Gen. Math. Notes, Vol. 26, No. 2, February 2015, pp. 79-96
ISSN 2219-7184; Copyright © ICSRS Publication, 2015
www.i-csrs.org
Available free online at http://www.geman.in
On Subclass of Meromorphic Univalent
Function Defined by Derivative Operator
A.R.S. Juma1, H.H. Ebrahim2 and Sh. I. Ahmed3
1
Department of Mathematics, Alanbar University, Ramadi – Iraq
E-mail: dr_ juma@hotmail.com
2,3
Department of Mathematics, Tikrit University, Tikrit – Iraq
2
E-mail: hassan1962pl@yahoo.com
3
E-mail: Ibrahim.shamil@yahoo.com
(Received: 14-11-14 / Accepted: 20-1-15)
Abstract
In the present paper, we introduce a new class of meromorphic univalent
functions defined by derivative operator. We obtain some geometric properties
like coefficient inequality, distortion and growth theorems. Hadmard product or
(convolution), radii of starlikeness and convexity, partial sums, integral operator
and closure theorem for the functions in the class , ( , , , ).
Keywords: Meromorphic univalent function, Hadmard product, Convex
function, Starlike function.
1
Introduction
Historically [1], the classical polylogarithm function was invented in 1696, by
Leibniz and Bernoulli, as mentioned in [2]. For |z|˂1 and c a natural number with
c≥2, the polylogari-thm function (which is also known as Jonquiere's function) is
defined by the absolutely convergent series: Li ( ) = ∑∞
.
80
A.R.S. Juma et al.
Later on, many mathematicians studied the polylogarithm function such as Euler,
Spence, Able, Lobachevsky, Rogers, Ramanujun and many others [5], where they
discovered many functional identities by using polylogarithm function. However,
the work employing polylogarithm has been stopped many decades, later. During
the past four decades, the work using polygarithm has again been in intensified
vividly due to its importance in many fields of math-ematic, such as complex
analytic, algebra, geometry, topology and mathematical physics (quantum field
theory) ([3], [9], [10]). In 1996, Ponnusamy and Sabapathy discussed the
geometric mapping properties of the generalized polylogarithm [10]. Recently,
Al-Shaqsi and Darus [13] generalized Ruscheweyh and salagean operators, using
polylogarithm function on class
of analyticfunctions in the open unit disk =
z ∈ ℂ: |z|˂1". By making use of the generalized operator they introduced related
properties.
A year later, same authors again employed the #th order .Polylogarithm function
to define a multiplier transformation on the class in [14].
Now, we will redefine the polylogarithm function to be on meromorphic type.
LetΩ denoted the class of function of the form
1
%( ) =
+'(
∞
,
Which are analytic and meromorphic univalent in the unit disk
∗
=
∈ℂ: 0 ≤ | |˂1" =
∖ 0".
Let - be a subclass of Ω of functions of the form
%( ) =
1
+'(
(( ≥ 0, / ∈ ℕ) . (1)
∞
A function % ∈ - is meromorphicstarlike function of order1 , (0 ≤ 1 < 1) if
−ℜ 5
%′( )
6 ≥ 1( ∈
%( )
∗ ).
The class of all meromorphically starlike functions is denoted by -∗ (1) . A
function % ∈ - is meromorphic convex function of order1 , (0 ≤ 1 < 1) if
−ℜ 51 +
%′′( )
6 ≥ 1( ∈
%′( )
∗ ).
On Subclass of Meromorphic Univalent…
81
The Hadmard product or (convolution) of two functions, f given by (1) and
is defined
7( ) =
1
+'8
∞
(% ∗ 7)( ) =
1
(8 ≥ 0, / ∈ ℕ) ,
+'( 8
, (3)
∞
:
∗
see [12], which are analytic and univalent in
(2)
.
Liu and Srivastava to defined a function ℎ= ( . , … . . ? ; , … … . A ; )by
multiplying the well-known generalized hyprogometric function BCA with D= as
follows:
ℎ= E
., … . .
