Hindawi Publishing Corporation
International Journal of Mathematics and Mathematical Sciences
Volume 2011, Article ID 103521, 10 pages
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 hesam@yahoo.com
Received 3 June 2011; Accepted 25 August 2011
Academic Editor: Stanisława R. Kanas
Copyright q 2011 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
Let Tj be the class of analytic functions f of the form
fz z −
∞
ak zk ,
ak ≥ 0, j ∈ N {1, 2, . . .} ,
1.1
kj1
defined in the open unit disc U {z ∈ C : |z| < 1}.
Let Ω be the class of functions ω analytic in U such that ω0 0, |ωz| < 1.
For any two functions f and g in Tj, f is said to be subordinate to g that is denoted
f ≺ g, if there exists an analytic function ω ∈ Ω such that fz gωz 1
.
Definition 1.1 see 2
. For n ∈ N and λ ≥ 0, the Al-Oboudi operator Dλn : Tj → Tj
is defined as Dλ0 fz fz, Dλ1 fz 1 − λfz λzf z Dλ fz, and Dλn fz Dλ Dλn−1 fz.
For λ 1, we get Sǎlǎgean differential operator 3
.
2
International Journal of Mathematics and Mathematical Sciences
Further, if fz z −
∞
kj1
ak zk , then
Dλn fz z −
∞
1 k − 1λ
n ak zk
ak ≥ 0.
kj1
1.2
For any function f ∈ Tj and δ ≥ 0, the j, δ-neighborhood of f is defined as
⎫
⎧
∞
∞
⎬
⎨
Nj,δ f gz z −
bk zk ∈ T j :
k|ak − bk | ≤ δ .
⎭
⎩
kj1
kj1
1.3
In particular, for the identity function ez z, we see that
⎧
⎨
⎫
∞
⎬
Nj,δ e gz z −
bk zk ∈ T j :
k|bk | ≤ δ .
⎩
⎭
kj1
kj1
∞
1.4
The concept of neighborhoods was first introduced by Goodman 4
and then generalized by
Ruscheweyh 5
.
Definition 1.2. A function f ∈ Tj is said to be in the class Tj n, m, A, B, λ if
Dλnm fz 1 Az
≺
,
Dλn fz
1 Bz
z ∈ U,
1.5
where n ∈ N0 , m ∈ N, λ ≥ 1, and −1 ≤ B < A ≤ 1.
We observe that Tj n, m, 1 − 2α, −1, 1 ≡ Tj n, m, α 6
, Tj 0, 1, 1 − 2α, −1, 1 ≡
Sj α 7
, the class of starlike functions of order α and Tj 1, 1, 1 − 2α, −1, 1 ≡ Cj α 7
, the
class of convex functions of order α.
2. Neighborhoods for the Class Tj n, m, A, B, λ
Theorem 2.1. A function f ∈ Tj belongs to the class Tj n, m, A, B, λ if and only if
∞
1 k − 1λ
n 1 − B1 k − 1λ
m − 1 − A ak ≤ A − B
kj1
2.1
for n ∈ N0 , m ∈ N, λ ≥ 1, and −1 ≤ B < A ≤ 1.
Proof. Let f ∈ Tj n, m, A, B, λ. Then,
Dλnm fz 1 Az
≺
,
Dλn fz
1 Bz
z ∈ U.
2.2
International Journal of Mathematics and Mathematical Sciences
3
Therefore,
ωz Dλn fz − Dλnm fz
BDλnm fz − ADλn fz
.
2.3
Hence,
Dn fz − Dnm fz λ
λ
|ωz| BDλnm fz − ADλn fz ∞
n
m
k
kj1 1 k − 1λ
1 k − 1λ
− 1 ak z
< 1.
