Gen. Math. Notes, Vol. 22, No. 1, May 2014, pp. 71-85
ISSN 2219-7184; Copyright © ICSRS Publication, 2014
www.i-csrs.org
Available free online at http://www.geman.in
On a Subclass of Univalent Functions Defined
by Multiplier Transformations
Waggas Galib Atshan1 and Thamer Khalil Mohammed2
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
Department of Mathematics, College of Education
University of Al-Mustansirya, Baghdad – Iraq
E-mail: thamerk76@yahoo.com
(Received: 23-10-13 / Accepted: 14-2-14)
Abstract
In the present paper, we introduce a subclass of univalent functions with
positive coefficients defined by multiplier transformations in the open unit disk
U={z ∈ ℂ: | | < 1}. We obtain some geometric properties, like coefficient
inequality,
closure
theorem,
neighborhoods
for
the
subclass
W ℓ, , , , , , , radii of starlikeness, convexity and close-to-convexity,
weighted mean, arithmetic mean, linear combination and integral representation.
Keywords: Univalent function, Multiplier transformations, Neighborhoods,
Radius of starlikeness, Weighted mean, Arithmetic mean, Linear combination,
Closure Theorem, Integral representation.
72
Waggas Galib Atshan et al.
1
Introduction
Let S be the class of functions of the form:
∞
= +
,
1
which are analytic and univalent in the unit disk U = {z ∈ ℂ: |z| < 1}.
∈ S is said to be starlike of order
A function
!{
#$ ˊ #
$ #
} >
#$ ˊˊ #
$ˊ #
< 1) if and only if
, (z ∈ U .
(2)
Denote the class of all starlike functions of order
f ∈ S is said to be convex of order β if and only if
Re {1 +
(0 ≤
}>
in U by S*( ). A function
, (0≤ β <1, z ∈ U)
Denote the class of all convex functions of order
(3)
in U by 1
A function f ∈ S is said to be close – to – convex of order
Re {
ˊ
}>
, 0≤
< 1,
∈3
.
if and only if
(4)
Let TH be subclass of S consisting of functions of the form:
=
+
∞
,
For the functions
=
+
≥0
∈ 67 given by 5 and ; ∈ 67 de<ined by
∞
,
= ≥0
Define the convolution (or Hadamard product) of
∗;
=
+
5
∞
=
.
6
and ; by
7
ℓ
For any integer m, we define the multiplier transformations AB
(see [4, 5] of
functions ∈ 67 by
On a Subclass of Univalent Functions Defined…
ℓ
AB
+ ∑∞
=
=
+ℓ
+ℓ+C−1
∞
+
F C, , ℓ
,
73
ℓ ≥ 0,
B
> 0, ∈ 3 ,
(8)
Where
F C, ⍺, ℓ =
+ℓ /
+ℓ+C−1
B
.
Definition 1: Let g be a fixed function defined by (6). The function ∈ 67 given
by (5) is said to be in the new class W ℓ, , , , , ,
J and only if
│
ℓ
∗;
AB
ℓ
AB ∗ ;
ˊˊ
∗;
+
ˊ
∗;
ℓ
AB
ℓ
AB
ˊˊ
ˊ
│ <
+
,
9
Where ℓ ≥ 0, ⍺ >0, m ∈ Z, 0 < η < 1, 0 < γ <1, 0 ≤ γ <1 and 0 < λ < 1. The
following interesting geometric properties of this function subclass were studied
by several authors for another classes , like, Altintas et al. [1] Atshan and Buti [2],
Atshan and Kulkarni [3], Kanas et al. [7,8], Murugusundaramoorthy and Magesh
[9,10] and Murugusundaramoorthy and Srivastava [11].
2
Coefficient Inequality
We obtain the necessary and sufficient condition for a function f to be in the class
W ℓ, , , , , , .
Theorem 1: M!N
∞
∈ 67. 6ℎ!C
CT C − 1 1 −
where ℓ ≥ 0, ⍺ > 0,
−
∈P
+
∈ Z, 0 <
ℓ, ⍺,
, ,
UF C, , ℓ
< 1, 0 <
,
,
J
= ≤
< 1, 0 ≤
CQ RCSy if
+
,
< 1 CQ 0 <
10
<1.
