Hindawi Publishing Corporation
Journal of Inequalities and Applications
Volume 2009, Article ID 759251, 12 pages
doi:10.1155/2009/759251
Research Article
Certain Classes of Harmonic Multivalent Functions
Based on Hadamard Product
Om P. Ahuja,1 H. Özlem Güney,2 and F. Müge Sakar2
1
Department of Mathematical Sciences, Kent State University, 14111 Claridon-Troy Road,
Burton, OH 44021, USA
2
Department of Mathematics, Faculty of Science and Arts, Dicle University,
21280 Diyarbakır, Turkey
Correspondence should be addressed to H. Özlem Güney, ozlemg@dicle.edu.tr
Received 25 February 2009; Accepted 12 September 2009
Recommended by Yong Zhou
We define and investigate two special subclasses of the class of complex-valued harmonic
multivalent functions based on Hadamard product.
Copyright q 2009 Om P. Ahuja 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.
1. Introduction
A continuous function f uiv is a complex-valued harmonic function in a complex domain
C if both u and v are real harmonic in C. In any simply connected domain D ⊆ C, we can write
f h g, where h and g are analytic in D. We call h the analytic part and g the co-analytic
part of f. Note that f h g reduces to h if the coanalytic part g is zero.
For p ≥ 1, denote by Hp the set of all multivalent harmonic functions f h g
defined in the open unit disc U, where h and g defined by
hz zp ∞
an zn ,
∞
gz npt
bn zn ,
bpt−1 < 1,
t ∈ N : {1, 2, . . .}
1.1
npt−1
are analytic functions in U.
Let F be a fixed multivalent harmonic function given by
Fz Hz Gz zp ∞
|An |zn npt
∞
|Bn |zn ,
npt−1
Bpt−1 < 1,
t ∈ N.
1.2
2
Journal of Inequalities and Applications
A function f ∈ Hp is said to be in the class HF p, t, α, k if
z f ∗ F z
z f ∗ F z
Re
≥
k
−
p
pα ,
z f ∗ F z
z f ∗ F z
1.3
where f ∗ F is a harmonic convolution of f and F. Note that z ∂/∂θr eiθ , f z ∂/∂θfr eiθ . Using the fact
Re w > kw − p pα ⇐⇒ Re 1 keiθ w − kpeiθ ≥ pα,
1.4
it follows that f ∈ HF p, t, α, k if and only if
Re
1 keiθ z f ∗ F z
iθ
≥ pα,
− kpe
z f ∗ F z
0 ≤ α < 1.
1.5
A function f in HF p, t, α, k is called k-uniformly multivalent harmonic starlike
function associated with a fixed multivalent harmonic function F. The set HF p, t, α, k is
a comprehensive family that contains several previously studied subclasses of Hp; for
example, if we let
Fz Iz :
zp
1−z
zp
1−z
∞
zp zn np1
∞
zn ,
1.6
np
then
HI p, α, 0 ≡ S∗H p, α :
z f z
≥ pα ,
f ∈ H p : Re
z fz
1.7
see 1, 2;
HI p, 1, 0, 0 ≡ S∗H p, 0 ≡ S∗H p ,
1.8
HI 1, 1, α, 0 ≡ S∗H 1, α ≡ S∗H α,
1.9
HI 1, 1, 0, 0 ≡ S∗H 0 ≡ S∗H ,
1.10
see 3;
see 4;
Journal of Inequalities and Applications
3
see 5, 6;
HI 1, 1, α, k ≡ GH k, α :
f ∈ H1 : Re
1 keiθ zf z
iθ
≥α ,
− ke
z fz
1.11
see 7;
HI p, 1, α, 1 ≡ MH p, α :
zf z
− peiθ ≥ pα ,
f ∈ H p : Re 1 eiθ z fz
1.12
see 8.
Finally, denote by T Hp the subclass of functions fz hz gz in Hp where
hz zp −
∞
|an |zn ,
gz npt
∞
|bn |zn .
1.13
npt−1
Let HF p, t, α, k : T Hp ∩ HF p, t, α, k.
In this paper, we investigate coefficient conditions, extreme points, and distortion
bounds for functions in the family HF p, t, α, k. We observe that the results so obtained for
this main family can be viewed as extensions and generalizations for various subclasses of
Hp and H1.
