Gen. Math. Notes, Vol. 20, No. 2, February 2014, pp. 79-94
ISSN 2219-7184; Copyright © ICSRS Publication, 2014 www.i-csrs.org
Available free online at http://www.geman.in
Waggas Galib Atshan
1
, A. H. Majeed
2
and Kassim A. Jassim
3
1
Department of Mathematics
College of Computer Science and Mathematics
University of Al-Qadisiya, Diwaniya-Iraq
E-mail: waggashnd@gmail.com; waggas_hnd@yahoo.com
2,3
Department of Mathematics, College of Science
University of Baghdad, Baghdad-Iraq
2
Email: ahmajeed6@yahoo.com
3
E-mail: kasim.hameed@yahoo.com
(Received: 16-10-13 / Accepted: 9-12-13)
Abstract
In this paper , we have studied a certain subclass of univalent functions defined by linear operator
, ,
by using differential subordination property.We obtain some geometric properties, like , coefficient inequality , neighborhoods of the class K(
, , , , , , , ),
convolution properties and integral mean inequalities for the fractional integral for this class.
Keywords: Univalent Function, Hypergeometric Function, Linear Operator,
Differential Subordination, Convolution, Integral Mean, Fractional Integral.

80 Waggas Galib Atshan et al.
Let S denote the class of functions of the form:-
) = + , 1) which are analytic in the open unit disk U={z
∈ ¢: | | < 1}
. Also, denoted by K the subclass of S consisting of functions of the form:-
) = − | | , 2) which are univalent and normalized in U.
For ∈ S, and of the form 1)and 4 ) ∈ S given by 4 ) = + : ,
We define the Hadamard product (or convolution)
∗ 4) ) = + : . 3)
For positive real values of
, … . ,
and
? , … . , ?
(
?
1,2, … . ),
@
≠ 0, −1, … . , C =
The generalized hypergeometric function
δ
F m
(z) is defined by
D
F
E z) ≡
D
F
E
, … , ; ? , … , ? ; )
=
) … )
? ) … ? )
(
≤ + 1
;
, ∈ N
I
I
= N ∪ P0}; ∈ Q), where
)
is the Pochhammer symbol defined by
J! 4)
+ 1) + 2) … + J − 1), J ∈ N.
S 5)
The notation
δ
F m
is quite useful for representing many well-known functions such as the exponential, the Binomial, the Bessel and Laguerre polynomial.

Some Geometric Properties of a Certain... 81
H[ , ,
W be a linear operator defined by
… . , ; ? , ? , ?
W
… . , ? ]: K → K
H[
, … , ; ? , … , ? ] ) = z
δ
F m
(
, , … , ; ? , ? , … , ? ; ) ∗ )
∞
= − [ ; ; )| | , 6) where,
[ ; ; ) =
? )
)
]
]
… )
… ? )
]
]
1
J − 1)! . 7)
For notational simplicity, we use shorter notation
_ [ ] for
H[
, … , ; ? , … , ?
].
In the sequel .It follows from (6) that
_
I
[1] ) = ), _
I
[2] ) = ` ).
The linear operator
_ [ ]
is called Dziok-Srivastava operator (see [5]) introduced by Dziok and Srivastava which was subsequently extended Dziok and
Raina [4] by using the generalized hypergeometric function, recently Srivastava et. al. [13] defined the linear operator
,
, ,
as follows:-
I,
, ,
) = )
,
, ,
) = 1 − )_ [ ] ) + a_ [ ] )b
`
=
, ,
), 8)
,
, ,
) =
, , a ,
, ,
)b, 9) and in general,
,
, ,
) =
, , a ] ,
, ,
)b , 0 ≤ ≤ 1, ≤ + 1; , ∈ N
I
= N ∪ P0} ; ∈ Q). 10)
If the function f(z) is given by (2), then we see from (6),(7),(8) and (10) that
,
, ,
) = − [ ; ; ; )| | , 11) where

