European Journal of Scientific Research ISSN 1450-216X / 1450-202X Vol. 136 No 2 November, 2015, pp.114-121 http://www.europeanjournalofscientificresearch.com
On Harmonic Univalent Functions Defined by Integral Convolution Waggas Galib Atshan Department of Mathematics ,College of Computer Science and Mathematics University of Al-Qadisiya,Diwaniya-Iraq E-mail:
[email protected] ,
[email protected] Enaam Hadi Abd Department of Computer ,College of Science University of Kerbala , Kerbala-Iraq. Department of Mathematics ,College of Science University of Baghdad, Baghdad-Iraq. E-mail:
[email protected] Abstract In this paper,we introduce and study a subclass of harmonic univalent functions defined by integral convolution (, )in the unit disk U={ :| |< 1}. We obtain some geometric properties, like coefficient inequality, extreme point and convex combination and convolution (or Hadamard product) for functions in the class (, ) are given. Keywords: harmonic univalent functions, Bernardi operator, Jung-Kim-Srivastava integral operator, Hadamardproduct. AMS Subject Classification: 30C45. A continuous complex valued function = + defined in a simply connected complex domain D is said to be harmonic in D, if both u and v are real harmonic in D. In any simply connected domain we can write = ℎ + ̅ , 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 sense-preserving in D is that |ℎ ( )| > | ( )| in D . Let denote the class of functions = ℎ + ̅ that are harmonic univalent and sensepreserving in the unit disk U={ :| |< 1} for which (0) = ℎ(0) = (0) − 1 = 0. For = ℎ + ̅ ∈ we may express the analytic functions h and g as
1. Introduction.
!
ℎ( ) = + , "#
!
( ) = $ |$% | < 1 , 0 < ≤ 1 "%
(1)
The class ( is defined as the subclass of consisting of all functions = ℎ + ̅ where h and g are given by !
ℎ( ) = + | | , "#
!
( ) = |$ | |$% | < 1 , 0 < ≤ 1 "%
(2)
On Harmonic Univalent Functions Defined by Integral Convolution
115
The harmonic function = ℎ + ̅ for ≡ 0 reduces to an analytic univalent function ≡ ℎ. In 1984, Clunie and Sheil-Small [4] investigated the class and as well as its geometric subclasses and obtained some coefficient bounds . Since then, there have been several related papers on and its subclass such as Avci and Zlotkiewicz[2] Silverman[7], Silverman and Silvia[8] and Jahangiri[5] studied the harmonic univalent functions . The convolution of two functions of form ( ) = + ∑!"# and,( ) = + ∑!"# - is defined as !
( ∗ ,)( ) = ( ) ∗ ,( ) = + - .
(3)
"#
The integral convolution is defined by ! - ( ⋄ ,)( ) = + . (4) 2 "# We consider the subclass (, )of functions of the form (1) that satisfy the condition: 7(ℎ( ) ⋄ 8( )) + (( ) ⋄ 9( )): 45 6 ;> , 0 ≥0. Let ( (, ) denote the subclass of (, ) consisting of functions = ℎ + ̅ ∈ such that h and g are of the form (2) . Lemma(1)[1]:Let ≥ 0 . Then 45@AB > if and only if |A − (1 + )| < |A + (1 − )|, where w be any complex number . 2. Coefficient Estimates We begin with asufficient condition for function in the class (, ) . Theorem(2.1): Let = ℎ + ̅ defined in eq(1). If ! ! 2 − = 2 − > C D | | + C D |$ | − ( − ) ≤ 0 , 2 2 "#
"%
where 0 < < ≤ 1 , then f is harmonic Univalent sense-preserving in U and ∈ (, ) . Proof: For proving ∈ (, ) , we must show that (5) holds true .If 7(ℎ( ) ⋄ 8( )) + (( ) ⋄ 9( )): -( ) A=6 ;= , ) F( ) ℎ( ) + (( ) then by lemma(1) 45(A) ≥ if and only if |A − (1 + )| < |A + (1 − )| , It suffices to show that |-( ) − (1 + )F( )| − |-( ) + (1 − )F( )| ≤ 0 . whereG(H) = 7(ℎ( ) ⋄ 8( )) + (( ) ⋄ 9( )): ) I(H) = ℎ( ) + (( ) Now find |G(H) − (J + K)I(H)|
! ! ! ! ! ! = L MNO + P ⋄ O + = PQ + RO $ P ⋄ O > PST − (1 + ) R + + O $ PSL !
= U N +
"# !
"#
"%
!
