Hindawi Publishing Corporation
International Journal of Mathematics and Mathematical Sciences
Volume 2012, Article ID 416875, 10 pages
doi:10.1155/2012/416875
Research Article
A New Subclass of Harmonic Univalent Functions
Associated with Dziok-Srivastava Operator
A. L. Pathak,1 K. K. Dixit,2 and R. Agarwal1
1
2
Department of Mathematics, Brahmanand College, The Mall, Kanpur 208004, India
Department of Mathematics, GIIT, Gwalior 474015, India
Correspondence should be addressed to A. L. Pathak, alpathak.bnd@gmail.com
Received 30 March 2012; Accepted 29 October 2012
Academic Editor: Attila Gilányi
Copyright q 2012 A. L. Pathak 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.
The purpose of the present paper is to study a certain subclass of harmonic univalent functions
associated with Dziok-Srivastava operator. We obtain coefficient conditions, distortion bounds,
and extreme points for the above class of harmonic univalent functions belonging to this class
and discuss a class preserving integral operator. We also show that class studied in this paper is
closed under convolution and convex combination. The results obtained for the class reduced to
the corresponding results for several known classes in the literature are briefly indicated.
1. Introduction
A continuous complex-valued function f u iv defined in a simply connected domain D, is
said to be harmonic in D if both u and v are harmonic in D. In any simply connected domain
D we can write f hg, where h and g are analytic in D. A necessary and sufficient condition
for f to be locally univalent and sense preserving in D is that |h z| > |g z|, z ∈ D. See
Clunie and Sheil-Small 1.
Denote by SH the class of functions f h g that are harmonic univalent and sense
preserving in the unit disk U {z : |z| < 1} for which f0 fz 0 − 1 0. Then for f h g ∈
SH , we may express the analytic function h and g as
hz z ∞
ak zk ,
k2
gz ∞
bk zk ,
|b1 | < 1.
1.1
k1
Note that SH , is reduced to S the class of normalized analytic univalent functions if
the coanalytic part of f h g is identically zero.
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International Journal of Mathematics and Mathematical Sciences
For more basic results on harmonic univalent functions, one may refer to the following
introductory text book by Duren 2 see also 3–5.
For αj ∈ C, j 1, 2, 3, . . . , q and βj ∈ C − {0, −1, −2, −3, . . .}, j 1, 2, 3, . . . , s, the
generalized hypergeometric functions are defined by
k
∞ α α · · · α
1 k
2 k
q z
k ,
q Fs α1 , α2 , . . . , αq ; β1 , β2 , . . . , βs ; z k0 β1 k β2 k · · · βs k k!
q ≤ s 1; q, s ∈ N0 {0, 1, 2, . . .} ,
1.2
where αk is the Pochhammer symbol defined by
⎧
⎨ Γα k αα 1 · · · α k − 1,
Γα
αk ⎩
1,
if k 1, 2, 3, . . . ,
if k 0.
1.3
Corresponding to the function
h α1 , α2 , . . . , αq ; β1 , β2 , . . . , βs ; z zq Fs α1 , α2 , . . . , αq ; β1 , β2 , . . . , βs ; z .
1.4
The Dziok-Srivastava operator 6, 7 Hq,s α1 , α2 , . . . , αq ; β1 , β2 , . . . , βs is defined for f ∈
S by Hq,s α1 , α2 , . . . , αq ; β1 , β2 , . . . , βs fz hα1 , α2 , . . . , αq ; β1 , β2 , . . . , βs ; z ∗ fz
∞ α 1 k−1 α2 k−1 . . . αq k−1
zk
,
z
ak
k − 1!
k2 β1 k−1 β2 k−1 . . . βs k−1
1.5
where ∗ stands for convolution of two power series.
To make the notation simple, we write
Hq,s α1 fz Hq,s α1 , α2 , . . . , αq ; β1 , β2 , . . . , βs fz.