?;
,…….
A
; F=
D=
BCA E
., … . .
?;
,…….
A
; F.
Where
, … … . A (GH IJKLHM parameters and B ≤ N + 1. K ∈ ℕ.
., … . . ? ;
analogous to Liu and Srivastava work [6] and corresponding function ∅ ( ) given
by
∞
1
1
∅ ( ) = D: PQ ( ) = + '
(4)
(/ + 2)
S
We consider liner operator Σ %( ): Ω → Ω
Which is defined the following Hadmard product (or convolution):
Σ %( ) = ∅ ( ) ∗ %( )
=
1
+'
∞
S
1
(
(/ + 2)
.
(5)
In [7], N.E, Cho and H. M. Srivastava a liner operator of the form:
V(J, W)%( ) =
1
L+/ Z
+'X
Y (
L+1
∞
J ∈ ℤ , L ≥ 0. (6)
Further , denoted by h : S → S iℎH j ℎHkHlℎ mHGnQiQnH I%IGmHG λ
defined by [11]:
h
,
%( ) =
D
(1 + )
o
∗ %( ) , > −1
82
A.R.S. Juma et al.
=
1
+ ' (q r(W, /)
∞
q :
q
, ( > −1, %( )∈s),
kℎHGHr(W, /) = t
W+/−1
u,
W
This function yields the following family of liner operators
=
1
v %( ) = (h
L+/ Z
+'X
Y r(W, /)(
L+1
,
∞
∗ V(J, W)%( )
J ∈ ℤ , L ≥ 0.
(7)
Now, we define the linear operator by
x %(z) = Ω %( ) ∗ v %( )
=
1
L+/ Z
1
+'X
Y
r(W, /)(
L + 1 (/ + 2)
∞
(J ∈ ℤ , L ≥ 0).
(8)
Now we define the following:
Definition 1.1: A function % ∈ - of the form (1) is in the class
satisfies the following condition:
z
z
X{ |,} ~(•)Y
′′
X{ |,} ~(•)Y
′
X{ |,} ~(•)Y
′′
X{ |,} ~(•)Y
where ≥ −1 0 ≤
′
+2
+2
z
z
< ,
< 1, 0 <
,
( , , , ) if
(9)
≤1 , ≥2 .
Special cases of this class was studied by many researchers like A.R .S. Juma and
H. Zirar [4]. Also Kh. R. Alhindi and Maslina Darus [1] investigate in the same
filed.
,
In this paper we obtain coefficient estimates for the class
( , , , ),
Hadmard product, growth and distortion theorems, radii of starlikeness and
convexity.
On Subclass of Meromorphic Univalent…
2
83
Coefficient Inequality
Theorem 2.1: A function %defined by (1) is in the class
only if
,
( , , , ), if and
L + / Z r(W, /)
'X
Y
/[/(1 + ) + E1 + (2 − 1)F]( ≤ 2 (1 − ) .
L + 1 (/ + 2)
∞
(10)
The result is sharp for the function
% ( )=
1
+
(/ + 2) 2 (1 − )
Z
„o
(11)
,/ ≥ 1
t„o u r(W, /)/[/(1 + ) + E1 + (2 − 1)F]
Proof: To proof the sufficient part, let the inequality (10) holds true and let| | =
1, by (9), we have
z
… tx
,
tx
tx
%(z)u
%(z)u
%(z)u + 2 tx
′′
= †' /(/ + 1) X
∞
,
,
,
′
′′
+ 2z −
%(z)u … −
′
L + / Z r(W, /)
Y
(
L + 1 (/ + 2)
z
tx
tx
,
… tx
D
†
,
%(z)u
%(z)u
,
′′
′
+ 2 z < 0,
%(z)u + 2 Xtx
′′
2(1 − )
L + / Z r(W, /)
)
− †
+
'
/(/
−
1
+
2
X
Y
(
:
L + 1 (/ + 2)
D
∞
≤ 'X
∞
,
%(z)u Y…,
′
†,
L + / Z r(W, /)
Y
/‡/(1 + ) + E1 + (2 − 1)Fˆ( − 2 (1 − ) ≤ 0 .