The result is sharp for the function
=
+
C C−1 1−
−
+
+
.
F C, ⍺, ℓ
Proof: Suppose that (10) is true for z ∈ 3 and |z| = 1. Then , we have
ℓ (
│ (AB
∗ ; ( )))ˊˊ │ −
│
ℓ
YAB
T ∗ ;( )UZ ˊˊ + (
+
11
ℓ
) YAB
T ∗ ;( )UZ ˊ│
74
Waggas Galib Atshan et al.
=│
∞
C(C − 1)F(C, , ℓ)
+(
=│
+
−
=
∞
\
∞
∞
∞
+
=
│− │
[
)(1 +
∞
C( (C − 1) + (
≤
\
n(n − 1)θ(n, α, ℓ)a\ b\ z \[
\
CF(C, , ℓ)
C(C − 1)F(C, , ℓ)
∞
∞
=
=
))F(C, ⍺, ℓ)
+
)│
│ − λ│ (
[
)
+
=
[
│
n(n − 1)θ(n, ⍺, ℓ)a\ b\ |z|\[
nλ η n − 1 + Tγ + γ U θ n, ⍺, ℓ a\ b\ │z│
C C−1 1−
[
−
+
F C, ⍺, ℓ
\[
–λ γ + γ
= −
+
≤ 0,
by hypothesis.
Hence, by maximum modulus principle,
Conversely, assume that
│
=│
∑∞
\
∑∞
\
^P
ℓ
AB
ℓ
AB
ℓ, ⍺,
∈P
, , ,
ℓ
∗;
AB
ℓ
AB ∗ ;
ˊˊ
∗;
+
ˊ
∗;
n n − 1 θ n, ⍺, ℓ
∑∞
\
, , ,
,
.
. Then from (9), we have
ˊˊ
ˊ
+
│
a\ b\ z \[
nTη n − 1 + γ + γ Uθ n, ⍺, ℓ a\ b\ z \[ + Tγ + γ U
Since Re (z) ≤ │z│for all z (z ∈ 3 , we get
!`
,
ℓ, ⍺,
∑∞
\
n n − 1 θ n, ⍺, ℓ
a\ b\ z \[
nTη n − 1 + γ + γ Uθ n, ⍺, ℓ a\ b\ z \[ +
+
│˂ λ .
a< .
12
On a Subclass of Univalent Functions Defined…
75
We choose the value of z on the real axis so that
ℓ
c(AB
T ∗ ;( )Uˊˊ
ℓ (
(AB
∗ ;)( )))ˊ
Jd e! S
Letting z → 1[ through real values, we obtain inequality (10).
Finally, sharpness follows if we take
( )=
+
( +
)− (
C((C − 1)(1 −
)
+
))F(C, ⍺, ℓ)=
,
(13)
n=2, 3….
The proof is complete.
Corollary 1: Let
\
, , ,
g(hi jhk )
≤
3
∈ P ( ℓ, ⍺,
, ). Then
, C = 2, 3, ….
(( [ )( [lg)[g(hi jhk ))m( ,⍺,ℓ)no
(14)
Closure Theorem
Theorem 2: Let the functions
q(
)=
+
q
defined by
∞
,T
,q
,q
≥ 0, r = 1,2, … . , sU,
be in the class WA( ℓ, ⍺, m, η,γ , γ , λ) for every k = 1,2, … µ . Then the function
h defined by
h(z) = z +
∞
\
e\ z \ , (e\ ≥ 0),
also belongs to the class WA( ℓ, ⍺, m, η,γ , γ , λ) , where
1
e\ =
µ
dJCz!
q
∈ P ( ℓ, ⍺,
, , ,
µ
y
a\,y , (n ≥ 2).
, ), then by Theorem 1, we hav!
76
Waggas Galib Atshan et al.