1.6
Special cases of the Dziok-Srivastava operator includes the Hohlov operator 8, the
Carlson-Shaffer operator La, c 9, the Ruscheweyh derivative operator Dn 10, and the
Srivastava-Owa fractional derivative operators 11–13.
We define the Dziok-Srivastava operator of the harmonic functions f h g given by
1.1 as
1.7
Hq,s α1 fz Hq,s α1 hz Hq,s α1 gz.
Recently, Porwal 14, Chapter 5 defined the subclass MH β ⊂ SH consisting of
harmonic univalent functions fz satisfying the following condition:
MH β f ∈ SH : Re
zh z − zg z
hz gz
<β ,
1<β≤
4
, z ∈ U.
3
1.8
International Journal of Mathematics and Mathematical Sciences
3
He proved that if f h g is given by 1.1 and if
∞ k−β
∞ kβ
|ak | |bk | ≤ 1,
k2 β − 1
k1 β − 1
15.
1<β≤
4
.
3
1.9
For g ≡ 0 the class of MH β is reduced to the class Mβ studied by Uralegaddi et al.
Generalizing the class MH β, we let MH α1 , β denote the family of functions f hg
of form 1.1 which satisfy the condition
Re
⎫
⎧ ⎨ z Hq,s α1 hz − zHq,s α1 gz ⎬
⎩
Hq,s α1 hz Hq,s α1 gz
⎭
< β,
1.10
where 1 < β ≤ 4/3 and Γα1 , k |α1 k−1 α2 k−1 · · · αq k−1 /β1 k−1 β2 k−1 · · · βs k−1 |.
Further, let MH α1 , β be the subclass of MH α1 , β consisting of functions of the form
fz z ∞
∞
|ak |zk − |bk | zk .
k2
1.11
k1
In this paper, we give a sufficient condition for f h g, given by 1.1 to be in
MH α1 , β, and it is shown that this condition is also necessary for functions in MH α1 , β.
We then obtain distortion theorem, extreme points, convolution conditions, and convex
combinations and discuss a class preserving integral operator for functions in MH α1 , β.
2. Main Results
First, we give a sufficient coefficient bound for the class MH α1 , β.
Theorem 2.1. If f h g ∈ SH is given by 1.1 and if
∞ k−β
∞ kβ
Γα1 , k
Γα1 , k
|ak | |bk | ≤ 1,
−
1!
k
k − 1!
β
−
1
β
−
1
k2
k1
2.1
∞ k−β
∞ kβ
Γα1 , k
Γα1 , k
|ak | |bk | ≤ 1.
k − 1!
k − 1!
k2 β − 1
k1 β − 1
2.2
then f ∈ MH α1 , β.
Proof. Let
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International Journal of Mathematics and Mathematical Sciences
It suffices to show that
z Hq,s α1 hz − z Hq,s α1 gz / Hq,s α1 hz Hq,s α1 gz − 1
< 1,
z H α hz − zH α gz / H α hz H α gz − 2β − 1 q,s 1
q,s 1
q,s 1
q,s 1
z ∈ U.
2.3
We have
z Hq,s α1 hz − z Hq,s α1 gz / Hq,s α1 hz Hq,s α1 gz − 1
z H α hz − zH α gz / H α hz H α gz − 2β − 1 q,s 1
q,s 1
q,s 1
q,s 1
≤ 2 β−1 −
≤ 2 β−1 −
∞
k−1
k−1
∞
k2 k − 1A|ak ||z|
k1 k 1A|bk ||z|
∞ k−1
k−1
− ∞
k2 k − 2β 1 A|ak ||z|
k1 k 2β − 1 A|bk ||z|
∞
k2 k − 1A|ak | k1 k 1A|bk |
∞ ∞ k2 k − 2β 1 A|ak | −
k1 k 2β
∞
2.4
,
− 1 A|bk |
where A denotes Γα1 , k/k − 1!.