L + 1 (/ + 2)
Thus by the maximum modulus theorem, we have % ∈
Conversely, suppose the% of the form (1) is in the class
(9) we have
,
( , , , ).
,
( , , , ), then by
84
A.R.S. Juma et al.
z
z
‰
:( D‹)
Œ
∑∞
+ ∑∞
X{ |,} ~(•)Y
′′
X{ |,} ~(•)Y
′
X{ |,} ~(•)Y
′′
X{ |,} ~(•)Y
′
/(/ + 1) t
ℜ“
:( D‹)
Œ
+ ∑∞
„o
u
+2
z
z
< ,
Z Š(q, )
( D
( o:)
„o Z Š(q, )
/(/ − 1 + 2 ) t„o u
Since|ℜ( )| ≤ | |for all , we get
∑∞
„o
+2
/(/ + 1) t
„o
„o
u
( o:)
Z Š(q, )
( D
( o:)
„o Z Š(q, )
/(/ − 1 + 2 ) t„o u
( o:)
(
(
D
D
‰< .
”< .
(12)
X{ |,} ~(•)Y
′′
Now, by choosing the value of z on the real axis so that the value of
X{ |,} ~(•)Y
′
is
real, then clearing the denominator of (12) and letting → 1 − through real value,
we get the inequality (10).
The result is sharp for the function
%( )=
1
+
(/ + 2) 2 (1 − )
Z
t„o u r(W, /)/[/(1 + ) + E1 + (2 − 1)F]
„o
Corollary 2.1: If % ∈
( ≤
,
Z
( , , , ) , then
(/ + 2) 2 (1 − )
,/ ≥ 1
t„o u r(W, /)/[/(1 + ) + E1 + (2 − 1)F]
„o
where ≥ −1 0 ≤
Corollary 2.2: If % ∈
and equality holds for
,
< 1,
( , , , ),
( ≤
0<
(3) 2 (1 − )
,
r(W, 1)(1 +
)
% ( ) = + Š(q,
(•) :–( D‹)
z
)( o –‹)
≤1 , ≥2 .
,
On Subclass of Meromorphic Univalent…
Corollary 2.3: If % ∈
and equality holds for
%( )=
( ≤
Corollary 2.4: If % ∈
and equality holds for
%( )=
3
+
:
,
t
+
:
,
t
(/ + 2) 2 (1 − )
Z
u r(W, /)/[2(/ +
„o
„o
(/ + 2) 2 (1 − )
Z
u r(W, /)/[2(/ +
„o
„o
( , 1,2, ) ,
( ≤
1
Corollary 2.5: If % ∈
and equality holds for
( , 1, , ) ,
,
1
85
t
Z
u r(W, /)/[2(/ +
„o
(/ + 2): 2 (1 − )
Z
t„o u r(W, /)/[2(/ +
„o
( , 1,2, ) ,
( ≤
%( ) =
1
,/ ≥ 1
)]
(/ + 2) : 2 (1 − )
„o
,
)]
,
)]
,/ ≥ 1
)]
18(1 − )
,
r(W, 1)[2(1 + ) ]
+
18(1 − )
z
r(W, 1)[2(1 + ) ]
Growth and Distortion Theorems
A growth and distortion property for the function% ∈
follows
Theorem 3.1: A function % defined by (1) is in the class
0 < | | = G < 1, we have
,
( , , , ) is given as
,
( , , , ) , then for
1
(3) (1 − )
1
(3) (1 − )
−
G ≤ |%( )| ≤ +
G
G r(W, 1)(1 + )
G r(W, 1)(1 + )
86
A.R.S. Juma et al.
with equality for
Proof: Since ∈
,
%( ) =
1
+
(3) (1 − )
z
r(W, 1)(1 + )
.
