A NOTE ON FUNCTIONS DEFINED WITH RELATED TO
(
j
,
k
)
-SYMMETRIC POINTS
FUAD. S. M. ALSARARI*
1, S. LATHA
21
Department of Studies in Mathematics,
University of Mysore, Manasagangotri, Mysore 570 006, India.
2
Department of Mathematics,
Yuvaraja’s College, University of Mysore, Mysore 570 005, India.
(Received On: 09-06-15; Revised & Accepted On: 21-08-15)
ABSTRACT.
I
n this paper using the concept of(
j
,
k
)
-symmetrical functions which generalizes the notion of even, odd andk
-symmetric functions. We consider the quotient of analytic represe- ntation of starlikeness and convexity. We present the criteria that embed a normalized analytic function in the class of functions that are starlike with respect to)
,
(
j
k
-symmetric points.Keywords and phrases. Convex functions, starlike functions,
(
j
,
k
)
-symmetric points.2010 AMS Subject Classification: 30C45.
1. INTRODUCTION
Let
A
denote the class of functions of form,
=
)
(
2 =
n n n
z
a
z
z
f
∑
∞
+
(1)which are analytic in the open unit disk
1},
|<
|
:
{
=
z
z
C
and
z
U
∈
andS
denote the subclass ofA
consisting of all function which are univalent inU
.We define a function
f
∈
A
to be strongly starlike orderα
,(0
<
α
≤
1)
if.
,
2
<
)
(
)
(
U
∈
′
z
z
f
z
f
z
arg
απ
The family of all strongly starlike functions is denoted
(
)
*
α
S
.
We now introduce the concept of
(
j
,
k
)
-symmetrical functions which generalizes the concept of even, odd andk
-symmetric functions. Consider,( 1)
=2
(
) =
,
.
i i i n n
n n
e
αf e z
αz
a e
αz
α
R
∞
−
+
∑
−∈
(2)
and
...
]
1)
(
[2
2
1
=
)]
(
[
2 1 2 3 22 2 11
+
−
−
+
+
k+ k−k k
z
a
k
ka
k
z
k
a
z
z
f
(3)© 2015, IJMA. All Rights Reserved 2 where
k
is a positive integer. The transformation (2) is a rotation off
because it rotates the unit disc in thez
-plane through an angleα
and rotates the image domain in thew
-plane in the reverse direction through an angle of the samemagnitude. In (3) the transformation
u
=
z
k mapsU
ontok
copies ofU
andf
(z
)
carries this surface ontok
copies off
(
U
)
joined by a suitable branch point atw
=
0
. It is intuitively clear that thek
th root merely unwinds the symmetry. Precisely, we haveDefinition 1.1: Let
k
be a positive integer. A domainD
is said to bek
-fold symmetric if a rotation ofD
about theorigin through an angle
k
π
2
carries
D
onto itself. A functionf
is said to bek
-fold symmetric inU
if for everyz
in
U
).
(
=
)
(
2 2
z
f
e
z
e
f
ki k
i π
π
The family of all
k
-fold symmetric function is denoted byS
k and fork
=
2
we get the odd univalent function.The notion of
(
j
,
k
)
-symmetrical functions( = 2, 3,...; = 0,1, 2,...,
k
j
k
−
1)
is a generalization of the notion of even, odd,k
-symmetrical functions and generalize the well-known result that each function defined on a symmetrical subset can be uniquely expressed as the sum of an even function and an odd function.The theory of
(
j
,
k
)
-symmetrical functions has many interesting applications, for instance in the investigation of the set of fixed points of mappings, for the estimation of the absolute value of some integrals, and for obtaining some results of the type of Cartan uniqueness theorem for holomorphic mappings [9].Definition 1.2: Let
=
(
)
2
k i
e
πε
andj
=
0,1,2,..,
k
−
1
wherek
≥
2
is a natural number. A functionf
:
U
C
is called(
j
,
k
)
-symmetrical if.