The last expression is bounded above by 1, if
∞
k2
k − 1
∞
∞
Γα1 , k
Γα1 , k
Γα1 , k
k − 2β 1
|ak | k 1
|bk | ≤ 2 β − 1 −
|ak |
k − 1!
k − 1!
k − 1!
k1
k2
−
∞
k1
Γα1 , k
k 2β − 1
|bk |,
k − 1!
2.5
which is equivalent to
∞ k−β
∞ kβ
Γα1 , k
Γα1 , k
|ak | |bk | ≤ 1.
−
1!
k
k − 1!
k2 β − 1
k1 β − 1
2.6
But 2.6 is true by hypothesis.
Hence
z Hq,s α1 hz − z Hq,s α1 gz / Hq,s α1 hz Hq,s α1 gz − 1
< 1,
z H α hz − zH α gz / H α hz H α gz − 2β − 1 q,s 1
q,s 1
q,s 1
q,s 1
z ∈ U,
2.7
and the theorem is proved.
International Journal of Mathematics and Mathematical Sciences
5
In the following theorem, it is proved that the condition 2.1 is also necessary for
functions f h g ∈ MH α1 , β that are given by 1.11.
Theorem 2.2. A function fz z ∞
k−β
k2
∞
k2
|ak |zk −
∞
k1
|bk |zk is in MH α1 , β, if and only if
∞
Γα1 , k
Γα1 , k
kβ
|ak | |bk | ≤ β − 1 .
k − 1!
k − 1!
k1
2.8
Proof. Since MH α1 , β ⊂ MH α1 , β, we only need to prove the “only if” part of the theorem.
For this we show that f ∈
/ MH α1 , β if the condition 2.8 does not hold.
Note that a necessary and sufficient condition for f h g given by 1.9 is in
MH α1 , β if
⎫
⎧ ⎨ z Hq,s α1 hz − zHq,s α1 gz ⎬
<β
Re
⎭
⎩
Hq,s α1 hz Hq,s α1 gz
2.9
is equivalent to
Re
∞ k
k
β−1 z− ∞
k2 k − β Γα1 , k/k − 1!|ak |z −
k1 k β Γα1 , k/k − 1!|bk |z
∞
k
k
z ∞
k2 Γα1 , k/k − 1!|ak |z −
k1 Γα1 , k/k − 1!|bk |z
≥ 0.
2.10
The above condition must hold for all values of z, |z| r < 1. Upon choosing the
values of z on the positive real axis where 0 ≤ z r < 1, we must have
k−1
k−1
β−1 − ∞
− ∞
k2 k − β Γα1 , k/k − 1!|ak |r
k1 k β Γα1 , k/k − 1!|bk |r
≥ 0.
∞
k−1
1 k2 Γα1 , k/k − 1!|ak |r k−1 − ∞
k1 Γα1 , k/k − 1!|bk |r
2.11
If the condition 2.8 does not hold then the numerator of 2.11 is negative for r and
sufficiently close to 1. Thus there exists a z0 r0 in 0,1 for which the quotient in 2.11
is negative. This contradicts the required condition for f ∈ MH α1 , β and so the proof is
complete.
Next, we determine the extreme points of the closed convex hulls of MH α1 , β,
denoted by clco MH α1 , β.
Theorem 2.3. f ∈ clco MH α1 , β, if and only if
fz ∞
xk hk z yk gk z ,
k1
2.12
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International Journal of Mathematics and Mathematical Sciences
where
h1 z z,
β − 1 k − 1! k
z , k 2, 3, . . .,
hk z z k − β Γα1 , k
∞
β − 1 k − 1! k
z , k 1, 2, 3, . . .,
gk z z − xk yk 1,
Γα
,
k
kβ
1
k1
2.13
xk ≥ 0, and yk ≥ 0. In particular the extreme points of MH α1 , β are {hk } and {gk }.
