doi:10.4236/am.2010.12011 Published Online July 2010 (http://www.SciRP.org/journal/am)
A Subclass of Harmonic Functions Associated with
Wright’s Hypergeometric Functions
Gangadharan Murugusundaramoorthy, Kalliyapan Vijaya
School of Advanced Sciences, Vellore Institute of Technology University, Vellore, India E-mail: [email protected], [email protected]
Received May 7, 2010; revised May 21, 2010; accepted June 25, 2010
Abstract
We introduce a new class of complex valued harmonic functions associated with Wright hypergeometric functions which are orientation preserving and univalent in the open unit disc. Further we define, Wright generalized operator on harmonic function and investigate the coefficient bounds, distortion inequalities and extreme points for this generalized class of functions.
Keywords:Harmonic Univalent Starlike Functions, Harmonic Convex Functions, Wright Hypergeometric Functions, Raina-Dziok Operator, Distortion Bounds, Extreme Points
1. Introduction
A continuous function f = u + iv is a complex-valued harmonic function in a complex domain G if both u and v are real and harmonic in G. In any simply-connected domain D ⊂ G, we can write f hg, where h and g are analytic in D. We call h the analytic part and g the co-analytic part of f. A necessary and sufficient condition for f to be locally univalent and orientation preserving in D is that |h'(z)||g'(z)| in D (see [1]).
Denote by H the family of functions
g h
f (1) which are harmonic, univalent and orientation preserving in the open unit disc U{z:|z|1} so that f is normal-ized by f(0)h(0) fz(0)10. Thus, for f hg
∈ H, we may express
, 1 | | , )
( 1
1 2
b z b z a z z f
m m m m
m
m (2)
where the analytic functions h and g are in the forms
1
2 1
( ) m, ( ) m (| | 1).
m m
m m
h z z a z g z b z b
We note that the family H of orientation preserving, normalized harmonic univalent functions reduces to the well known class S of normalized univalent functions if the co-analytic part of f is identically zero, that is g ≡0.
Let the Hadamard product (or convolution) of two po- wer series
2 )
(
m m mz
z
z
(3) and
2 ) (
m m mz
z
(4) in S be defined by
2 ) )( * (
m
m m m z
z
.
For positive real parameters 1, A1,...,p, Ap and 1
, B1,...,q Bq(p, q N = 1, 2, 3, ...) such that
. ,
0 ,
1
1 1
U z A
B p
m m q
m
m
(5) The Wright’s generalized hypergeometric function [2]
1 1 1 1
1, 1,
, , , , ; , , ;
, , ;
q p p q q
q m m p m m q
p A A B B z
p A B z
is defined by
1, 1,
1
0 0 0
, , ;
, .
!
p q t t p t t q
p q
t t t t
m t t
m
A B z
mA mB
z
z U m
the relationship:
, ! ) ( , ) (
) ( , ) (
; , ; ,
; , ,
0 1 1
1 1
, 1 ,
1
m z
z F
z B A
m
m m q m
m p m
q p
q p
q t t p t t q p
(6)
) }; 0 { .
; 1
(pq pqN0N zU is the generalized
hypergeometric function (see for details [3]) where N denotes the set of all positive integers and ()n is the
Pochhammer symbol and
q
t t p
t t 0 1 0
(7)
By using the generalized hypergeometric function Dz- iok and Srivastava [3] introduced the linear operator. In [4] Dziok and Raina extended the linear operator by us-ing Wright generalized hypergeometric function. First we define a function
A B z
z B A
q t t p t t q p
q t t p t t q p
; , ,
; , ,
, 1 ,
1 , 1 ,
1
Let W
t,At
1,p t,Bt
1,q
:SS be a linear operatordefined by
1, 1, 1, 1,
, , ( )
: , , ; * ( )
t t p t t q
p q t t p t t q
W A B z
z A B z z
We observe that, for f(z) of the form (1), we have
1,
1,1 2
, , ( )
: ( ) ,
t t p t t q
n m m m
W A B z
z z
(8)
where m(1) is defined by
)) 1 ( ( )) 1 ( ( )! 1 (
)) 1 ( (
)) 1 ( (
) (
1 1
1 1 1
m B m
B m
m A m
A
q q
p p m
(9)
If, for convenience, we write
( ), ( );( , ) ( , )
( ) )(
1 1 , 1 , 1 1
z B B
A A
W
z W
q q p
p
(10)
introduced by Dziok and Raina [4].