),
(
=
)
(
z
f
z
z
∈
U
f
ε
ε
jThe family of all
(
j
,
k
)
-symmetrical functions is denoted beS
(j,k) .S
(0,2),S
(1,2) andS
(1,k) are respectively the classes of even, odd andk
-symmetric functions. We have the following decomposition theorem.Theorem 1.3: [9] For every mapping
f
:
U
C
, there exists exactly the sequence of(
j
,
k
)
-symmetrical functionsk j
f
, ,)
(
=
)
(
,1
0 =
z
f
z
f
jk kj
∑
−where
)
(
1
=
)
(
1
0 =
,
f
z
k
z
f
vj vk
v k
j
ε
ε
− −
∑
(4) .1)
0,1,2,...,
=
1,2,...;
=
;
(
f
∈
A
k
j
k
−
Definition 1.4: A function
f
inA
is said to belong to the classS
*(j,k) if.
0,
>
)
(
)
(
,
U
∈
′
ℜ
z
z
f
z
f
z
k j
where
f
j,k(
z
)
is defined by (4)Definition 1.5: A function
f
inA
is said to belong to the classK
(j,k), if.
0,
>
)
(
)
)
(
(
,
U
∈
′
′
′
ℜ
z
z
f
z
f
z
k j
In our work we study the classes
S
*(j,k) andK
(j,k) and obtain conditions for starlikeness with respect to(
j
,
k
)
-symmetric points, in terms of the operator..
)
(
)/
(
)
(
)/
(
=
)
;
,
,
,
,
(
,,
z
f
z
f
z
z
f
z
f
bz
a
z
b
a
k
j
f
I
k j k j′
′
′′
+
For special choices
a
=
b
=
1
we get.
)
(
)/
(
)
(
/
]
)
(
[
=
)
(
)/
(
)
(
)/
(
1
=
)
,1,1;
,
,
(
, , ,,
z
f
z
f
z
z
f
z
f
z
z
f
z
f
z
z
f
z
f
z
z
k
j
f
I
k j k j k j k j′
′
′
′
′
′
′′
+
which is the quotient of analytic representations of starlikeness and convexity with respect to
(
j
,
k
)
-symmetric points.For special choices of the parameters we get classes studied by Tuneski, Irmak, Singh, Obradovic, Sakaguchi and Silverman.
2. MAIN RESULTS
The following lemma is due to Miller and Mocanu [2]
Lemma 2.1: [2] Let
Ω
be a subset of the complex planC
and let the functionψ
:
C
2×
U
→
C
satisfyψ
(
ix
,
y
;
z
)
∈/
Ω
for all real
2
1
,
2x
y
x
≤
−
+
and for allz
∈
U
, if the functionp
(z
)
is analytic inU
,p
(0)
=
1
andΩ
∈
′
(
);
)
),
(
(
p
z
z
p
z
z
ψ
for allz
∈
U
thenℜ
{
p
(
z
)}
>
0
,z
∈
U
.
Lemma 2.2: Let
f
∈
A
,
k
≥
2
is a natural number,j
=
0,1,2,..,
k
−
1,
anda,
b
∈
R
. Also, letΩ
=
C
\
Ω
1, where
≥
∈
∈
+
−
′
+
Ω
(
)
:
,
,
|
|
1
)
(
)
(
=
, ,1
a
b
bui
z
u
u
z
f
z
z
f
b
k j k jR
U
IfU
∈
Ω
∈
z
z
b
a
k
j
f
I
(
,
,
,
,
;
)
,
thenf
∈
S
*(j,k).
Proof: Let
)
(
)
(
=
)
(
,z
f
z
f
z
z
p
k j′
then
p
(z
)
is analytic inU
andp
(0)
=
1
. Further, for the the function
)
(
)
(
)
(
=
)
;
,
(
, ,r
s
b
b
a
z
f
z
z
f
b
z
s
r
k j k j+
−
′
+
ψ
then
).