"%
!
"%
= $ > ( ̅) Q − (1 + ) N + + $ ( ̅) QU + 2 2 "%
!
"#
"#
!
!
"%
!
$ > = ( ̅) − (1 + ) − (1 + ) − (1 + ) $ ( ̅) U = U + + 2 2 "#
"%
"#
"%
"%
(7)
116
!
Waggas Galib Atshan and Enaam Hadi Abd
!
= > = U[ − (1 + )] − C(1 + ) − D − C(1 + ) − D $ ( ̅) U 2 2 !
"#
!
"%
2 − = + 2 2 − > + 2 = U[ − 1 − ] − C D − C D $ ( ̅) U 2 2 "# !
"%
!
2 − = + 2 2 − > + 2 D | || | + C D |$ || ̅| ≤ [1 + − ]| | + C 2 2
|G(H) + (J − K)I(H)| "#
"%
! ! ! ! ! ! = L MNO + P ⋄ O + = PQ + RO $ P ⋄ O > PST + (1 − ) R + + O $ PSL !
= U N +
"# !
"#
"%
!
"%
!
"%
= $ > ( ̅) Q + (1 − ) N + + $ ( ̅) QU + 2 2 "%
"#
!
"#
!
!
"%
"%
!
= $ > ( ̅) + (1 − ) + (1 − ) + (1 − ) $ ( ̅) U = U + + 2 2 "#
"%
!
"#
!
= > = U[ + (1 − )] + C(1 − ) + D + C(1 − ) + D $ ( ̅) U 2 2 !
"#
!
"%
"%
2 − = − 2 2 − > − 2 = U[ + 1 − ] − C D − C D $ ( ̅) U 2 2 "#
"%
!
!
2 − = − 2 2 − > − 2 ≥ [ + 1 − ]| | − C D | || | − C D |$ || ̅| 2 2 "#
"%
|G(H) − (J + K)I(H)| − |G(H) + (J − K)I(H)| ! ! 2 − = + 2 2 − > + 2 D | || | + C D |$ || ̅| Q ≤ N[1 + − ]| | + C 2 2 "# !
"% !
2 − = − 2 2 − > − 2 − N[ + 1 − ]| | − C D | || | − C D |$ || ̅| Q 2 2 "# !
= −2( − ) + 2 C !
= 2 C "#
"#
!
"%
2 − = 2 − > D | | + 2 C D |$ | 2 2 "%
!
2 − = 2 − > D | | + 2 C D |$ | − 2( − ) ≤ 0 2 2 "%
So we have ! ! 2 − = 2 − > = C D | | + C D |$ | ≤ ( − ) 2 2 "#
"%
X YZ ( ̅) , ( ) = + + 2 − = 2 − > "# "% 2 2 where !
!
!
| = ( − ) |X | + |Y "# !
!
"%
!
!
!
2 − = 2 − > | = ( − ) C D | | + C D |$ | = |X | + |Y 2 2 "# "% "# "% Theorem (2.2): Let = ℎ + ̅ with h and g given by (2). Then ∈ ( (, ) if and only if
On Harmonic Univalent Functions Defined by Integral Convolution !
117
!
2 − > 2 − = C D | | + C D |$ | ≤ ( − ) , 2 2 "#
(7)
"%
Proof: from theorem (2.1) to prove the necessary part let us assume that ( (, ) using (5),
we get 7(ℎ( ) ⋄ 8( )) + (( ) ⋄ 9( )): ; 45 6 ) ℎ( ) + (( ) = > C + ∑!"# | | + ∑!"% |$ |( ̅) D 2 2 = 45 [ \ ! ! + ∑ "#| | + ∑ "%|$ | = 45 [
= 45 [
> = | | + ∑!"% |$ |( ̅) D 2 2 \> + ∑!"#| | + ∑!"%|$ |
C + ∑!"#
> = | | + ∑!"% |$ | ( ̅) D 2 2 − \ > 0 + ∑!"#| | + ∑!"%|$ |
C + ∑!"#
If we choose z to be real and let → 1^ , we obtain the condition (7) and the proof is complete .
In the following theorem, we obtain the extreme points of the class ( (, ) . Theorem(3.1): Let f be given by (2) . Then ∈ ( (, ) iff f can be expressed as
3. Extreme Points !
( ) = _` ℎ ( ) + a ( )b , "%
whereℎ% ( ) = , ℎ ( ) = + (e
(d^e)
^dfg )
, 2 = 2,3, … , ( ) = + (e
1,2, … , ∑!"%(` + a ) = 1, (` ≥ 0, a ≥ 0). The extreme point of( (, ) are {ℎ },{ } . Proof: Assume that can be expressed by (8). Then , we have !