It is of interest to note that, if At = 1 (t = 1, 2, ..., p), Bt
= 1 (t = 1, 2, ...,q) in view of the relationship (6) the linear operator (8) includes the Dziok-Srivastava operator (see [3]), for more details on these operators see [3,4,6,7] and
[8]. It is interesting to note that Wright generalized hy-pergeometric function contains, Dziok-Srivastava opera-tor as its special cases, further other linear operaopera-tors the Hohlov operator, the Carlson-Shaffer operator [6], the
Ruscheweyh derivative operator [7], the generalized Bernardi-Libera-Livingston operator, the fractional de-rivative operator [8], and so on. For example if p = 2 and q = 1 with 11 ( 1), 21, 11, then
) ( * ) 1 ( ) (
) ( ) 1 ; 1 , 1 (
1 2
1
z z
z z
f D
z W
is called Ruscheweyh derivative of order δ (δ > −1). From (8) now we define, Wright generalized hyperge- ometric harmonic function f hg of the form (1), as
) ( ] [ ) ( ] [ ) ( ]
[ 1 f z W 1 hz W 1 g z
W p
q p
q p
q (11)
and we call this as Wright generalized operator on har-monic function.
Motivated by the earlier works of [1,5,9-13] on the subject of harmonic functions, we introduce here a new subclass WSH([1],) of H.
For 0 1, let WSH([1],) denote the subfam-ily of starlike harmonic functions f H of the form (1) such that
)) ( ] [ (argW 1 f z
p
q (12)
equivalently
) ( ] [ ) ( ] [
)) ( ' ] [ ( )) ( ' ] [ ( Re
1 1
1 1
z g W z h W
z g W z z h W z
p q p
q
p q p
q
(13)
where Wqp[1]f(z) is given by (11) and zU.
We also let WVH([1],)WSH([1],)VH where H
V the class of harmonic functions with varying argu-ments introduced by Jahangiri and Silverman [10], con-sisting of functions f of the form (1) in H for which there exists a real number φ such that
, 2 , 0
) 1 ( ), 2 (mod
) 1 (
m
m m
m m
(14)
where marg(am) and marg(bm).
In this paper we obtain a sufficient coefficient condi-tion for funccondi-tions f given by (2) to be in the class
) ], ([1 H
89 )
], ([1 H
WV . Further, distortion results and extreme points for functions in WVH([1],) are also obtained.
2. The Class
WS
H
α γ
1,
We begin deriving a sufficient coefficient condition for the functions belonging to the class WSH([1],).
Theorem 1. Let f hg be given by (2). If
2
1 1
| | | |
1 1
1 ( ) 1
1 m m m m m m a b b
(15) , 10 Then f WSH([1],).
Proof. We first show that if the inequality (15) holds for the coefficients of f hg , then the required con-dition (13) is satisfied. Using (11) and (13), we can write
) ( ) ( Re ) ( ] [ ) ( ] [ )) ( ' ] [ ( )) ( ' ] [ ( Re 1 1 1 1 z B z A z g W z h W z g W z z h W z p q p q p q p q where )) ( ' ] [ ( )) ( ' ] [ ( )
(z zW 1 h z zW 1 g z
A p
q p
q
and ) ( ] [ ) ( ] [ )
(z W 1 h z W 1 g z
B p
q p
q
In view of the simple assertion that Re(w) if and only if 1 w 1 w , it is sufficient to show that
( ) (1 ) ( ) ( ) (1 ) ( ) 0.
A z B z A z B z (16) Substituting for A(z) and B(z) the appropriate expres-sions in (16), we get
( ) (1 ) ( ) ( ) (1 ) ( )
A z B z A z B z
1 2
1 1
(2 ) ( 1 ) ( )
( 1 ) ( )
m m
m
m m
m m
m
z m a
z m b z
1 2 1 2( 1 ) ( )
( 1 ) ( )
m m
m
m m
m m
m
z m a
z m b z
1 2 1 1 1 22(1 ) 1 ( )
1 ( ) 1 m m m m m m m m m z a m
z b z
1 1 1 2 1 2(1 ) 11
( ) ( ) 0.
1 m m 1 m m
m z b m m a b
by virtue of the inequality (15). This implies that )
], ([1 H WS
f .
Now we obtain the necessary and sufficient condition for function f hg be given with condition (14).
Theorem 2. Let f hg be given by (2) and for 1
0 , then fWVH([1],) if and only if
1 1 2 1 1 1 ) ( | | 1 | |
1 b b
m a m
m
m m m
(17) Proof. Since WVH([1],)WSH([1],), we only need to prove the necessary part of the theorem. Assume that fWVH([1],), then by virture of (11) to (13), we obtain . 0 ) ( ] [ ) ( ] [ )) ( ' ] [ ( )) ( ' ] [ ( Re 1 1 1 1 z g W z h W z g W z z h W z p q p q p q p qThe above inequality is equivalent to
mm m m
m
m m m
m
m m m
m
m m m
z b z a z z b m z a m z 1 1 2 1 1 1 2 1 ) ( ) ( ) ( ) ( ) ( ) ( Re
= 0.
