_±_r_.

5.7

Proof. We have

f_z_≤₁ ∞

kj1

kak|z|k−1≤1rj

∞

kj1

kak. 5.8

In view of Theorem2.1,

∞

kj1

kak≤ A−B

1jλn−11−B1jλm−1−A. 5.9

Thus,

_f_z_≤₁ A−B

j_.

5.10

On the other hand,

f_z_≥₁₋ ∞

kj1

kak|z|k−1≥1−rj

∞

kj1

kak

≥1− A−B

j_.

5.11

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6. Radii of Starlikeness and Convexity

In this section, we find the radius of starlikeness of orderρ and the radius of convexity of

orderρfor functions in the classTjn, m, A, B, λ.

Theorem 6.1. If f ∈ Tjn, m, A, B, λ, then f is starlike of order ρ, 0 ≤ ρ < 1 in |z| <

r1n, m, A, B, λ, ρ, where

r1

n, m, A, B, λ, ρinfk

1 k−1λn1−B 1 k−1λm−1−A1−ρ

k−ρA−B

1/k−1

.

6.1

Proof. It is suﬃcient to show that |zfz/fz − 1| ≤ 1 − ρ 0 ≤ ρ < 1 for |z| <

r1n, m, A, B, λ, ρ. We have

zfz

fz −1

≤

_∞

kj1k−1ak|z|k−1

1−∞_k_j₁ak|z|k−1

. 6.2

Thus,|zfz/fz−1| ≤1−ρif

∞

kj1

k−ρak|z|k−1

1−ρ ≤1. 6.3

Hence, by Theorem2.1,6.3will be true if

k−ρ|z|k−1

1−ρ ≤

1 k−1λn1−B 1 k−1λm−1−A

A−B 6.4

or if

|z| ≤

1 k−1λn1−B 1 k−1λm−1−A1−ρ

k−ρA−B

1/k−1

k≥j1. 6.5

This completes the proof.

Theorem 6.2. If f ∈ Tjn, m, A, B, λ, then f is convex of order ρ, 0 ≤ ρ < 1 in |z| <

r2n, m, A, B, λ, ρ, where

r2

n, m, A, B, λ, ρinf

k

1 k−1λn1−B 1 k−1λm−1−A1−ρ

k−ρA−B

1/k−1

.

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Proof. It is suﬃcient to show that |zfz/fz| ≤ 1 − ρ 0 ≤ ρ < 1 for |z| <

r1n, m, A, B, λ, ρ. We have

zfz

f_z

≤

_∞

kj1kk−1ak|z|k−1

1−∞_k_j₁kak|z|k−1

. 6.7

Thus,|zfz/fz| ≤1−ρif

∞

kj1

kk−ρak|z|k−1

1−ρ ≤1. 6.8

Hence, by Theorem2.1,6.8will be true if

kk−ρ|z|k−1

1−ρ ≤

1 k−1λn1−B 1 k−1λm−1−A

A−B 6.9

or if

|z| ≤

1 k−1λn1−B 1 k−1λm−1−A1−ρ

kk−ρA−B

1/k−1

k≥j1. 6.10

This completes the proof.

Acknowledgment

The author wish to thank the referee for his valuable suggestions.

References

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2 F. M. Al-Oboudi, “On univalent functions defined by a generalized S˘al˘agean operator,” International

Journal of Mathematics and Mathematical Sciences, no. 25–28, pp. 1429–1436, 2004.

3 G. S˘al˘agean, “Subclasses of univalent functions,” in Complex analysis—Fifth Romanian-Finnish Seminar,

Part 1 (Bucharest, 1981), vol. 1013 of Lecture Notes in Math., pp. 362–372, Springer, Berlin, Germany, 1983.

4 A. W. Goodman, “Univalent functions and nonanalytic curves,” Proceedings of the American

Mathematical Society, vol. 8, pp. 598–601, 1957.

5 S. Ruscheweyh, “Neighborhoods of univalent functions,” Proceedings of the American Mathematical

Society, vol. 81, no. 4, pp. 521–527, 1981.

6 M. K. Aouf, “Neighborhoods of certain classes of analytic functions with negative coeﬃcients,”

International Journal of Mathematics and Mathematical Sciences, vol. 2006, Article ID 38258, 6 pages, 2006.

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