Proof. For functions f of the form 2.12, we have
fz ∞
xk hk z yk gk z
k1
∞ β−1
∞ β−1
k − 1!
k − 1!
k
xk z −
yk zk .
z
Γα
,
k
Γα
,
k
k
−
β
k
β
1
1
k2
k1
2.14
Then
∞ k−β
∞ kβ
β − 1 k − 1!
β − 1 k − 1!
Γα1 , k
Γα1 , k
xk yk
k − 1!
k − 1!
k − β Γα1 , k
k β Γα1 , k
k2 β − 1
k1 β − 1
∞
∞
xk yk
k2
2.15
k1
1 − x1 ≤ 1,
and so f ∈ clco MH α1 , β.
Conversely, suppose that f ∈ clco MH α1 , β. Set xk k − β/β − 1k −
1!/Γα1 , k|ak |, k 2, 3, 4, . . . and yk k β/β − 1k − 1!/Γα1 , k|bk |, k 1, 2, 3, . . ..
Then note that by Theorem 2.2, 0 ≤ xk ≤ 1, k 2, 3, 4, . . . and 0 ≤ yk ≤ 1, k ∞
1, 2, 3, . . .. We define x1 1 − ∞
k2 xk − k1 yk , and by Theorem 2.2, x1 ≥ 0.
Consequently, we obtain fz ∞
k1 xk hk z yk gk z as required.
Theorem 2.4. If f ∈ MH α1 , β, then
β−1
β1
−
|b1 | r 2 ,
2−β
2−β
β−1
β1
1
fz ≥ 1 − |b1 |r −
−
|b1 | r 2 ,
Γα1 , 2
2−β
2−β
fz ≤ 1 |b1 |r 1
Γα1 , 2
|z| r < 1,
2.16
|z| r < 1.
International Journal of Mathematics and Mathematical Sciences
7
Proof. We only prove the right hand inequality. The proof for left hand inequality is similar
and will be omitted. Let f ∈ MH α1 , β. Taking the absolute value of f, we have
∞
fz ≤ 1 |b1 |r |ak | |bk |r k
k2
≤ 1 |b1 |r ∞
|ak | |bk |r 2
k2
∞
2 − β Γα1 , 2
β−1
2 − β Γα1 , 2
1 |ak | |bk | r 2
2 − β Γα1 , 2 k2
β−1
β−1
∞
k − β Γα1 , k
β−1
k β Γα1 , k
1 |ak | |bk | r 2
2 − β Γα1 , 2 k2
β − 1 k − 1!
β − 1 k − 1!
β1
β−1
1
1− |b1 | r 2
2 − β Γα1 , 2
β−1
β−1
β1
1
−
|b1 | r 2 .
Γα1 , 2
2−β
2−β
2.17
1 |b1 |r ≤ 1 |b1 |r ≤ 1 |b1 |r 1 |b1 |r For our next theorem, we need to define the convolution of two harmonic functions.
For harmonic functions of the form
fz z ∞
|ak |zk −
k2
∞
|bk |zk ,
k1
2.18
∞
∞
Fz z |Ak |zk − |Bk |zk .
k2
k1
We define the convolution of two harmonic functions f and F as
∞
∞
f ∗ F z fz ∗ Fz z |ak Ak |zk − |bk Bk |zk .
k2
2.19
k1
Using this definition, we show that the class MH α1 , β is closed under convolution.
Theorem 2.5. For 1 < γ ≤ β ≤ 4/3, let f ∈ MH α1 , γ and F ∈ MH α1 , β. Then
f ∗ F ∈ MH α1 , γ ⊆ MH α1 , β .
2.20
∞
∞
k
k
k
Proof. Let fz z ∞
k2 |ak |z −
k1 |bk |z be in MH α1 , γ, and Fz z k2 |Ak |z −
∞
k
k1 |Bk |z be in MH α1 , β.