( ) = _` ℎ ( ) + a ( )b "% !
!
!
2( − ) 2( − ) = (` + a ) + ` + a ( ̅) (2 − = ) (2 − > ) "%
!
"#
"%
!
2( − ) 2( − ) = + ` + a ( ̅) (2 − = ) (2 − > ) "#
"%
Therefore, ! ! (2 − = ) 2( − ) 2( − ) 2 − > j kl m` + l ml ma (2 − = ) 2( − ) 2( − ) (2 − > ) "#
!
"%
= O (` + a ) − `% P = (1 − `% ) ≤ − "%
So ∈ ( (, ) . Conversely, let ∈ ( (, ). Then
(d^e)
^dig )
( ̅) , 2 =
(8)
118
Waggas Galib Atshan and Enaam Hadi Abd
(2 − = ) 2( − ) 2( − ) , $ ≤ no ` = , (2 − = ) (2 − > ) 2( − ) k=2,3,… and k=1,2,…,respectively we define ≤
!
a =
!
2 − > 2( − )
`% = 1 − ` − a . "#
"%
Then, note that 0 ≤ ` ≤ 1 (2 = 2,3, … ) , 0 ≤ a ≤ 1 (2 = 1,2, … ) . Hence , !
!
( ) = + + $ ( ̅) "# !
"%
!
2( − ) 2( − ) ` + a ( ̅) = + (2 − > ) (2 − = ) "# !
!
"%
= + (ℎ ( ) − )` + ( ( ) − ) a "# !
"% !
!
!
= O1 − ` − a P + ` ℎ ( ) + a ( ) "#
!
"%
"# !
"% !
= `% ℎ% ( ) + ` ℎ ( ) + a ( ) = _` ℎ ( ) + a ( )b , "#
"%
and the proof is complete .
"%
Now, we show ( (, ) is closed under convex combination of its members . Theorem(4.1): The class ( (, ) is closed under convex combination . Proof: For p = 1,2,3, … let q ∈ ( (, ) where q is given by
4. Convex Combination !
!
q ( ) = + ,q + $ ,q ( ̅) . "#
"%
Then, by (7), we have ! ! 2 − = 2 − > r ,q r + r$ ,q r ≤ 1 . 2( − ) 2( − ) "#
"%
For ∑! q"% 5q = 1 , 0 ≤ 5q ≤ 1 ,the convex combination of q may be written as
!
!
!
!
!
5q q ( ) = + t 5q r ,q ru + t 5q r$ ,q ru ( ̅) . q"% !
!
"#
q"%
Then, by (9) , we have
"%
!
q"% !
2 − = 2 − > t 5q r ,q ru + t 5q r$ ,q ru 2( − ) 2( − ) "#
q"%
"%
q"%
(2 − = ) 2 − > = 5q v r ,q r + r$ ,q rw ≤ 5q (1) = 1 . 2( − ) 2( − ) !
q"%
!
"#
Therefore,∑! q"% 5q q ( )
!
"%
∈( (, ) .
!
q"%
(9)
On Harmonic Univalent Functions Defined by Integral Convolution
119
5. Bernardi Operator
Definition(1)[3]: The Bernardi operator is defined by z+1 xy (2( )) = y { | y^% 2(|)o| , z ~ = @1,2, … B } If 2( ) = + ∑!"# , then ! z+1 xy (2( )) = + z+2 "#
Remark(1):If = ℎ + ̅ , we have !
!
( ) = $ ( ̅)
ℎ( ) = + , "#
"%
, ( ≥ 0,
(10) $ ≥ 0)
Then xy (( )) = xy (ℎ( )) + x y _( )b Theorem(5.1): If ∈ ( (, ), then xy () (z ~) is also in the class ( (, ) . Proof: By (10) and (11), we get !
!
(11)
xy (( )) = xy O + + $ ( ̅) P !
"#
"% !
z+1 z+1 = + + $ ( ̅) . z+2 z+2 "# "% Since ∈ ( (, ), then by theorem (2.2), we have
!
!
2 − = 2 − > | | + |$ | ≤ 1 . 2( − ) 2( − ) "#
"%
Sincez ∈ ~ , then
!
y%
y
≤ 1 , therefore !
2 − = z + 1 2 − > z + 1 l m | | + l m |$ | 2( − ) z + 2 2( − ) z + 2 "#
!