2. Main Results
Theorem 2.1. Let f h g be such that h and g are given by 1.1. Then f ∈ HF p, t, α, k if the
inequality
∞
∞
n1 k − pk α
n1 k pk α
1
|an An | |b B | ≤
p1 − α − 1 n n
2
p1 − α 1 − p1 − α − 1
p1
−
α
1
−
npt−1
npt
2.1
is satisfied for some k k ≥ 0, p p ≥ 1, α 0 ≤ α < 1, and t t ≥ 1.
Proof. In view of 1.5, we need to prove that Rew > 0, where
ke 1 zh ∗ H z − z g ∗ G z − p keiθ α h ∗ Hz g ∗ G z
w
iθ
h ∗ Hz g ∗ G z
:
Az
.
Bz
2.2
Using the fact that Rew > 0 ⇔ |1 w| ≥ |1 − w| , it sufficies to show that
|Az Bz| − |Az − Bz| ≥ 0.
2.3
4
Journal of Inequalities and Applications
Therefore, we obtain
|Az Bz| − |Az − Bz|
≥
∞
p1 − α 1 − p1 − α − 1 |z|p −
n1 k − pk α 1 |an An ||z|n
npt
∞
∞
n1 k pk α − 1 |bn Bn ||z|n −
n1 k − pk α − 1 |an An ||z|n
−
npt
npt−1
∞
n1 k pk α 1 |bn Bn ||z|n
−
npt−1
p1 − α 1 − p1 − α − 1 |z|p
∞
∞
2 n1 k − pk α |an An ||z|n −
2 n1 k pk α |bn Bn ||z|n
−
npt
npt−1
p1 − α 1 − p1 − α − 1 |z|p
⎧
∞
⎨
2 n1 k − pk α
|an An |
× 1−
⎩
npt p1 − α 1 − p1 − α − 1
∞
−
npt−1
⎫
⎬
2 n1 k pk α
|bn Bn | ≥ 0.
⎭
p1 − α 1 − p1 − α − 1
2.4
By hypothesis, last expression is nonnegative. Thus the proof is complete.
The coeficient bounds 2.1 is sharp for the function
∞
p1 − α 1 − p1 − α − 1
Xn zn
fz z 2
n1
k
−
pk
α
npt
∞
p1 − α 1 − p1 − α − 1
Yn zn ,
2 n1 k pk α Bn
npt−1
p
where
∞
npt
|Xn | ∞
npt−1
2.5
|Yn | 1.
Corollary 2.2. For p ≥ 1/1 − α, 0 ≤ α < 1, if the inequality
∞
∞
n1 k − pk α |an An | n1 k pk α |bn Bn | ≤ 1
npt
holds, then f ∈ HF p, t, α, k .
npt−1
2.6
Journal of Inequalities and Applications
5
Corollary 2.3. For 0 ≤ α < 1 and 1 ≤ p ≤ 1/1 − α, if the inequality
∞
∞
n1 k − pk α |an An | n1 k pk α |bn Bn | ≤ p1 − α
npt
2.7
npt−1
holds, then f ∈ HF p, t, α, k.
Theorem 2.4. Let f h g be such that h and g are given by 1.13. Also, suppose that k ≥ 0 and
0 ≤ α < 1. Then
i for 1 ≤ p ≤ 1/1 − α , f ∈ HF p, t, α, k if and only if
∞
∞
n1 k − pk α |an An | n1 k pk α |bn Bn | ≤ p1 − α;
npt
2.8
npt−1
ii for p1 − α ≥ 1 , f ∈ HF p, t, α, k if and only if
∞
∞
n1 k − pk α |an An | n1 k pk α |bn Bn | ≤ 1.
npt
2.9
npt−1
Proof. According to Corollaries 2.2 and 2.3, we must show that if the condition 2.9 does not
hold, then f /
∈ HF p, t, α, k, that is, we must have
⎛
⎜
Re⎜
⎝
1 ke
iθ
⎞
zh ∗ H z − z g ∗ G z − p keiθ α h ∗ Hz g ∗ G z
⎟
⎟ ≥ 0.