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International Journal of Mathematics and Mathematical Sciences
Then the convolution f ∗ F is given by 2.19. We wish to show that the coefficients of
f ∗ F satisfy the required condition given in Theorem 2.2. For Fz ∈ MH α1 , β we note that
|Ak | ≤ 1 and |Bk | ≤ 1. Now, for the convolution function f ∗ F, we obtain
∞
k − α Γα1 , k
k2
α − 1 k − 1!
≤
|ak Ak | ∞
k α Γα1 , k
k1
∞
k − α Γα1 , k
|ak | α − 1 k − 1!
|bk Bk |
∞
k α Γα1 , k
α − 1 k − 1!
α − 1 k − 1!
k2
k1
≤ 1,
since f ∈ MH α1 , γ .
|bk |
2.21
Therefore f ∗ F ∈ MH α1 , γ ⊆ MH α1 , β.
Theorem 2.6. The class MH α1 , β is closed under convex combination.
Proof. For i 1, 2, 3, . . ., let fi z ∈ MH α1 , β, where fi z is given by
fi z z ∞
|aki |zk −
k2
∞
|bki |zk .
2.22
k1
Then by Theorem 2.2,
∞ k−β
∞ kβ
Γα1 , k
Γα1 , k
|aki | |bk | ≤ 1.
k − 1!
k − 1! i
k2 β − 1
k1 β − 1
For
∞
i−1 ti
2.23
1, 0 ≤ ti ≤ 1, the convex combination of fi may be written as
∞
∞
∞
∞
∞
k
ti fi z z ti |aki | z −
ti |bki | zk .
i−1
k2
i−1
k1
2.24
i−1
Then by 2.23,
∞ k−β
∞
∞ kβ
∞
Γα1 , k Γα1 , k ti |aki | ti |bki |
k − 1! i−1
k − 1! i−1
k2 β − 1
k1 β − 1
∞
∞ k−β
∞ kβ
Γα1 , k
Γα1 , k
ti
|ak | |bk |
k − 1! i k1 β − 1 k − 1! i
i−1
k2 β − 1
≤
∞
ti 1.
i−1
This is the condition required by Theorem 2.2 and so
∞
i−1 ti fi z
∈ MH α1 , β.
2.25
International Journal of Mathematics and Mathematical Sciences
9
3. A Family of Class Preserving Integral Operator
Let fz hz gz be defined by 1.1; then Fz defined by the relation
Fz c1
zc
z
tc−1 htdt 0
c1
zc
z
tc−1 gtdt,
c > −1.
3.1
0
Theorem 3.1. Let fz hz gz ∈ SH be given by 1.11 and f ∈ MH α1 , β, then Fz is
defined by 3.1 also belong to MH α1 , β.
Proof. Let fz z ∞
k2
|ak |zk −
∞
k1
|bk |zk be in MH α1 , β, then by Theorem 2.2, we have
∞ k−β
∞ kβ
Γα1 , k
Γα1 , k
|ak | |bk | ≤ 1.
−
1!
k
k − 1!
β
−
1
β
−
1
k2
k1
3.2
By definition of Fz, we have
Fz z ∞
c 1
k2
c k
|ak |zk −
∞
c 1
k1
c k
|bk |zk .
3.3
Now we have
∞ k−β
∞ kβ
Γα1 , k c 1
Γα1 , k c 1
|ak | |bk |
k − 1! c k
k − 1! c k
k2 β − 1
k1 β − 1
∞ k−β
∞ kβ
Γα1 , k
Γα1 , k
≤
|ak | |bk |
k − 1!
k − 1!
k2 β − 1
k1 β − 1
3.4
≤ 1.
Thus Fz ∈ MH α1 , β.
Acknowledgments
The first author is thankful to University Grants Commission, New Delhi Government
of India for supporting this paper financially under research project no. 8-2 223/2011
MRP/Sc/NRCB. The authors are also thankful to the reviewers for giving fruitful
suggestions and comments.
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