82 Waggas Galib Atshan et al.
[ ; ; ; ) = e
? )
)
]
]
… )
… ? )
]
]
[1 + J − 1)]
J − 1)! f , 12)
J ∈ N\P1}, ∈ N
I ) .
Unless otherwise stated .We note that when
= 1 and
= 0
, the linear operator
,
, ,
would reduce to the familier Dziok-Srivastava linear operator given by(see[5]), includes (as its special cases) various other linear operators introduced and studied by Carlson and Shaffer [3], Owa [9] and Ruscheweyh [10].
For two analytic functions f,
4 ∈
K , we say that f is subordinate to
4
written
f(z)
≺ 4
(z) if there exists a schwarz function w(z), which (by definition) is analytic in U with w(0)=0 and |w(z)|<1 for all z
∈ Q such that f(z)=
4
(w(z)),z
∈ Q
.
Furthermore, if the function
4
(z) is univalent in U, then we have the following equivalence (see [8]):
f(z)
≺ 4
(z)
↔
f(0)=
4
(0) and f(U) ⊂
4
(U).
Definition (1): For any function f
∈
K and
∅ ≥ 0
, the
∅ − Jlm4ℎ:opℎooq
f is
defined as,
N
,∅
) = P4 ) = − |: | ∈ r; J|| | − |: || ≤ ∅}. (13)
In particular, for the function e(z)=z, we see that,
N
,∅
(l) = {4( ) = − |: | ∈ r; J| : | ≤ ∅}. (14)
The concept of neighborhoods was first introduced by Goodman [6] and then generalized by Ruscheweyh [11] , and studied by some authors, like Atshan [1] and Atshan and Kulkarni [2].
Definition (2): For fixed parameters A and B, with -1
≤ < ≤ 1
, we say that f
∈ r
is in class K(
, , , , , , , ), if it satisfies the following subordination
condition:
1 +
(
( st,
, ,
,
, ,
( ))′′
( ))′
≺
1 +
1 +
. (15)
In view of the definition of subordination, (15) is equivalent to the following condition:

Some Geometric Properties of a Certain... 83 v v
(
( st,
, ,
,
, ,
(
(
For convenience, we write st,
, ,
,
, ,
( ))′′
( ))′
( ))′′
( ))′ + ( − ) v v < 1, ( ∈ Q).
K(
, , , , , , 1 − 2w, −1) = K( , , , , , , w), where
K( , , , , , , w) denotes the class of functions in K satisfying the inequality:
Re y1 +
( a st,
, ,
,
, ,
( ))′′
( )b
` z > w, 0 ≤ w < 1; ∈ Q).
|, }, ~
•
, €, •, ‚, ƒ, „)
The following theorem gives a necessary and sufficient condition for a function f to be in the class K(
, , , , , , , ).
Theorem (1): A function f
∈ r
belong to the class K(
, , , , , , , )
if and only if
J(2J − 1) [ ( ; ; ; ) a(1 − )[ t ( ; ; ; ) + ( − )b | |
≤ − ) , 16) for
, , ∈ N
I
, ≤ + 1, ≥ 0 Jq − 1 ≤ < ≤ 1.
Proof: Let f
∈
K(
, , , , , , , )
. Then
1 + st,
, ,
,
, ,
))′′
))′
≺
1 +
1 +
, ∈ Q . 17)
Therefore, there exists an analytic function w such that
… ) = st,
, , st,
, ,
))′′
))′′ + − ) ,
, ,
))′
. 18)

84 Waggas Galib Atshan et al.
Hence,
|…( )| = †
( st,
, ,
( st,
, ,
( ))′′
( ))′′ + ( − )( ,
, ,
( ))′
†
= ‡
− ∑ J(J − 1) [ st ( ; ; ; )| |
( − ) − ∑ J [ ( ; ; ; ) a [ t ( ; ; ; ) + ( − )b | |
< 1.
Thus,
‰
∑ J(J − 1) [ st ( ; ; ; )| |
( − ) + ∑ J [ ( ; ; ; )( [ t ( ; ; ; ) + ( − ))| |
‡
‰ < 1.
Therefore,
Šl ‹
∑ J(J − 1) [ st ( ; ; ; )| |
( − ) + ∑ J [ ( ; ; ; )( [ t ( ; ; ; ) + ( − ))| |
Œ
< 1. (19)
Taking |z|=r, for sufficiently small r with 0<r<1, since w(z) is analytic for
|z|=1.Then , the inequality (19) yields
J(J − 1) [ st ( ; ; ; )| |p
< ( − )p + J [ ( ; ; ; ) a [ t ( ; ; ; ) − ( − )b | |p .
Equivalently,
J(2J − 1) [ ( ; ; ; )((1 − ) [ t ( ; ; ; ) + ( − ))| |p
≤ ( − )p and (16) follows upon letting r
→
1.
Conversely, for |z|=r, 0<r<1, we have r n
<r. That is,
J(2J − 1) [ ( ; ; ; )((1 − ) [ t ( ; ; ; ) + ( − ))| |p