⎠
h ∗ Hz g ∗ G z
2.10
Choosing the values of z r on positive real axis where 0 ≤ z r < 1, and using Re−eiθ ≥
−|eiθ | −1 , the inequality 2.10 reduces to
⎧
⎫
∞
n
n
⎨ 1 keiθ pzp − ∞
− A⎬
npt n|an An |z −
npt−1 n|bn Bn |z
Re
∞
⎩
⎭
n
n
p zp − ∞
npt |an An |z npt−1 |bn Bn |z
⎧
⎫
∞
n
n
⎨ 1 k pr p − ∞
− B⎬
npt n|an An |r −
npt−1 n|bn Bn |r
≥
∞
⎩
⎭
n
n
p rp − ∞
npt |an An |r npt−1 |bn Bn |r
⎧
⎫
n
⎨ p1 − αr p − ∞
⎬
npt n1 k − pk α |an An |r − C
,
∞
⎩
⎭
n
n
p rp − ∞
npt |an An |r npt−1 |bn Bn |r
2.11
6
Journal of Inequalities and Applications
∞
n
p
n
where A denotes k eiθ αpzp − ∞
npt |an An |z npt−1 |bn Bn |z , B denotes kαp r −
∞
∞
∞
n
n
n
npt |an An |r npt−1 |bn Bn |r , and C denotes
npt−1 n1 k pk α|bn Bn |r .
Letting r → 1− , we obtain
&
%
∞
∞
p1 − α −
npt n1 k − pk α |an An | npt−1 n1 k pk α |bn Bn |
∞
≥ 0.
1− ∞
npt |an An | npt−1 |bn Bn |
2.12
If the condition 2.10 does not hold, then the numerator in 2.12 is negative for r sufficiently
close to 1. Hence there exists z0 r0 in 0, 1 for which 2.12 is negative. Therefore, it follows
that f /
∈ HF p, t, α, k and so the proof is complete.
Theorem 2.5. If f ∈ HF p, t; α, k, then for |z| r < 1 , |Apt | ≤ |An | ≤ |Bn |, and Apt /
0,
⎧
⎪
1 bpt−1 r pt−1
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
p1 − α
⎪
⎪
⎪
⎪
⎪
p t 1 k − pk α Apt ⎪
⎪
⎪
⎪
⎪
⎪
⎪
p1
p t − 1 1 k pk α ⎪
⎪
⎪
bpt−1 Bpt−1 r ; p1 − α ≤ 1
− ⎪
⎪
p t 1 k − pk α Apt ⎨
fz ≤
⎪
⎪
⎪
1 bpt−1 r pt−1
⎪
⎪
⎪
⎪
⎪
⎪
⎪
1
⎪
⎪
⎪
⎪
⎪
p t 1 k − pk α Apt ⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
p1
p t − 1 1 k pk α ⎪
⎪
bpt−1 Bpt−1 r ; p1 − α ≥ 1,
⎪
− ⎩
p t 1 k − pk α Apt fz ≥
⎧
⎪
1 − bpt−1 r pt−1
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
p1 − α
⎪
⎪
− ⎪
⎪
Apt ⎪
p
t
k
−
pk
α
1
⎪
⎪
⎪
⎪
⎪
⎪
⎪
p1
p t − 1 1 k pk α ⎪
⎪
⎪
bpt−1 Bpt−1 r ;
− ⎪
⎪
⎨
p t 1 k − pk α Apt ⎪
⎪
⎪
1 − bpt−1 r pt−1
⎪
⎪
⎪
⎪
⎪
⎪
⎪
1
⎪
⎪
⎪
− ⎪
⎪
p
t
k
−
pk α Apt ⎪
1
⎪
⎪
⎪
⎪
⎪
⎪
⎪
p1
p
t
−
1
k
pk
α
1
⎪
⎪
bpt−1 Bpt−1 r ;
⎪
− ⎩
p t 1 k − pk α Apt p1 − α ≤ 1
p1 − α ≥ 1.
2.13
These bounds are sharp.