;
,
,
,
,
(
=
)
(
)
(
)
(
)
(
=
)
);
(
),
(
(
, ,z
b
a
k
j
f
I
z
p
z
p
z
b
b
a
z
f
z
z
f
b
z
z
p
z
z
p
k j k j
′
+
−
′
+
′
ψ
So, by Lemma 2.1 for proving
f
∈
S
*(j,k) or equivalentlyℜ
p
(
z
)
>
0
,z
∈
U
, it is enough to show that1
)
;
,
(
ix
y
z
∈
Ω
ψ
for all real2
1
,
2x
y
x
≤
−
+
andz
∈
U
. For1 , ,
)
(
)
(
=
)
;
,
(
∈
Ω
−
+
′
+
i
x
y
b
b
a
z
f
z
z
f
b
z
y
ix
k j k jψ
since
|
y
/
x
|
≥
1
.© 2015, IJMA. All Rights Reserved 4
Theorem 2.3: Let
f
∈
A
,
k
≥
2
is a natural number ,j
=
0,1,2,..,
k
−
1
, anda,
b
∈
R
.(i) If
(
)
(0
<
<
1)
*,
α
α
S
∈
k j
f
,| |
>
2
b
tan
a b
απ
−
whena
−
b
>
0
and if
1
| |
arctan
,
> 0
2
|
[ ( , , , , ; )
] |<
(1
)
,
0.
2
b
a b
a b
arg I f j k a b z
b
a b
π
α
λ
π
α
−
−
−
−
≡
−
− ≤
for all
z
∈
U
thenf
∈
S
*(j,k).
(ii) If
>
1
(
>
1)
)
(
)
(
,
,
µ
µ
z
f
z
f
z
k j
k j
′
and
|
I
(
f
,
j
,
k
,
a
,
b
;
z
)
−
b
|<
λ
2≡
µ
(
a
−
b
)
2+
b
2 for allz
∈
U
thenf
∈
S
*(j,k).
Proof: Let
Σ
1=
{
w
:|
arg
(
w
−
b
)
|<
λ
1}
andΣ
2=
{
w
:|
w
−
b
|<
λ
2}
be a subsets defined in the complex planC
. By Lemma 2.2, for proving(i
)
and(ii
)
it is enough to show thatΣ
1⊆
Ω
andΣ
2⊆
Ω
,
respectively.)
(i
we will show thatΣ
1⊆
Ω
by verifyingΣ
1
Ω
1=
∅
.
Ifw
∈
Ω
1 then for somez
∈
U
,|
u
|
≥
1
andu
∈
R
we have)
(
)
(
)
(
|=
)
(
|
, ,
bui
b
a
arg
z
f
z
z
f
arg
b
w
arg
k j
k
j
+
−
+
′
−
,
,
( )
|
(
) | .
( )
j kj k
f
z
arg
arg a b bui
zf
z
≥
−
− +
′
(5)For
a
−
b
>
0
we have2
|<
)
(
<|
|
|
arctan
arg
a
b
bui
π
b
a
b
−
+
−
and1
=
2
|
|
arctan
|
)
(
|
−
α
π
λ
−
≥
−
b
a
b
b
w
arg
i.e.
w
∈/
Σ
1. In the casea
−
b
≤
0
we havew
∈/
Σ
1 because.
=
2
)
(1
|
|
arctan
2
2
|
)
(
|
α
π
π
≥
−
α
π
λ
1
−
+
−
≥
−
b
b
a
b
w
arg
If
w
∈
Ω
1 thenw
∈/
Σ
2 because of,
,
( )
|
|=
.(
)
( )
j kj k
f
z
w b
a b bui
zf
z
−
− +
′
.
=
)
(
)
(
.
)
(
)
(
=
2 2 2 2 2 2,
,
µ
λ
b
b
a
u
b
b
a
z
f
z
z
f
k j
k
j
−
+
≥
−
+
′
Analogous to
(i
)
we can prove(ii
)
.Corollary 2.4: Let
f
∈
A
,
k
≥
2
is a natural number,j
= 0,1, 2,..,
k
−
1,
.(i) If
(
)
* ,
α
S
∈
k jf
for some0
<
α
<
1
and ,
2
)
(1
<
1
)
(
)/
(
)
(
/
]
)
(
[
,,
−
α
π
−
′
′
′
′
z
f
z
f
z
z
f
z
f
z
arg
k j k jfor all
z
∈
U
thenf
∈
S
*(j,k).