Some Geometric Properties of a Certain... 85
≤ J(2J − 1) [ ( ; ; ; )((1 − ) [ t ( ; ; ; ) + ( − )) | |p
≤ − )p
.
From (16), we have
† J J − 1) [ st ; ; ; )| | †
≤ J J − 1) [ st ; ; ; )| |p
< − )p + J [ ; ; ; ) [ t ; ; ; ) + − ))| |p
< | − ) + J [ ; ; ; ) [ t ; ; ; ) + − ))| | |.
This proves that
1 + st,
, ,
,
, ,
))′′
))′ and hence f
∈
K(
, , , , , , , )
.
≺
1 +
1 +
Theorem (2): If
, ∈ Q,
∅ =
3 e
) … )
? ) … ? ) 1 + )f • 1 − )
−
) … )
? ) … ? ) 1 + )) t + − )Ž then K(
, , , , , , , )
⊂
N
•,∅ e).
Proof: It follows from (16), that if f
∈
K(
, , , , , , , )
, then
, 20)
3 [ ; ; ; ) a1 − ) [ t ; ; ; ) + − )b J| | ≤ − .
Hence
3
) … )
? ) … ? ) 1 + )) ‘ 1 − ) e
) … )
? ) … ? ) 1 + )f t
+ − )’

86 Waggas Galib Atshan et al.
× J | | ≤ − , 21) which implies that
J| | ≤
3
−
) … )
? ) … ? ) 1 + )) ‘ 1 − ) ”
) … )
? ) … ? ) 1 + )• t
+ − )’
=
∅
(22)
Using (14), we get the result.
Definition (3): The function
4
defined by
4
(z)=
− ∑
∞
|: | member of the class K c f
∈
K(
, , , , , , , )
such that
is said to be a
(
, , , , , , , )
if there exist a function
–
4 )
) − 1– ≤ 1 − — ,
∈ Q, 0 ≤ — < 1). 23)
Theorem (3): If f
∈
K(
, , , , , , , ) and
— = 1 −
6 [
3 ∅ [ ; ; ; ) 1 − ) [ ; ; ; ) + − ))
; ; ; ) 1 − )[ ; ; ; ) + − )) − − )
, 24) then
N
•,∅
)
⊂ K c
(
, , , , , , , ).
Proof: Let
4 ∈ N
•,∅
).
Then we have from (13) that
J|| | − |: || ≤ ∅, which implies the coefficient inequality
|| | − |: || ≤
∅
2.
Also, since f
∈
K(
, , , , , , , )
, we have from (16)
| | ≤
6 [
−
, , , ) 1 − ) [ t ; ; ; ) + − ))
,

Some Geometric Properties of a Certain... 87 where
[
[ ( ; ; ; ) = (
( ) … ( )
(? ) … (? ) (1 + )) , t ( ; ; ; ) = (
( ) , … . , ( )
(? ) , … . , (? ) (1 + )) t , so that
–
4( )
( ) − 1– = ‰
∑ (| | − |: |)
− ∑ | |
‰ <
∑ || | − |: ||
1 − ∑ | |
≤ ∅.
3 [ ( ; ; ; )((1 − ) [
6 [ ( ; ; ; )((1 − ) [ t t ( ; ; ; ) + ( − ))
( ; ; ; ) + ( − )) − ( − )
=1-c.
Thus by Definition (3),
4 ∈
K c
(
, , , , , , , )
for c given by (24).
This completes the proof.
Theorem (4): Let the functions f j
(j=1,2) defined by
@
( ) = − |
,@
| , (C = 1,2), (25) be in the class K(
, , , , , , , )
.Then f
1 where
*f
2
∈ K( , , , , , , , ˜)
,
˜ ≤
J(2J − 1) [ ( ; ; ; )((1 − ) [ t
J(2J − 1) [ ( ; ; ; )((1 − ) [ t
( ; ; ; ) + ( − )) − ( − ) ( + [ t
( ; ; ; ) + ( − )) − ( − ) (1 + [ t
( ; ; ; ))
( ; ; ; ))
Proof: We must find the largest
˜
such that
J(2J − 1) [ ( ; ; ; )(a1 − ˜) [ t ( ; ; ; ) + ( − ˜)b
− ˜
™
,
™|
,
| ≤ 1.
Since @
∈ K( , , , , , , , )(C = 1,2), then
J(2J − 1) [ ( ; ; ; )(a1 − ) [ t ( ; ; ; ) + ( − )b
−
™
By Cauchy –Schwarz inequality, we get
,@
™ ≤ 1 , ( C = 1,2). (26)