Journal of Inequalities and Applications
7
Proof. Suppose p1 − α ≤ 1. Let f ∈ HF p, t, α, k and |Apt | ≤ |Bn |. In view of 1.13, we get
∞ p pt−1 fz z bpt−1 z
−
|an |zn − |bn |zn npt
∞
≤ 1 bpt−1 r pt−1 |an | |bn |r p1
npt
≤ 1 bpt−1 r pt−1 ≤ 1 bpt−1 r pt−1 p1 − α
p t 1 k − pk α Apt ∞
p t 1 k − pk α Apt ×
|an | |bn |r p1
p1
−
α
npt
p1 − α
p t 1 k − pk α Apt ⎛
⎞
∞
n1 k − pk α Apt n1 k pk α Apt ×⎝
|an | |bn |⎠r p1
p1
−
α
p1
−
α
npt
≤ 1 bpt−1 r pt−1 p1 − α
p t 1 k − pk α Apt ⎞
∞
n1 k − pk α
n1 k pk α
×⎝
|An an | |Bn bn |⎠r p1 .
p1 − α
p1 − α
npt
⎛
2.14
Using Theorem 2.4i, we obtain
fz ≤ 1 bpt−1 r pt−1 p1 − α
p t 1 k − pk α Apt p1
p t − 1 1 k pk α bpt−1 Bpt−1 r
× 1−
p1 − α
1 bpt−1 r pt−1
p1 − α
p t 1 k − pk α Apt p1
p t − 1 1 k pk α bpt−1 Bpt−1 r .
− p t 1 k − pk α Apt The proofs of other cases are similar and so are omitted.
2.15
8
Journal of Inequalities and Applications
Corollary 2.6. If f ∈ HF p, t, α, k, then
w : |wz| <
⎧
⎫
p1 − α
⎪
⎪
⎪
⎪
1
−
⎪
⎪
⎪
⎪
Apt ⎪
⎪
p
t
k
−
pk
α
1
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
b
B
p
t
k
−
pk
α
−
p
t
−
1
k
pk
α
1
1
⎪
⎪
pt−1
pt−1
⎪
⎪
⎪
⎪
−
;
⎪
⎪
⎪
⎪
⎪
⎪
A
p
t
k
−
pk
α
1
pt
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎨
⎬
p1 − α ≤ 1⎪
⎪
⎪
⎪
⎪
1
⎪
⎪
⎪
⎪
⎪
⎪
1
−
⎪
⎪
Apt ⎪
⎪
p
t
k
−
pk
α
1
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
b
B
p
t
k
−
pk
α
−
p
t
−
1
k
pk
α
1
1
pt−1
pt−1
⎪
⎪
⎪
⎪
⎪
⎪
−
;
⎪
⎪
Apt ⎪
⎪
p
t
k
−
pk
α
1
⎪
⎪
⎪
⎪
⎩
⎭
p1 − α ≥ 1
⊂ fU.
2.16
Theorem 2.7. Suppose An / 0 n p t, n p t 1, . . . and Bn / 0 n p t − 1, n p t, . . ..
Then f ∈ clco HF p, t, α, k if and only if
∞
Xn hn z Yn gn z ,
fz z ∈ U,
2.17
npt−1
where
hpt−1 z zp
⎧
p1 − α
⎪
⎪
zn ;
zp − ⎪
⎪
⎨
n1 k − pk α |An |
hn z ⎪
⎪
1
⎪
p
⎪
zn ;
⎩z − n1 k − pk α |An |
gn z n p t, p t 1, . . . , p1 − α ≤ 1,
n p t, p t 1, . . . , p1 − α ≥ 1,
⎧
p1 − α
⎪
⎪
zn ;
zp ⎪
⎪
⎨
n1 k pk α |Bn |
⎪
⎪
1
⎪
p
⎪
zn ;
⎩z n1 k pk α |Bn |
⎛
Xpt−1 ≡ Xp 1 − ⎝
∞
Xn npt
∞
n p t − 1, p t, . . . , p1 − α ≤ 1
n p t − 1, p t, . . . , p1 − α ≥ 1
⎞
Yn ⎠ ,
Xn ≥ 0, Yn ≥ 0.
npt−1
2.18
In particular, the extreme points of HF p, t, α, k are {hn } and {gn } .
Journal of Inequalities and Applications
9
Proof. Suppose p1 − α ≤ 1. For functions of the form 2.17, we can write
∞
fz zp −
npt
∞
p1 − α
p1 − α
Xn zn Yn zn . 2.19
n1 k − pk α |An |
n1
k
pk
α
|
|B
n
npt−1
On the other hand, for 0 ≤ Xp ≤ 1, we obtain
∞
n1 k − pk α |An |
p1 − α
Xn
p1 − α
n1 k − pk α |An |
npt
n1 k pk α |Bn |
p1 − α
Yn
p1 − α
n1 k pk α |Bn |
npt−1
∞
⎛
⎝
∞
Xn npt
∞
2.20
⎞
Yn ⎠ 1 − Xp ≤ 1.
npt−1
Thus f ∈ HF p, t, α, k, by Theorem 2.4.