∞
A − Bz kj1 1 k − 1λ
n B1 k − 1λ
m − A ak zk 2.4
1 k − 1λ
n 1 k − 1λ
m − 1 ak zk
R
< 1.
n
m
k
A − Bz ∞
kj1 1 k − 1λ
B1 k − 1λ
− A ak z
2.5
Thus,
∞
kj1
Taking |z| r, for sufficiently small r with 0 < r < 1, the denominator of 2.5 is positive and
so it is positive for all r with 0 < r < 1, since ωz is analytic for |z| < 1. Then, inequality 2.5
yields
∞
1 k − 1λ
n 1 k − 1λ
m − 1 ak r k
kj1
< A − Br B
∞
1 k − 1λ
nm ak r k − A
kj1
∞
2.6
1 k − 1λ
n ak r k .
kj1
Equivalently,
∞
1 k − 1λ
n 1 − B1 k − 1λ
m − 1 − A ak r k ≤ A − Br,
kj1
2.7
and 2.1 follows upon letting r → 1.
Conversely, for |z| r, 0 < r < 1, we have r k < r. That is,
∞
1 k − 1λ
n 1 − B1 k − 1λ
m − 1 − A ak r k
kj1
≤
∞
1 k − 1λ
1 − B1 k − 1λ
− 1 − A ak r ≤ A − Br.
kj1
n
m
2.8
4
International Journal of Mathematics and Mathematical Sciences
From 2.1, we have
∞
n
m
k
−
1
a
z
−
1λ
−
1λ
1
k
1
k
k
kj1
≤
∞
1 k − 1λ
n 1 k − 1λ
m − 1 ak r k
kj1
∞
< A − Br B1 k − 1λ
m − A 1 k − 1λ
n ak r k
2.9
kj1
∞
< A − Bz B1 k − 1λ
m − A 1 k − 1λ
n ak zk .
kj1
This proves that
Dλnm fz 1 Az
≺
,
Dλn fz
1 Bz
z ∈ U,
2.10
and hence f ∈ Tj n, m, A, B, λ.
Theorem 2.2. If
δ
1 λj
n−1 A − B
,
m
1 − B 1 λj − 1 − A
2.11
then Tj n, m, A, B, λ ⊂ Nj,δ e.
Proof. It follows from 2.1 that if f ∈ Tj n, m, A, B, λ, then
1 λj
∞
m
n−1 kak ≤ A − B,
1 − B 1 λj − 1 − A
kj1
2.12
which implies
∞
kak ≤ kj1
Using 1.4, we get the result.
1 λj
n−1 A − B
δ.
m
1 − B 1 λj − 1 − A
2.13
International Journal of Mathematics and Mathematical Sciences
5
3. Neighborhoods for the Classes Rj n, A, B, λ and Pj n, A, B, λ
Definition 3.1. A function f ∈ Tj is said to be in the class Rj n, A, B, λ if it satisfies
1 Az
Dλn fz ≺
1 Bz
z ∈ U,
3.1
where −1 ≤ B < A ≤ 1, λ ≥ 1 and n ∈ N0 .
Definition 3.2. A function f ∈ Tj is said to be in the class Pj n, A, B, λ if it satisfies
Dλn fz 1 Az
≺
z
1 Bz
z ∈ U,
3.2
where −1 ≤ B < A ≤ 1, λ ≥ 1 and n ∈ N0 .
Lemma 3.3. A function f ∈ Tj belongs to the class Rj n, A, B, λ if and only if
∞
1 − B1 k − 1λ
n1 ak ≤ A − B.
kj1
3.3
Lemma 3.4. A function f ∈ Tj belongs to the class Pj n, A, B, λ if and only if
∞
1 − B1 k − 1λ
n ak ≤ A − B.
kj1
3.4
Theorem 3.5. Rj n, A, B, λ ⊂ Nj,δ e, where
δ
A − B
.
n
1 λj 1 − B
3.5
Proof. If f ∈ Rj n, A, B, λ, we have
∞
n 1 λj
1 − Bkak ≤ A − B,
kj1
3.6
which implies
∞
kak ≤ kj1
A − B
δ.
n
1 λj 1 − B
3.7
Theorem 3.6. Pj n, A, B, λ ⊂ Nj,δ e, where
A − B
.