88 Waggas Galib Atshan et al.
J(2J − 1) [ ( ; ; ; )((1 − ) [ t
−
( ; ; ; ) + ( − ))
š™
,
™|
,
| ≤ 1 . (27)
We want only to show that
J(2J − 1) [ ( ; ; ; )((1 − ˜) [
− ˜
J(2J − 1) [ ( ; ; ; )((1 − ) [
− t t
( ; ; ; ) + ( − ˜))
( ; ; ; ) + ( − ))
™
š™
,
™|
,
™|
,
| ≤
,
|.
This equivalently to
š™
,
™|
,
| ≤
( − ˜)((1 − ) [
( − )((1 − ˜) [ t t
( ; ; ; ) + ( − ))
( ; ; ; ) + ( − ˜))
.
From (27), we have
š™
,
™|
,
| ≤
−
J(2J − 1) [ ( ; ; ; )((1 − ) [ t ( ; ; ; ) + ( − ))
.
Thus, it is sufficient to show that
−
J(2J − 1) [ ( ; ; ; )((1 − ) [ t ( ; ; ; ) + ( − ))
( − ˜)((1 − ) [
( − )((1 − ˜) [ t t
( ; ; ; ) + ( − ))
( ; ; ; ) + ( − ˜))
,
≤ which implies to
˜ ≤
J(2J − 1) [ ( ; ; ; )((1 − ) [
J(2J − 1) [ ( ; ; ; )((1 − ) [ t t ( ; ; ; ) + ( − )) − ( − ) ( + [ t
( ; ; ; ) + ( − )) − ( − ) (1 + [ t
( ; ; ; ))
( ; ; ; ))
This completes the proof.
Theorem (5): Let the functions f j
(j=1, 2) defined by (25) be in the class r( , , , , , , , ).
Then the function h defined by
ℎ( ) = − (|
,
| + |
,
| ) (28) belong to the class
K( , , , , , , , ›)
, where

Some Geometric Properties of a Certain... 89
› ≤
J(2J − 1) [ ( ; ; ; )((1 − ) [ t
J(2J − 1) [ ( ; ; ; )((1 − ) [ t
( ; ; ; ) + ( − )) − 2[ t ( ; ; ; )( − ) − 2( − )
( ; ; ; ) + ( − )) + 2 ( − ) + 2[ t ( ; ; ; )( − )
Proof: We must find the largest
›
such that
J(2J − 1) [ ( ; ; ; ) a(1 − ›) [ t ( ; ; ; ) + ( − ›)b
− ›
(™
,
™ + ™
,
™ ) ≤ 1.
Since @
∈ K( , , , , , , , ) (C = 1,2), we get e
J(2J − 1) [ ( ; ; ; )((1 − ) [
− t ( ; ; ; ) + ( − )) f ™
,
™
≤ •
J(2J − 1) [ ( ; ; ; )((1 − ) [ t
−
( ; ; ; ) + ( − ))
™
,
™ž ≤ 1 , (29) and e
J(2J − 1) [ ( ; ; ; )((1 − ) [
− t ( ; ; ; ) + ( − )) f ™
,
™
≤ •
J(2J − 1) [ ( ; ; ; )((1 − ) [
− t ( ; ; ; ) + ( − ))
™
Combining the inequalities (29) and (30), gives
1
2 e
J(2J − 1) [ ( ; ; ; )((1 − ) [
− t ( ; ; ; ) + ( − )) f ™
,
™ + ™
,
™ž ≤ 1. (30)
,
™ ≤ 1 . (31)
But ℎ ∈ K( , , , , , , , ›)
if and only if
(
J(2J − 1) [ ( ; ; ; )((1 − ›) [
− › t ( ; ; ; ) + ( − ›))
(™
,
™ + ™
,
™ ) ≤ 1 . (32)
The inequality (32) will be satisfied if
J(2J − 1) [ ( ; ; ; )((1 − ›) [
− › t ( ; ; ; ) + ( − ›))
≤
(J(2J − 1) [ ( ; ; ; )((1 − ) [ t
2( − )
( ; ; ; ) + ( − ))
, (J = 2,3, … ) , (33)