!
"%
2 − = 2 − > | | + |$ | ≤ 1 , ≤ 2( − ) 2( − ) "#
"%
and this gives the result .
6. Convolution (Hadamard Product) !
!
!
!
Define the convolution of two harmonic functions of the form : ( ) = + + $ ( ̅) , and
"#
"%
,( ) = + z + o ( ̅) . "#
"%
We define the convolution of two harmonic functions and F as !
!
( ∗ ,)( ) = ( ) ∗ ,( ) = + z + $ o ( ̅) "#
"%
Theorem(6.1): For 0 ≤ ≤ ≤ 1, let ∈ ( (, )and , ∈ ( (, ).Then ∗ , ∈ ( (, ). Proof: Since ∈ ( (, )and , ∈ ( (, ), then by Theorem (2.2), we have
120 !
Waggas Galib Atshan and Enaam Hadi Abd
!
2 − = 2 − > | | + |$ | ≤ 1 2( − ) 2( − ) "#
(12)
"%
And ! ! 2 − > 2 − = |z | + |o | ≤ 1 . 2( − ) 2( − ) "#
(13)
"%
From (13), we get the following inequalities 2( − ) 2( − ) z < , o < . 2 − = 2 − > Therefore ! ! 2 − > 2 − = | ||z | + |$ ||o | 2( − ) 2( − ) "#
"%
!
!
2 − = 2 − > | ||z | + |$ ||o | = |$% ||o% | + 2( − ) 2( − ) !
< $% +
"# !
"#
"%
!
2 − = 2 − > | | + |$ | 2 − = 2 − > "% !
2 − = 2 − > | | + |$ | ≤ 1 ≤ 2( − ) 2( − ) "# "% Then ∗ , ∈ ( (, ) ⊂ ( (, ) , and the proof is complete .
7. The June-Kim-Srivastava Integral Operator
Definition(2)[6]: The June-Kim-Srivastava integral operator is defined by 2 2( ) = { 2()o , > 0
() } ! If 2( ) = + ∑ "# , then ! 2 2( ) = + l m 2+1 "# , where Remark(2): If ( ) = ℎ( ) + ( ) !
!
ℎ( ) = + | |
, ( ) = |$ |( ̅)
"#
"%
,
$% < 1 ,
( ) then ( ) = ℎ( ) + Theorem(7.1): If ∈ ( (, ), then is also in ( (, ) . Proof: By (15) and (16), we obtain !
!
(14)
(15)
(16)
( ) = O + | | + |$ |( ̅) P
!
!
"#
"%
!
2 2 = + l m | | + l m |$ |( ̅) , 2+1 2+1 "# "% Since ∈ ( -(, ) , then by theorem(2.2), we have !
2 − = 2 − > | | + |$ | ≤ 1 , 2( − ) 2( − ) "#
"%
(17)
On Harmonic Univalent Functions Defined by Integral Convolution we must show ! ! 2 − = 2 2 − > 2 l m | | + l m |$ | ≤ 1 2( − ) 2 + 1 2( − ) 2 + 1 "#
"%
But in view of (17) the inequality in (18) holds true if ≤ 1 , % Since > 0 and 2 ≥ 1 , therefore (18) holds true and this gives the results . #
121 (18)
References [1] [2] [3] [4] [5] [6] [7] [8]
E. S. Aqlan, Some Problems Connected with Geometric Function Theory,Ph.D. Thesis(2004), Pune University ,Pune . Y. Avci and E. Zlotkiewicz ,On harmonic univalent mappings , Ann. Unv. MariaeCruieSklodowska Sect. A, 44(1990) 1-7 . S. D. Bernardi ,convex and starlike univalent function , Trans. Amer. Math. Soc. , 135(1969), 429-446 . J. Clunie, T. Shell-Small, Harmonic univalent functions, Ann. Acad. Sci. Fenn. Ser. Al. Math., 9, No. 3 (1984), 3-25 . J. M. Jahangiri, Harmonic functions starlike in the unit disk, J. math. Anal. Appl. 235, (1999), 470-477 . I. B. Jung, Y. C. Kim and H. M. Srivastava, The Hardy Space of analytic functions associated with one-parameter families of integral operators, J. Math. Appl. 176(1993), 138-197 . H. Silverman, Harmonic univalent functions with negative coefficients, Proc. Amer. Math. Soc. 51, (1998), 283-289 . H. Silverman and E. M. Silvia, Subclasses of harmonic univalent functions, New Zeal. J. Math. 28, (1999), 275-284 .