δ
n−1
1 λj
1 − B
3.8
6
International Journal of Mathematics and Mathematical Sciences
Proof. If f ∈ Pj n, A, B, λ, we have
∞
n−1 1 λj
1 − Bkak ≤ A − B,
3.9
kj1
which implies
∞
A − B
kak ≤ δ.
n−1
1 λj
1 − B
kj1
3.10
Thus, in view of condition 1.4, we get the required result of Theorem 3.6.
4. Neighborhood of the Class Kλj n, m, A, B, C, D
Definition 4.1. A function f ∈ Tj is said to be in the class Kλj n, m, A, B, C, D if it satisfies
fz
A−B
<
−
1
gz
1−B ,
z ∈ U,
4.1
for −1 ≤ B < A ≤ 1, −1 ≤ D < C ≤ 1, λ ≥ 1 and g ∈ Tj n, m, C, D, λ.
Theorem 4.2. For g ∈ Tj n, m, C, D, λ, one has Nj,δ g ⊂ Kλj n, m, A, B, C, D and
n−1 m
1 λj
1 − D 1 λj − 1 − C δ
1−A
1− ,
n m
1−B
1 λj 1 − D 1 λj − 1 − C − C − D
4.2
1−n −1
m
δ ≤ 1 − D 1 λj − C − D 1 λj
1 − D 1 λj − 1 − C .
4.3
where
Proof. Let f ∈ Nj,δ g for g ∈ Tj n, m, C, D, λ. Then,
∞
k|ak − bk | ≤ δ,
kj1
∞
bk ≤ kj1
C−D
.
n m
1 λj 1 − D 1 λj − 1 − C
4.4
International Journal of Mathematics and Mathematical Sciences
Consider
7
∞ |a − b |
fz
k
kj1 k
gz − 1 ≤ 1 − ∞ b
kj1 k
m
n 1 λj 1 − D 1 λj − 1 − C
δ
≤
n m
1 λj
1 λj 1 − D 1 λj − 1 − C − C − D
n−1 m
1 λj
1 − D 1 λj − 1 − C δ
n m
1 λj 1 − D 1 λj − 1 − C − C − D
4.5
A−B
.
1−B
This implies that f ∈ Kλj n, m, A, B, C, D.
5. Distortion and Covering Theorems
Theorem 5.1. If f ∈ Tj n, m, A, B, λ, then
A−B
r j1
n m
1 jλ 1 − B 1 jλ − 1 − A
A−B
≤ fz ≤ r r j1
n m
1 jλ 1 − B 1 jλ − 1 − A
r−
5.1
0 < |z| r < 1,
with equality for
A−B
r j1
n m
1 jλ 1 − B 1 jλ − 1 − A
fz z − z ±r.
5.2
Proof. In view of Theorem 2.1, we have
∞
m
n ak
1 jλ 1 − B 1 jλ − 1 − A
kj1
≤
∞
1 k − 1λ
1 − B1 k − 1λ
− 1 − A ak ≤ A − B.
n
m
5.3
kj1
Hence,
∞
∞
fz ≤ r ak r k ≤ r r j1
ak ≤ r kj1
kj1
A−B
r j1 ,
n m
1 jλ 1 − B 1 jλ − 1 − A
∞
∞
fz ≥ r −
ak r k ≥ r − r j1
ak ≥ r − kj1
kj1
A−B
r j1 .
n m
1 jλ 1 − B 1 jλ − 1 − A
5.4
This completes the proof.
8
International Journal of Mathematics and Mathematical Sciences
Theorem 5.2. Any function f ∈ Tj n, m, A, B, λ maps the disk |z| < 1 onto a domain that contains
the disk
A−B
.
n m
1 jλ 1 − B 1 jλ − 1 − A
|w| < 1 − 5.5
Proof. The proof follows upon letting r → 1 in Theorem 5.1.
Theorem 5.3. If f ∈ Tj n, m, A, B, λ, then
A − B
j
r
n−1 m
1 jλ
1 − B 1 jλ − 1 − A
1− ≤ f z ≤ 1 A−B
j
r
n−1 m
−
−
A
1 jλ
−
B
1
jλ
1
1
5.6
0 < |z| r < 1,
with equality for
A−B
j1
z
n−1 m
1 jλ
1 − B 1 jλ − 1 − A
fz z − z ±r.