( , , , ) , we have form Theorem 2.1 the inequality
L + / Z r(W, /)
'X
Y
/‡/(1 + ) + E1 + (2 − 1)Fˆ( ≤ 2 (1 − ) ,
L + 1 (/ + 2)
∞
1
|%( )| ≤
+'( | | ,
| |
then
∞
for 0 < | | = G < 1, we have
1
1
(3) (1 − )
|%( )| ≤ + G ' ( , ≤ +
G.
G
G r(W, 1)(1 + )
∞
1
1
(3) (1 − )
|%( )| ≥
−'( | | , ≥ −
G, | | = G.
| |
G r(W, 1)(1 + )
Also
∞
Hence the proof is complete.
Theorem 3.1: A function % defined by (1) is in the class
0 < | | = G < 1, we have
,
( , , , ), then for
(3) (1 − )
(3) (1 − )
1
1
−
≤ |%( )′| ≤ : +
G,
:
G
r(W, 1)(1 + )
G
r(W, 1)(1 + )
with equality for
%( ) =
1
+
Proof: Form Theorem 2.1 we have
'X
∞
(3) (1 − )
.
r(W, 1)(1 + )
L + / Z r(W, /)
Y
/‡/(1 + ) + E1 + (2 − 1)Fˆ( ≤ 2 (1 − ) .
L + 1 (/ + 2)
On Subclass of Meromorphic Univalent…
87
−1
|%( )′| ≤ … : … + ' /( | |
Thus
∞
for 0 < | | = G < 1 , we get:
and
,
D
(3) (1 − )
1
1
|%( )′| ≤ … : … + ' /( , ≤ … : … +
,
G
G
r(W, 1)(1 + )
|%( )′| ≥ …
−1
2
… − ' /(/ | |
∞
/−1
/=1
∞
(3) (1 − )
1
, |%( )′| ≥ … 2 … − ' /(/ , ≥ 2 −
.
G
G
r(W, 1)(1 + )
1
∞
/=1
This completes the proof
4
Hadmard Product
Theorem 4.1: Let functions
%, 7 ∈
,
( , , , ). Then(% ∗ 7) ∈
%( ) =
7( ) =
and
˜=
2
1
2
: (1
Proof: Since %, 7 ∈
'
∞
Z
,
,
+'8
,
∞
: (1
„o
+'(
1
− )(/ + 2 − 1) − t„o u
( , , , ) for
∞
1
(% ∗ 7)( ) =
where
,
+'( 8
∞
:
− )( / + 1)
Z Š(q, )
( o:)
,
/‡/(1 + ) + E1 + (2 − 1)Fˆ
( , , , ), then by Theorem 2.1 we have
t„o u r(W, /)/[/(1 + ) + E1 + (2 − 1)F]
„o
(/ + 2) 2 (1 − )
( ≤ 1,
:
88
A.R.S. Juma et al.
and
'
∞
t
Z
u r(W, /)/[/(1 + ) + E1 + (2 − 1)F]
„o
„o
(/ + 2) 2 (1 − )
We must find the largest rsuch that
'
∞
Z
t„o u r(W, /)/[/(1 + ) + E1 + (2 − 1)F]
„o
(/ + 2) 2 (1 − )
8 ≤ 1.
( 8 ≤ 1.