) ( ) ( 1 ) ( ) ( ) ( ) ( ) 1 ( Re 1 1 1 1 2 1 1 1 1 1
2 1
mm m m
m
m m m
m
m m m
m
m m m
This condition must hold for all values of z, such that |z| = r < 1. Upon choosing φ according to (14) we must have (18).
If (17) does not hold, then the numerator in (18) is negative for r sufficiently close to 1. Therefore, there exists a point zoro in (0, 1) for which the quotient in
(18) is negative. This contradicts our assumption that )
], ([1 H
WV
f . We thus conclude that it is both nec-essary and sufficient that the coefficient bound inequality (17) holds true when fWVH([1],). This completes the proof of Theorem 2.
If we put k
2 in (14), then Theorem 2 gives the following corollary.
Corollary 1. A necessary and sufficient condition for g
h
f satisfying (17) to be starlike is that arg(am) ,
/ ) 1 (
2m k
and arg(bm)2 2(m1)/k,(k1, ).
, 3 , 2
3. Distortion Bounds and Extreme Points
In this section we obtain the distortion bounds for the functions f WVH([1],) that lead to a covering re-sult for the family WVH([1],).
Theorem 3. If f WVH([1],) then 2
1 1
2 1
1 1 1
( ) (1 )
( ) 2 2
f z b r b r
and
2
1 1
2 1
1 1 1
( ) (1 ) .
( ) 2 2
f z b r b r
Proof. We will only prove the right-hand inequality of the above theorem. The arguments for the left-hand ineq- uality are similar and so we omit it. Let f WV H
) ],
([1 taking the absolute value of f, we obtain
1 2 ( )
(1 ) m
m m
m
f z
b r a b r
2 1
2
(1 ) m m .
m
b r r a b
This implies that
1
2 1
2
2 1 2 1
2
1 1
( ) (1 )
( ) 2
2 2
( ) ( )
1 m 1 m
m
f z b r
a b r
2
1 1
2 1
1 1 1
(1 ) 1
( ) 2 1
b r b r
2
1 1
2 1
1 1 1
(1 ) ,
( ) 2 2
b r b r
which establish the desired inequality.
As consequences of the above theorem and corollary 1, we state the following corollary.
Corollary 2. Let f hg and of the form (2) be so that fWVH([1],). Then
2 1 2 1 2 1
2 1 2 1
1 2 1
2 ( ) 1 [ ( ) 1] :
(2 ) ( ) 2 ( ) 1 [ ( ) 1]
( ). (2 ) ( )
w w
b f U
For a compact family, the maximum or minimum of the real part of any continuous linear functional occurs at one of the extreme points of the closed convex hull. Un- like many other classes, characterized by necessary and sufficient coefficient conditions, the family WVH([1],
)
is not a convex family. Nevertheless, we may still apply the coefficient characterization of the WVH([1],
)
to determine the extreme points.
Theorem 4. The closed convex hull of WVH([1],) (denoted by clcoWVH([1],)) is
1 . :, )
(
1 2
1 2
b b
a m
z b z a z z f
m m m
m m m m
m m
1 1
1 1 1
2
1
1 1 1
2 1
(1 ) (1 ) ( ) ( ) ( ) ( )
0.
1 ( ) ( )
m m
m m m m
m
m m m m m
m m
b m a r m b r
b a b r
91
By setting
) ( ) (
1
1
m m
m
and
) ( ) (
1
1
m m
m
,
then for b1 fixed, the extreme points for clcoWVH([ ],1
)
are
1
1
m m
m m
z xz b z z b z xz (19) where m2 and |x|1|b1|.
Proof. Any function f in clcoWVH([1],) be ex-pressed as
, )
(
2 1
2
m
m i m m
m i
m e z b z b e z
a z z
f m m
where the coefficient satisfy the inequality (15). Set ,
) (
1 z z
h g1(z)b1z, hm(z)zmeimzm, g zm( )b z1
,
m
i m
me z
for m = 2, 3,…Writing
m m
m
a X
, m
m m
b Y
m = 2, 3... and 1 ; 2
1
m m
X X
; 1
2
1
m m
Y
Y we get
. )] ( ) ( [ ) (
1
m m m m m
z g Y z h X z
f
In particular, putting
z b z z
f1( ) 1 and
m m m
m
m z z xz bz yz
f ( ) 1 ,
1
(m2, x y 1 b )
We see that extreme points of functions in clcoWVH
)} ( { ) ],
([1 fm z . To see that fm is not an extreme
point if both |x|0 and |y|0, we will show that it can then also be expressed as a convex linear combina-tions of funccombina-tions in clcoWVH([1],). Without loss of generality, assume |x| ≥ |y|. Choose 0 small enough so that
y x
. Set A1 and
y x B1 . We then see that both
m m m
mAxz bz Byz
z z
t1( ) 1
and
m m
m
m A xz bz B yz
z z
t2( ) (2 ) 1 (2 )
are in clcoWVH([1],) , and that { ( ) 2 1 )
(z t1 z
fm
)} (
2 z t
. The extremal coefficient bounds show that
functions of the form (19) are the extreme points for clco )
], ([1 H
WV , and so the proof is complete.