5.7
Proof. We have
∞
∞
f z ≤ 1 kak |z|k−1 ≤ 1 r j
kak .
kj1
kj1
5.8
In view of Theorem 2.1,
∞
A−B
.
n−1 m
1 jλ
1 − B 1 jλ − 1 − A
kak ≤ kj1
5.9
Thus,
f z ≤ 1 A−B
j
r .
n−1 m
1 jλ
1 − B 1 jλ − 1 − A
5.10
On the other hand,
∞
∞
f z ≥ 1 −
kak |z|k−1 ≥ 1 − r j
kak
kj1
kj1
A−B
j
≥1− r .
n−1 m
−
−
A
1 jλ
−
B
1
jλ
1
1
This completes the proof.
5.11
International Journal of Mathematics and Mathematical Sciences
9
6. Radii of Starlikeness and Convexity
In this section, we find the radius of starlikeness of order ρ and the radius of convexity of
order ρ for functions in the class Tj n, m, A, B, λ.
Theorem 6.1. If f ∈ Tj n, m, A, B, λ, then f is starlike of order ρ, 0 ≤ ρ < 1 in |z| <
r1 n, m, A, B, λ, ρ, where
r1 n, m, A, B, λ, ρ infk
1/k−1
1 k − 1λ
n 1 − B1 k − 1λ
m − 1 − A 1 − ρ
.
k − ρ A − B
6.1
Proof. It is sufficient to show that |zf z/fz − 1| ≤ 1 − ρ 0 ≤ ρ < 1 for |z| <
r1 n, m, A, B, λ, ρ.
We have
∞ k − 1a |z|k−1
f z
k
kj1
z
.
∞
fz − 1 ≤
1 − kj1 ak |z|k−1
6.2
Thus, |zf z/fz − 1| ≤ 1 − ρ if
∞
k − ρ ak |z|k−1
≤ 1.
1−ρ
kj1
6.3
Hence, by Theorem 2.1, 6.3 will be true if
k − ρ |z|k−1 1 k − 1λ
n 1 − B1 k − 1λ
m − 1 − A
≤
A − B
1−ρ
6.4
or if
1/k−1
1 k − 1λ
n 1 − B1 k − 1λ
m − 1 − A 1 − ρ
|z| ≤
k − ρ A − B
k ≥j 1 .
6.5
This completes the proof.
Theorem 6.2. If f ∈ Tj n, m, A, B, λ, then f is convex of order ρ, 0 ≤ ρ < 1 in |z| <
r2 n, m, A, B, λ, ρ, where
1/k−1
1 k − 1λ
n 1 − B1 k − 1λ
m − 1 − A 1 − ρ
r2 n, m, A, B, λ, ρ inf
.
k
k − ρ A − B
6.6
10
International Journal of Mathematics and Mathematical Sciences
Proof. It is sufficient to show that |zf z/f z| ≤ 1 − ρ 0 ≤ ρ < 1 for |z| <
r1 n, m, A, B, λ, ρ.
We have
∞ kk − 1a |z|k−1
f z k
kj1
z
.
∞
f z ≤
1 − kj1 kak |z|k−1
6.7
Thus, |zf z/f z| ≤ 1 − ρ if
k−1
∞ k k−ρ a
k |z|
≤ 1.
1−ρ
kj1
6.8
Hence, by Theorem 2.1, 6.8 will be true if
k k − ρ |z|k−1 1 k − 1λ
n 1 − B1 k − 1λ
m − 1 − A
≤
A − B
1−ρ
6.9
or if
1/k−1
1 k − 1λ
n 1 − B1 k − 1λ
m − 1 − A 1 − ρ
|z| ≤
k k − ρ A − B
k ≥j 1 .
6.10
This completes the proof.
Acknowledgment
The author wish to thank the referee for his valuable suggestions.
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1
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2
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G. Sălăgean, “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
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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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M. I. S. Robertson, “On the theory of univalent functions,” Annals of Mathematics. Second Series, vol. 37,
no. 2, pp. 374–408, 1936.
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