By Cauchy-Schwarz inequality, we get
'
∞
Z
t„o u r(W, /)/[/(1 + ) + E1 + (2 − 1)F]
„o
(/ + 2) 2 (1 − )
™( 8 ≤ 1,
(13)
To prove the theorem it is enough to show that
'
∞
≤'
∞
Z
t„o u r(W, /)/[/(1 + ˜) + E1 + ˜(2 − 1)F]
„o
Z
(/ + 2) 2˜(1 − )
t„o u r(W, /)/[/(1 + ) + E1 + (2 − 1)F]
„o
(/ + 2) 2 (1 − )
Which is equivalent to
™( 8 ≤
From (13), we have
™( 8 ≤
t
„o
„o
™( 8
˜[/(1 + ˜) + E1 + ˜(2 − 1)F]
∙
[/(1 + ) + E1 + (2 − 1)F]
Z
(/ + 2) 2 (1 − )
u r(W, /)/[/(1 + ) + E1 + (2 − 1)F]
We must show that
(/ + 2) 2 (1 − )
tL+1u r(W, /)/‡/(1 + ) + E1 + (2 − 1)Fˆ
L+/ J
( 8
≤
,
∙
˜‡/(1 + ˜) + E1 + ˜(2 − 1)Fˆ
‡/(1 + ) + E1 + (2 − 1)Fˆ
,
On Subclass of Meromorphic Univalent…
Which gives
˜≤
2
: (1 −
)(/ + 2 − 1) −
2
: (1
„o
t
„o
89
− )( / + 1)
Z Š(q, )
u
( o:)
/‡/(1 + ) + E1 + (2 − 1)Fˆ
:
∙
Hence the proof is complete.
Theorem 4.2: Let the functions %› (Q = 1, 2, … . ) defined by
∅=
: (1
4
,
'
7( ) =
1
, ((
,›
+ '((:
∞
( , , , ) , where
4
: (1
− )(/ + 2 − 1) − t „o u
Proof: Since %› ∈
∞
+'(
∞
,›
≥ 0, i = 1,2)
( , , , ) . Then the function7 defined by
,
be in the class
Is in the class
%› ( ) =
1
Z
,
„o
,
+(:
,: )
− )( / + 1)
Z Š(q, )
( o:)
/‡/(1 + ) + E1 + (2 − 1)Fˆ
( , , , ), (Q = 1,2), then by Theorem 2.1 we have:
t„o u r(W, /)/[/(1 + ) + E1 + (2 − 1)F]
„o
,
(/ + 2) 2 (1 − )
(
,›
≤ 1, Q = 1,2, ….
Hence
'œ
∞
≤ œ'
∞
Z
t„o u r(W, /)/[/(1 + ) + E1 + (2 − 1)F]
„o
Z
(/ + 2) 2 (1 − )
t„o u r(W, /)/[/(1 + ) + E1 + (2 − 1)F]
„o
(/ + 2) 2 (1 − )
:
• (:
:
,›
≤ 1,
( ,› • ≤ 1, Q = 1,2.
:
90
A.R.S. Juma et al.
Thus
:
Z
1 t u r(W, /)/[/(1 + ) + E1 + (2 − 1)F]
' œ „o
• ((:
2
(/ + 2) 2 (1 − )
„o
∞
To prove the theorem, we must find the largest ∅ such that
,
+ (:
,: )
≤ 1,
[/(1 + ∅) + E1 + ∅(2 − 1)F]
∅
:
„o Z
t„o u r(W, /)/‡/(1 + ) + E1 + (2 − 1)Fˆ
≤
,
(/ + 2) 4 (1 − )
So that
∅≤
: (1 −
4
)(/ + 2 − 1) −
Hence the proof is complete.
∑∞
: (1
„o
t„o
− )(/ + 1)
Z Š(q, )
u ( o:)
+ ∑∞
(
8 with|8 | ≤ 1 is in the class
( , , , ).
Theorem
,
v% %( ) =
4
4.3:
/‡/(1 + ) + E1 + (2 − 1)Fˆ
∈
,
,
( , , , ), ((Wm7( ) =
'X
Since
L + / Z r(W, /)
Y
/‡/(1 + ) + E1 + (2 − 1)Fˆ( ≤ 2 (1 − ) .
L + 1 (/ + 2)
'
∞
'
∞
'
∞
t
„o
„o
Z
u r(W, /)/[/(1 + ) + E1 + (2 − 1)F]
Z
(/ + 2) 2 (1 − )
t„o u r(W, /)/[/(1 + ) + E1 + (2 − 1)F]
„o
Z
(/ + 2) 2 (1 − )
t„o u r(W, /)/[/(1 + ) + E1 + (2 − 1)F]
„o
(/ + 2) 2 (1 − )
.