90 Waggas Galib Atshan et al. so that,
› ≤
J(2J − 1) [ ( ; ; ; )((1 − ) [ t
J(2J − 1) [ ( ; ; ; )((1 − ) [ t
( ; ; ; ) + ( − )) − 2[ ( ; ; ; )( − ) − 2( − )
( ; ; ; ) + ( − )) + 2 ( − ) + 2[ ( ; ; ; )( − )
This completes the proof.
Definition (4) [12]: The fractional integral of order s (s>0) is defined for a
function f by:
Ÿ ]¡ ( ) =
1
Γ
(¢) £
I
(¤)
( − ¤) ]¡ q¤ , where the function f is an analytic in a simply- connected region of the complex zplane containing the origin , and multiplicity of
( − ¤) ¡]
is removed by requiring log(z-t) to be real, when (z-t)>0.
In 1925, Littlewood [7] proved the following subordination theorem:-
Theorem (6) (Littlewood [7]): If f and
4
are analytic in U with f w > 0 and
= pl ¥¦
(0<r<1)
≺ 4
, then for
¨
£ | ( )| §
I
¨ q ≤ £ |4( )| §
I q
Theorem (7): Let f
∈ r( , , , , , , , )
and suppose that f n
is defined by
( ) = −
−
J(2J − 1) [ ( ; ; ; )((1 − ) [ t ( ; ; ; ) + ( − ))
, (J ≥ 2). (34)
Also, let
(m − ©)
ªs
|
¥
|
≤
( − )
Γ
(J + 1)
Γ
(¢ + © + 3)
J(2J − 1) [ ( ; ; ; )((1 − ) [ t ( ; ; ; ) + ( − ))
Γ
(J + ¢ + © + 1)
Γ
(2 − ©)
, (35) for
0 ≤ © ≤ m, ¢ > 0
, where
(m − ©) by
(m − ©)
ªs
= (m − ©)(m − © + 1) … m.
denote the Pochhammer symbol defined
If there exists an analytic function q defined by

Some Geometric Properties of a Certain... 91
«¬( )-
]
=
J(2J − 1) [ ( ; ; ; )(1 − ) [ t ( ; ; ; ) + ( − ))
Γ
(J + ¢ + © + 1)
( − )
Γ
(J + 1)
× (m − ©)
ªs
¥ where i
≥ ©
and
_(m)|
¥
| ¥] , (36)
_(m) =
Γ
(m − ©)
Γ
(m + ¢ + © + 1), (¢ > 0, m ≥ 2), (37) then , for z = pl
¨
£ |Ÿ
I
]¡]ª ( )| §
¥ ϕ and 0 < p < 1
¨ q ϕ
≤ £ |Ÿ
I
]¡]ª
Proof: Let
( ) = − ∑ |
¥
| ¥
( )|
.
§
For q ϕ
(¢ > 0, w > 0) . (38)
© ≥ 0
and Definition (4), we get
Ÿ ]¡]ª ( ) =
Γ
(2) ¡sªs
Γ
(¢ + © + 2) (1 −
¥
Γ
Γ
(m + 1)
(2)
Γ
Γ
(¢ + © + 2)
(m + ¢ + © + 1)
|
¥
| ¥] )
=
Γ
(2) ¡sªs
Γ
(¢ + © + 2) •1 −
¥
Γ
(¢ + © + 2)
Γ
(2)
(m − ©)
ªs
_(m)|
¥
| ¥] ž, where
_(m) =
Γ
(m − ©)
Γ
(m + ¢ + © + 1) ,
(¢ > 0, m ≥ 2)
Since H is a decreasing function of i, we have
0 < _(m) ≤ _(2) =
Γ
(2 − ©)
Γ
(¢ + © + 3).
Similarly, from (34) and Definition (4), we get
Ÿ ]¡]ª ( ) =
Γ
(2) ¡sªs
Γ
(¢ + © + 2) (1 −
( − )
Γ
(J + 1)
Γ
(¢ + © + 2)
J(2J − 1) [ ( ; ; ; )((1 − ) [ t ( ; ; ; ) + ( − ))
Γ
(J + ¢ + © + 1)
For w > 0, and z = pl ¥ ϕ (0 < p < 1)
, we must show that
] ).