Proof:
∞
C T(C − 1 )(1 −
)− (
+
)UF(C, ⍺, ℓ)
,q =
≤ (
+
),
(15)
for every k=1, 2,…..,µ. Hence
∞
=
1
=
s
{
q
(
)− (
C((C − 1)(1 −
)− (
+
C Y(C − 1)(1 −
)− (
+
))F(C, ⍺, ℓ)
By Theorem (1), it follows that ℎ ∈ P ( ℓ, ⍺,
4
))F(C, ⍺, ℓ)! =
+
1
)) F(C, ⍺, ℓ)= (
s
∞
∞
C((C − 1) (1 −
, , ,
,q =
, ).
Neighborhoods for the Class P ( ℓ, ⍺,
Definition 2: For any function
defined as:
}~ ( ) = {; ( ) =
+
∞
=
{
,q )
q
Z≤ (
, , ,
+
).
, )
∈ 67 CQ | ≥ 0, Nℎ! | −neighborhood of f is
∈ 67:
∞
C|
− = | ≤ |}.
16
In particular, for the function e(z)= z ,we see that,
}~ ! = {;
=
+
∞
=
∈ 67:
∞
C|= | ≤ |} .
17
The concept of neighborhoods was first introduced by Goodman [6] and then
general by Ruscheweyh [12].
Definition 3: A function f ∈ TH is said
P • ℓ, ⍺, , , , ,
if
there
exists
P ℓ, ⍺, , , , , , such that
│
;
−1 │<1−•
to
a
be in the class
function
g
∈
∈ 3, 0 ≤ • < 1 .
On a Subclass of Univalent Functions Defined…
Theorem 3: If g ∈ P ( ℓ, ⍺,
• =1−
, , ,
77
, ) and
|((1 − ) + ( + ))F(2, ⍺, ℓ)
2((1 − ) + ( + ))F(2, ⍺, ℓ) − (
Then N„ (g) ⊂ WA† ( ℓ, ⍺, m, η, γ , γ , λ).
Proof: Let
+
)
.
(18)
∈ }δ (;). We want to find from (16) that
∞
C│
− = │ ≤ |,
which readily implies the following coefficient inequality
∞
|
−= |≤
Next, since ; ∈ P
∞
= ≤
|
.
2
ℓ, ,
2 1−
So that
│
≤|
;
− 1│ ≤
1−
2 1−
+
, ,
,
+
+
+
,
, we have from Theorem 1
.
F 2, ⍺, ℓ
∑∞ │ − = │
1 − ∑∞ =
+
+
Thus by Definition (3),
+
F 2, , ℓ
F 2, , ℓ [
∈P
•
ℓ, ,
, ,
,
+
,
= 1 − •.
19
for • given by 18 .
This completes the proof.
5
Radii of Starlikeness, Convexity and Close–to–
Convexity
Using the inequalities (2), (3), (4) and Theorem 1, we can compute the radii
starlikeness, convexity and close - to - convexity.
Theorem 4: If f ∈ P ℓ, , , , , , , then f is univalent starlike of order
Š 0 ≤ Š < 1) in the disk │z│< e , where
78
Waggas Galib Atshan et al.
e (ℓ, ,
, ,
,
C(1 − Š)T(C − 1)(1 − ) − (
, , Š) = JC {
(C − Š) ( +
)UF(C, ⍺, ℓ)=
+
)
}
[
,
C≥2
Proof: It is sufficient to show that
ˊ( )
− 1│ ≤ 1 – Š , (0 ≤ Š < 1),
( )
│
for │z│ < e (ℓ, ,
we have
│
, ,
,
, , Š),
∑∞ (C − 1)
ˊ( )
−1 │= │
( )
1 + ∑∞
[
[
≤
∑∞ (C − 1)
1 − ∑∞
│ │
│ │
[
[
.
Thus
│
#$ˊ(#)
$(#)
− 1│ ≤ 1 – Š,
if
∞
(C − Š) │ │
1−Š
[
≤1.
(20)
Hence, by Theoremn1, (20) will be true if
(C − Š)│ │
1−Š
[
≤
CT(C − 1)(1 −
)− (
( +
)UF(C, , ℓ)=
+
)
,
Equivalently if
│ │≤{
( [‹)(( [ )( [gl)[g(hi jhk ))m(C, ,ℓ)no
( [‹)g(hi jhk )
i
}oŒi , n ≥ 2
Setting │z│= r1, we get the desired result.