+
( , , , ), iℎHW%( ) ∗ 7( ) ∈
Proof: From Theorem 2.1 we have
∞
:
|( 8 | ,
( |8 | ,
( ≤ 1.
On Subclass of Meromorphic Univalent…
Thus%( ) ∗ 7( ) ∈
Corollary 4.1: v% %( ) =
∑∞
,
5
+ ∑∞
(
91
,
8 with 0 ≤ 8 < 1 is in the class
( , , , ).
( , , , ).
∈
,
,
( , , , ). , (Wm7( ) =
+
( , , , ), iℎHW%( ) ∗ 7( ) ∈
Radii of Starlikeness and Convexity
Theorem 5.1: Let the function %( ) defined by (1) be in the clas , ( , , , )
.Then f is meromorphically starlike of order 1(0 ≤ 1 < 1) in the disk | | <
G , ( , , , ), kℎHGH
G ( , , , , 1) = inf “
„o
t„o
Z
u r(W, /)/[/(1 + ) + E1 + (2 − 1)F](1 − 1)
(/ + 2) 2 (/ + 2 − 1)(1 − )
The result is sharp for the function is given by (11).
Proof: It is enough to show that
…
…
%′( )
+ 1… ≤ 1 − 1 ,
%( )
∑∞ (/ + 1)(
%′( )
+ 1… = ¡ D
+ ∑∞ (
%( )
This will be bounded by 1 − 1,
¡≤
∑∞ (/ + 1)( | | o
.
1 − ∑∞ ( | | o
∑∞ (/ + 1)( | | o
≤1−1,
1 − ∑∞ ( | | o
'(/ + 2 − 1)( | |
∞
o
≤1−1,
By theorem 2.1, we have
'
∞
Z
t„o u r(W, /)/[/(1 + ) + E1 + (2 − 1)F]
„o
(/ + 2) 2 (1 − )
( ≤ 1.
”
}
Ÿ }
.
92
A.R.S. Juma et al.
Hence
| |
o
| |≤“
≤
t
„o
„o
t
Z
u r(W, /)/[/(1 + ) + E1 + (2 − 1)F](1 − 1)
„o
„o
Z
(/ + 2) 2 (/ + 2 − 1)(1 − )
u r(W, /)/[/(1 + ) + E1 + (2 − 1)F](1 − 1)
(/ + 2) 2 (/ + 2 − 1)(1 − )
,
}
Ÿ¢}
”
.
This completes the proof of the theorem.
Theorem 5.1: Let the function %( ) defined by (1) be in the class , ( , , , )
.Then f is meromorphically convex of order r(0 ≤ r < 1) in the disk | | <
G: ( , , , , r), kℎHGH
G: ( , , ˜, , r) = inf “
„o
t„o
Z
u r(W, /) (1 − r)[/(1 + ) + E1 + (2 − 1)F]
(/ + 2) 2 (/ + 2 − r)(1 − )
”
}
Ÿ }
.
The result is sharp for the function is given by (11).
Proof: By using the same technique in the proof of Theorem 5.1 we can show that
…
%′′( )
+ 2… ≤ 1 − r , (0 ≤ r < 1).
%′( )
For | | < G: with the aid of Theorem 2.1, we have the assertion of Theorem 5.2.
6
Closure Theorem
Theorem 6.1: Let the function
%q ( ) =
1
+'(
∞
, E(
,q
,q
≥ 0, k ∈ ℕ = 1,2, … "F,
be in the class ℛ( , , ˜, ) for every n=1,2,….i. Then the function ℎ defined by
ℎ( ) =
also belong to the class
,
1
+'H
∞
(Hq ≥ 0 , / ∈ ℕ)
( , , , ), where
On Subclass of Meromorphic Univalent…
1
H = '(
J
ℓ
Proof: Since%q ∈
'X
∞
,
,q
93
(/ = 1,2, … ).