(ii) If
µ
1
>
)
(
)
(
, ,z
f
z
f
z
k j k j′
for some
µ
>
1
and1
<
µ
)
(
)/
(
)
(
/
]
)
(
[
, ,−
′
′
′
′
z
f
z
f
z
z
f
z
f
z
k j k jfor all
z
∈
U
thenf
∈
S
*(j,k).
Corollary 2.5: Let
f
∈
A
,
k
≥
2
is a natural number,
j
=
0,1,2,..,
k
−
1,
.(i) If
(
)
* ,α
S
∈
k jf
for some0
<
α
<
1
and
2
)
(1
<
1
)
(
)
(
)
(
)
(
,,
−
α
π
−
′
′
′′
z
f
z
f
z
f
z
f
arg
k j k jfor all
z
∈
U
thenf
∈
S
*(j,k).
(ii) If
µ
1
>
)
(
)
(
, ,z
f
z
f
z
k j k j′
for some
µ
>
1
and1
<
2
)
(
)
(
)
(
)
(
,,
−
µ
′
′
′′
z
f
z
f
z
f
z
f
k j k jfor all
z
∈
U
thenf
∈
S
*(j,k).
Corollary 2.6: Let
f
∈
A
,
k
≥
2
is a natural number,
j
=
0,1,2,..,
k
−
1,
.(i) If
(
)
* ,
α
S
∈
k jf
for some0
<
α
<
1
such that<
1
2
2
tan
απ
and if
2
2
1
arctan
<
1
)
(
)/
(
)
(
)/
(
1
, ,απ
−
+
′
′
′′
−
z
f
z
f
z
z
f
z
f
z
arg
k j k jfor all
z
∈
U
thenf
∈
S
*(j,k).
(ii) If
µ
1
>
)
(
)
(
, ,z
f
z
f
z
k j k j′
for some
µ
>
1
and
1
<
5
)
(
)/
(
)
(
)/
(
1
, ,µ
+
′
′
′′
−
z
f
z
f
z
z
f
z
f
z
k j k jfor all
z
∈
U
thenf
∈
S
*(j,k).
Remark 2.7: For
j
=
1
andk
=
N
we get the results of Nikola Tuneski in [10]REFERENCES
1. Duren. P.L, Univalent functions. New York: Springer-Verlag. 1983.
2. Miller, S. And Mocanu, P. T. Differential Subordinations. Theory and Applications. Marcel Dekker, New and York-Basel, 2000.
3. Obradovic, M. and Tuneski, N. On the starlike criteria defined by Silverman, Zeszyty nauk. Politech. Rzeszowskiej Mat. 24 (181), 59-64, 2000.
4. Sakaguchi, K. On a certain univalent mapping, J. Math. Soc. Japan. 11, 72-75, 1959. 5. Silverman, H. Convex and starlike criteria. Int. J. Math. Math. Sci. 22(1), 75-79, 1999.
6. Singh, V. and Tuneski, N. On a Criteria for starlikeness and convexity of analytic functions, Acte Math.Sci. 24(B4), 597-602, 2004.
7. Tuneski, N. On a Criteria for starlikeness of analytic functions, Integral Transform. Spec. Funct. 14(3), 263-270, 2003.
© 2015, IJMA. All Rights Reserved 6 9. Piotr, L and Jrezy, P, On
(
j
,
k
)
-symmtrical functions, Mathematica Bohemica, 120 13-25, 1995.10. Nikola, T . Some results univalent functions defined with respect to N-symmetric points. Novi Sad J. Math. 38 (3), 91-95, 2008.
11. Tuneski, N. Irmak, H. Starlikeness and convexity of a class of analytic functions. Interat. J. Math. Math. Sci. Art ID 38089, 8 PP, 2006.
12. Singh, P. Tugel, M. Convex and starlike criteria. Indian J. Pure. Apl. Math. 12(4), 513-520, 1981.
13. Fuad Al-Sarari and S. Latha, A few results on functions that are Janowski starlike related to (j; k)-symmetric opints, Octo. Math. Maga 21( 2), (October 2013), 556-563.
Source of support: Nil, Conflict of interest: None Declared