Theorem 5: If f ∈ P (ℓ, , , , , , ), then f is univalent convex of order Š
(0 ≤ Š <1) in the disk │z│< e , where
e (ℓ, ,
, ,
,
, , Š) =JC
{
( [‹)(( [ )( [gl)[g(hi jhk ))m( ,•,ℓ)no
( [‹)g(hi jhk )
i
}oŒi , n ≥ 2.
On a Subclass of Univalent Functions Defined…
79
Proof: It is sufficient to show that
│
#$ ˊˊ(#)
$ ˊ(#)
│ ≤ 1- Š ,
│z│ < r (ℓ, α, m, η, γ , γ , λ, ψ),
for
We have
│
Thus
│
#$ ˊˊ(#)
#$ ˊ(#)
(0 ≤ ψ < 1),
ˊˊ (
)
∑∞ C(C − 1)
│= │
ˊ( )
1 + ∑∞ C
[
[
≤
∑∞
C(C − 1)
1 − ∑∞
│ │
C
│ │
[
[
.
│≤ 1- Š ,
If
∞
C(C − Š)
│ │
(1 − Š)
[
≤ 1.
(21)
Hence by Theorem 1, (21) will be true if
C(C − Š) │ │
(1 − Š)
[
≤
C((C − 1)(1 −
)− (
( +
+
)
))F(C, , ℓ)=
Equivalently if
│z│ ≤ {
(1 − Š)((C − 1)(1 − ) − ( +
(C − Š) ( + )
))F(C, , ℓ)=
}
[
,
n ≥ 2.
Setting │z│= r2, we get the desired result.
Theorem 6: Let a function f ∈ P (ℓ, , , , , , ). Then f is univalent close –
to - convex of order Š (0 ≤ Š < 1) in the disk │z│ < eŽ , where
eŽ (ℓ, ,
, ,
,
, , Š)
(1 − Š)((C − 1)(1 −
= JC {
)− ( +
( + )
Proof: It is sufficient to show that
│ ˊ( ) − 1│ ≤ 1−Š , (0 ≤ Š < 1),
))F(C, , ℓ)=
}
[
,
n ≥ 2
80
Waggas Galib Atshan et al.
for
│z│< eŽ (ℓ, ,
, ,
, ),
,
We have
│ ˊ( ) − 1│ = │
∞
C
[
│≤
∞
C
[
│ │
.
Thus
│ ˊ( ) − 1│ ≤ 1− Š ,
If
∞
C
│ │
1−Š
[
≤ 1.
(22)
Hence, by Theorem 1, (22) will be true if
oŒi
│#│
≤
[‹
(( [ )( [gl)[g(hi jhk ))m( , ,ℓ)no
g(hi jhk )
,
Equivalently if
( [‹)(( [ )( [gl)[g(hi jhk ))m( , ,ℓ)no
│z│ ≤ {
g(hi jhk )
i
}oŒi , C ≥ 2
Setting │z│ = r3, we get the desired result.
6
Weighted Mean and Arithmetic Mean
Definition 4: Let
weighted mean •• of
V’ (z) = [(1-j)
and
and
(z) + (1+j)
be in the class WA(ℓ, ,
is given by:
(z)],
, ,
,
, ). Then the
0< j <1
Theorem 7: Let
and be in the class WA (ℓ, , , , , , ) . Then the
weighted mean •• of and is also in the class WA (ℓ, , , , , , ).
Proof: By Definition (4), we have
•• ( ) =
“ (1 − ”)
( ) + (1 + ”) ( )•
(23)
On a Subclass of Univalent Functions Defined…
1
= “ (1 − ”) – +
2
∞
= +
∞
— + (1 + ”) – +
,
1
˜(1 − ”)
2
Since f1 and f2 are in the class WA (ℓ, ,
∞
81
,
+ (1 + ”)
, ,
,
,
™
∞
—
,
.