( , , , ) it follows from Theorem 2.1 that
L + / Z r(W, /)
Y
/‡/(1 + ) + E1 + (2 − 1)Fˆ(
L + 1 (/ + 2)
for every n=1,2,…... J
,q
≤ 2 (1 − ) ,
Hence
'X
∞
L + / Z r(W, /)
Y
/‡/(1 + ) + E1 + (2 − 1)FˆH
L + 1 (/ + 2)
L + / Z r(W, /)
1
'X
Y
/‡/(1 + ) + E1 + (2 − 1)Fˆ ¥ ' (
L + 1 (/ + 2)
J
ℓ
∞
,q
¦
1
L + / Z r(W, /)
= ' §' X
Y
/‡/(1 + ) + E1 + (2 − 1)Fˆ(
J
L + 1 (/ + 2)
ℓ
∞
≤ 2 (1 − ).
By Theorem 2.1, it follows that ℎ ∈
7
,
,q ¨
( , , , )
Integral Operator and Partial Sums
Next, we consider some properties have been found the other class in [8].
Theorem 7.1: The% ∈
C(©) =
is in the class
o ,
,
( , , , ) if and only if the function C given by
o
ª i %(i)mi , ( > −1,
S
( , , , ).
Proof: By using of (14), we have
%(©) = ( + 1)C( ) + C ′ ( ),
Which, in the right hand of (8), implies
(14)
(15)
94
A.R.S. Juma et al.
(x
,
∗
%(z)) = ( + 1)(x
Therefore, we have
x
,
,
∗
%(z) = x
,
o ,
∗
C(z)) + (x
,
C(z))′ =
(x
o ,
∗
C(z)).
C(z)
and the desired result follows at once.
Theorem 7.2: Let% ∈ - given by (2) and define the partial sums N ( ) and
Nq ( ) 8l
N ( )=
1
(Wm Nq ( ) =
1
qD
+'(
,
Suppose also that
' m/ (/ ≤ 1, ¥m/ =
∞
/=1
tL+1u r(W, /)/‡/(1 + ) + E1 + (2 − 1)Fˆ
L+/ J
(/ + 2) 2 (1 − )
¦ .
(16)
Then we have
ℛH 5
%( )
1
Nq ( )
m
6 > 1 − (WmℛH «
¬>1−
.
Nq ( )
m
%( )
1−m
Each of the bounds in (17) is the best possible for/ ∈ ℕ .
(17)
Proof: For the coefficient m given by (16), it is not difficult to verify that
m
o
> m > 1, / = 1,2 … . .. .
Therefore, by using the hypothesis (16), we have
qD
' ( + mq ' ( ≤ ' m ( ≤ 1 .
∞
∞
q
%( )
1
mq ∑ ∞ q (
7 ( ) = mq §
− X1 − Y¨ = 1 +
Nq ( )
mq
1 + ∑qD (
Be setting
(18)
o
o
(19)
On Subclass of Meromorphic Univalent…
95
And applying (18) we find that
¡
7 ( )−1
mq ∑ ∞ q (
¡≤
7 ( )+1
2 − 2 ∑qD ( − mq ∑∞
Which readily yields the left assertion (17). If we take
then
%( ) =
1
−
q
mq
,
q
%( )
1
= 1−
→ 1 − ( → 1 −),
Nq ( )
mq
mq
q(
(20)
(21)
which shows that the bound in (17) is the best possible for each/ ∈ ℕ .
Similarly, if we put
Nq ( )
mq
(1 + mq ) ∑∞ q (
7: ( ) = (1 + mq ) −
®=1−
%( ) 1 + mq
1 + ∑qD (
o
o
and make use of (18) we obtain
(1 + mq ) ∑∞ q (
7: ( ) − 1
¡
¡≤
7: ( ) + 1
2 − 2 ∑qD ( + (1 − mq ) ∑∞
q(
≤1
(21)
which leads us to the assertion (17).The bounds given in the right of (17)is sharp
with the function given by (21).The proof of the theorem is complete.
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