Conversely, suppose that f ∈ HF p, t, α, k. Then, it follows Theorem 2.4 that
|an | ≤ p1 − α
,
n1 k − pk α |An |
p1 − α
.
|bn | ≤ n1 k pk α |Bn |
2.21
Setting
Xn n1 k − pk α |an An |
,
p1 − α
Yn n1 k pk α |bn Bn |
,
p1 − α
2.22
and defining
⎛
Xp 1 − ⎝
∞
Xn npt
∞
⎞
Yn ⎠ ,
npt−1
2.23
10
Journal of Inequalities and Applications
where Xp ≥ 0, we obtain
fz zp −
∞
npt
zp −
zp −
∞
|an |zn |bn |zn
npt−1
∞
∞
p1 − αXn
p1 − αYn
zn zn
n1
k
−
pk
α
n1
k
pk
α
|
|
|A
|B
n
n
npt
npt−1
∞
∞
zp − gn z Yn
zp − hn zXn −
npt
⎛
⎛
⎝1 − ⎝
∞
Xn npt
Xp zp 2.24
npt−1
∞
∞
⎞⎞
Yn ⎠⎠zp npt
hn zXn npt
npt−1
Xn hn z ∞
∞
∞
gn zYn
npt−1
Yn gn z.
npt−1
Thus f can be expressed as 2.17. The proof for the case p1 − α ≥ 1 is similar and hence is
omitted.
Theorem 2.8. The class HF p, t, α, k is closed under convex combinations.
Proof. For j 1, 2, . . . , let the functions fj given by
fj z zp −
∞ ∞ aj zn bj zn
n
n
npt
2.25
npt−1
are in HF p, t, α, k. Also suppose the given fixed harmonic functions are given by
Fj z zp ∞ ∞ Aj zn Bj zn .
n
n
npt
For
∞
j1
2.26
npt−1
μj 1, 0 ≤ μj ≤ 1 the convex combinations of fj can be expressed as
∞
j1
μj fj z zp −
∞
⎞
⎞
⎛
⎛
∞
∞
∞
⎝ μj ajn ⎠zn ⎝ μj bjn ⎠zn .
npt
j1
npt−1
j1
2.27
Journal of Inequalities and Applications
11
Since
∞
npt
∞
n1 k − pk α ajn Ajn n1 k pk α bjn Bjn npt−1
≤
⎧
⎨p1 − α
if p1 − α ≥ 1
⎩1
if p1 − α ≤ 1,
2.28
2.27 yields
∞
∞
∞
∞
μj ajn Ajn μj bjn Bjn n1 k − pk α
n1 k pk α
npt
j1
npt−1
j1
⎫
⎧
∞
∞
∞
⎨
⎬
μj
n1 k − pk α ajn Ajn n1 k pk α bjn Bjn ⎭
⎩npt
j1
npt−1
≤
⎧
∞
⎪
⎪
μj p1 − α
p1
−
α
⎪
⎪
⎪
⎪
j1
⎪
⎨
if p1 − α ≤ 1
⎪
⎪
∞
⎪
⎪
⎪
⎪
μj 1
⎪
⎩
if p1 − α ≥ 1.
j1
2.29
Thus the coefficient estimate given by Theorem 2.4 holds. Therefore, we obtain
∞
μj fj z ∈ HF p, t, α, k .
2.30
j1
Acknowledgment
This present investigation is supported with the Project no. DÜBAP-07-02-21 by Dicle
University, The committee of the Scientific Research Projects.
References
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Mathematical Sciences Research Journal, vol. 7, no. 9, pp. 347–352, 2003.
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Journal of Inequalities in Pure and Applied Mathematics, vol. 7, no. 5, article 190, pp. 1–9, 2006.
3 O. P. Ahuja and J. M. Jahangiri, “Multivalent harmonic starlike functions,” Annales Universitatis Mariae
Curie-Skłodowska. Sectio A, vol. 55, no. 1, pp. 1–13, 2001.
4 J. M. Jahangiri, “Harmonic functions starlike in the unit disk,” Journal of Mathematical Analysis and
Applications, vol. 235, no. 2, pp. 470–477, 1999.
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Journal of Inequalities and Applications
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