92 Waggas Galib Atshan et al.
¨
£ †1 −
I
Γ
(¢ + © + 2)
Γ
(2)
(m − ©)
ªs
_(m) |
¥
| ]¥ † q ϕ
¨
≤ £ |1 −
¥
I
(A − B)
Γ
(J + 1)
Γ
(¢ + © + 2)
J(2J − 1) [ ( ; ; ; )((1 − ) [ t ( ; ; ; ) + ( − ))
Γ
(2)
Γ
(J + ¢ + © + 1)
] | § q ϕ
.
By applying Littlewood’s subordination theorem, it would suffice to show that
1 − ∑
¥
Γ
(¡sªs )
Γ
( )
(m − ©)
ªs
_(m) |
¥
| ¥] ≺
1-
(A − B)
Γ
(J + 1)
Γ
(¢ + © + 2)
J(2J − 1) [ ( ; ; ; )((1 − ) [ t ( ; ; ; ) + ( − ))
Γ
(2)
Γ
(J + ¢ + © + 1)
] .
By setting
1 −
Γ
(¢ + © + 2)
Γ
(2)
(m − ©)
ªs
_(m) |
¥
| ¥]
¥
=
(A − B)
Γ
(J + 1)
Γ
(¢ + © + 2)
J(2J − 1) [ ( ; ; ; )((1 − ) [ t ( ; ; ; ) + ( − ))
Γ
(2)
Γ
(J + ¢ + © + 1)
«¬( )we find that
«¬( )-
]
=
J(2J − 1) [ ( ; ; ; )((1 − ) [ t ( ; ; ; ) + ( − ))
Γ
(J + ¢ + © + 1)
(A − B)
Γ
(J + 1)
]
,
× (m − ©)
ªs
¥
_(m) |
¥
| ¥] , which readily yields w(0)=0.For such a function q , we obtain
«¬( )-
]
≤
J(2J − 1) [ ( ; ; ; ) a(1 − ) [ t ( ; ; ; ) + ( − )b
Γ
(J + ¢ + © + 1)
(A − B)
Γ
(J + 1)
× (m − ©)
ªs
¥
≤
_(m) |
¥
|| | ¥]
J(2J − 1) [ ( ; ; ; ) a(1 − ) [ t ( ; ; ; ) + ( − )b
Γ
(J + ¢ + © + 1)
(A − B)
Γ
(J + 1)
× _(2)| | (m − ©)
ªs
¥
|
¥
|

Some Geometric Properties of a Certain... 93
= | |
J(2J − 1) [ ( ; ; ; )((1 − ) [ t ( ; ; ; ) + ( − ))
Γ
(J + ¢ + © + 1)
(A − B)
Γ
(¢ + © + 3)
Γ
(J + 1)
×
Γ
(2 − ©) (m − ©)
ªs
¥
This completes the proof.
|
¥
| ≤ | | < 1.
By taking
τ = 0
, in the Theorem 7, we have the following corollary:
Corollary (1): Let f
∈ r( , , , , , , , )
, and suppose that is defined by
(34). Also let m|
¥
| ≤
(A − B)
Γ
(J + 1)
Γ
(¢ + 3)
J(2J − 1) [ ( ; ; ; )((1 − ) [ t ( ; ; ; ) + ( − ))
Γ
(2)
Γ
(J + ¢ + 1)
¥ n
≥ 2
. If there exists an analytic function q defined by
«¬( )-
]
=
J(2J − 1) [ ( ; ; ; ) a(1 − ) [
(A − B) t
Γ
( ; ; ; ) + ( − )b
Γ
(J + ¢ + 1)
(J + 1)
× m_(m) |
¥
¥
| ¥] , where
Then for
z = pl ¥ ϕ
¨
£ |Ÿ
I
]¡
_(m) =
and 0 < p < 1
( )| § q ϕ
Γ
(m)
Γ
(m + ¢ + 1) , ( ¢ > 0, m ≥ 2).
≤ £ |Ÿ
I
¨
]¡ ( )| § q (¢ > 0, w > 0) .
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