, ) so by Theorem 1, we get
CT(C − 1)(1 −
)− (
+
)UF(C, , ℓ)
,
= ≤ (
+
),
CT(C − 1)(1 −
)− (
+
)UF(C, , ℓ)
,
= ≤ (
+
).
and
∞
Hence
∞
CT(C − 1)(1 −
+ ”)
1
=
(1 − ”)
2
1
(1 + ”)
+
2
≤
,
∞
∞
)−(
1
) UF(C, , ℓ) ( “ (1 − ”)
2
+
•)=
CT(C − 1)(1 −
CT(C − 1)(1 −
1
(1 − ”) (
2
Therefore, •• ∈ P (ℓ, ,
+
, ,
)+
,
)− (
)− (
1
(1 + ”) (
2
, ).
+
+
+
+ (1
,
)UF(C, , ℓ)
)UF(C, , ℓ)
)= (
+
,
,
=
=
).
The proof is complete.
Theorem 8: Let
š(
)=
+
(z),
∞
,š
(z) ,…,
,
T
{(
,š
) defined by
≥ 0, J = 1,2, … , s , C ≥ 2U
(24)
82
Waggas Galib Atshan et al.
be in the class P (ℓ, ,
1,2, … , s)de<ined by
1
h z =
s
, ,
, ). Then the arithmetic mean of
,
{
) (J =
25
š
š
š(
is also in the class P
ℓ, ,
, ,
,
,
.
Proof: By (24), (25), we can write
1
=
s
ℎ
Since
š
∈P
ℓ, ,
{
∞
+
š
, ,
∞
,
=
,š
,
1
s
+
{
.
,š
š
for every (i=1,2,…,µ) so by using Theorem 1,
We prove that
∞
CT C − 1 1 −
1
=
s
{
+
UF C, , ℓ
{
š
=
,š
∞
CT C − 1 1 −
š
{
1
≤
s
7
−
1
s
+
š
=
−
+
+
UF C, , ℓ
,š =
. The proof is complete.
Linear Combination
In the following theorem, we prove a linear combination for the class
P
ℓ, , m, ,
,
,
.
Theorem 9: Let
š
∞
=
+
,š
,
,š
≥ 0, J = 1,2, … , s, C ≥ 2
belong to the class WATℓ, α, m, η, γ , γ , λU. Then
On a Subclass of Univalent Functions Defined…
Ÿ
=
where
{
zš
š
∈ P
š
{
ℓ, ,
83
, ,
,
,
,
zš = 1.
š
Proof: By Theorem (1), we can write for every i ∈ "1,2, … , s
∞
CT C − 1 1 −
−
+
Therefore
Ÿ
=
{
+
zš
š
+
{
+
š
However
∞
=
{
š
CT C − 1 1 −
∞
zš [
Then F (z) ∈ P
−
+
, ,
,
,
≤ 1.
,š
zš
,š
+
.
{
UF C, , ℓ
š
CT C − 1 1 −
ℓ, ,
,š =
∞
∞
=
UF C, , ℓ
−
+
+
zš
UF C, , ℓ
,š
,š =
=
] ≤ 1.
.
So the proof is complete.
8
Integral Representation
In the following theorem, we obtain integral representation for the function f.
Theorem 10: Let f ∈ P
ℓ
AB
ℓ, ,
∗;
where │ F N │˂ 1 , ∈ 3.
, ,
,
,
ˊ=!
. Then
¢
£
hi jhk m ¡ g
¤¡
¡ [lm ¡ g
,
84
Waggas Galib Atshan et al.
Proof: By putting
ℓ
YAB
T
∗;
UZ ˊˊ
ℓ
YAB
T ∗;
UZ ˊ
=¥
,
in (9), we have
│
¦ #
l¦ # j hi jhk
│<
,
or equivalently
¥
¥
+
= F
+
,│ F
│ < 1,
∈ 3.
We get
ℓ T$∗© # UZˊˊ
Y§¨
ℓ T$∗©
Y§¨
# UZˊ
=
hi jhk m # g
#
[lm # g
,
After integration, we have
ℓ
log ((AB
T ∗;
U ˊ =
# hi jhk m ¡ g
ª ¡ [lm # g
QN.
Therefore,
ℓ
( AB
T ∗;
U ˊ=!
¢
£
«i ¬«k - ® ¯
¤¡
® iŒ°- ® ¯
.
and this gives the required result.
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