4. Inclusion Relation
Following Avici and Zlotkiewicz [9] (see also Rusch- eweyh [14]), we refer to the δ-neighborhood of the func-tion f(z) defined by (2) to be the set of functions F for which
2 2
1 1 2
( ) : ( ) ,
.
m m
m m
m m
m m m m m
N f F z z A z B z
m a A b B b B
(20)
In our case, let us define the generalized δ− neighbor-hood of f to be the set
1 2
1 1
( ) : : ( )[( )
( ) ] (1 ) (1 ) .
m m m
m
m m
N f F m a A
m b B b B
(21)
Theorem 5. Let f be given by (2). If f satisfies the conditions
, 1 ) ( ) (
) ( ) (
1 1
1 2
m m m
m
m m
b m m
a m m
(21)
1
0 and 0 1, then N(f)WSH([1],). Proof. Let f satisfy (22) and F(z) be given by
, )
(
2
2
m m m m
m
mz B z
A z z
F
which belongs to N(f). We obtain
) ( ) ( ) ( | | ) 1
( 1
2
1
m m
m m
B m A m
B
1 1 1
1 2
1 2
(1 ) (1 )
( )( )
( )
( ) ( ) ( )
m m m
m
m m
m m m
m
B b b
m A a
m B b
m a m b
1
1 2
(1 ) (1 ) 1
( ) ( ) ( )
2 m m m m
b
m m a m b
(1 ) (1 )
1 . 21 | | ) 1 ( ) 1
( 1 1
b b
Hence for
| |
1 1 1 2 1
1
b
, we infer that
) ], ([1 H
WS which concludes the proof of Theorem 5. Now, we will examine the closure properties of the class WVH([1],) under the generalized Bernardi-Lib- era-Livingston integral operator Lc(f) which is de-fined by
z c
c
c t f t dt c
z c f L
0
1 () , 1.
1 )
(
Theorem 6. Let f WVH([1],). Then Lc(f)
) ], ([1 H
WV .
Proof. From the representation of Lc(f), it follows that
z c
c
c t h t g t dt
z c f L
0
1 ( ) ( ) 1
)
(
z
m m m c
z
m m m c
c
dt t b t
dt t a t t z c
0 1
1
0 2
1
1
1
2 m
m m m
m
mz B z
A
z ,
where
m m a
m c
c A
1
; m bm
m c
c B
1
. Therefore
) ( | | 1 1
| | 1
1 1
1
m m
m
m b
m c
c m a m c
c m
) ( | | 1 | |
1 1
1
m
m m m
b m a m
2(1).
Since fWVH([1],), therefore by Theorem 2, )
], ([ )
( H 1
c f WV
L .
Theorem 7. For 0 1, let fWVH([1],) and FWVH([1],). Then (f*F)WVH([1],)
) ], ([1 H
WV .
Proof. Let
) ], ([ )
( 1
1 2
H m
m m m
m
m z b z WV
a z z
f
and
) ], ([ )
( 1
1 2
H m
m m m
m
m z B z WV
A z z
F
Then
1 2
) ( * ) (
m
m m m m
m m
ma z B b z
A z z F z
f
For (f*F)WVH([1],) we note that Am 1
and Bm 1. Now by Theorem 2, we have
1
1 2
1
| || | 1
) ( ) (
| || | 1
) ( ) (
m m m
m
m m m
m
B b m
A a m
1
1 2
1 | |
1
) ( ) ( | | 1
) ( ) (
m
m m m
m
m a m b
m
. 1 | | 1
) ( ) ( | | 1
) ( ) (
1
1 2
1
m m
m
m m
m a m b
m
Therefore (f*F)WVH([1],)WVH([1],) and since the above inequality bounded by 2(1) while
) 1 ( 2 ) 1 (
2 .
5. Concluding Remarks
The various results presented in this paper would provide interesting extensions and generalizations of those con-sidered earlier for simpler harmonic function classes (see [10,12,13]). The details involved in the derivations of such specializations of the results presented in this paper are fairly straight-forward.
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