Available online throug
ISSN 2229 – 5046
ON THE EXISTENCE OF GENERALISED FIX POINTS
DIBYENDU BANERJEE*
1, BISWAJIT MANDAL
21
Department of Mathematics, Visva - Bharati, Santiniketan-731235, West Bengal, India.
2
Bunia Dangal High School, P.O. –Bunia, Labpur-731303, West Bengal, India.
(Received On: 17-12-15; Revised & Accepted On: 27-01-16)
ABSTRACT
U
sing the idea of generalised iterations of functions we prove fix point theorems for certain class of complex functions.Key words: Generalised fix points, Exact factor order, Complex functions.
AMS Subject Classification: 30D60.
1. INTRODUCTION AND DEFINITIONS
A single valued complex function
f
(
z
)
is said to belong to class I iff
(
z
)
is entire transcendental and class II if it is regular in the complex plane punctured ata
,b
(
a
≠
b
)
and has an essential singularity atb
and a singularity ata
and if
f
(
z
)
omits the valuesa
andb
except possible ata
.For arbitrary complex function
f
(
z
)
, the iterations are defined inductively byz
z
f
0(
)
=
andf
n+1(
z
)
=
f
(
f
n(
z
))
;n
=
0
,
1
,
2
,...
.A point
α
is called a fix point off
(
z
)
of ordern
ifα
is a solution off
n(
z
)
=
z
and a fix point of exact ordern
if
α
is a solution off
n(
z
)
=
z
but not a solution off
k(
z
)
=
z
,k
=
1
,
2
,
3
,...,
n
−
1
. With this definition ofiteration, for functions of class I, Baker [1] proved the following theorem.
Theorem: A [1] If
f
(
z
)
belongs to class I, thenf
(
z
)
has fix points of exact ordern
except for at most one value ofn
.In [4], Bhattacharyya extended Theorem A to functions in class II as follows.
Theorem: B [4] If
f
(
z
)
belongs to class II, thenf
(
z
)
has an infinity of fix points of exact ordern
, for every positive integern
.In 1997, Lahiri and Banerjee [7] introduced a new concept of iteration called relative iterations (defined below) and using this, proved the result of Bhattacharyya [4].
Let
f
(
z
)
andϕ
(
z
)
be functions of the complex variablez
. Let
)))
(
(
(
)))
(
(
(
)
(
))
(
(
))
(
(
)
(
)
(
)
(
1 3
1 2
1
z
f
f
z
f
f
z
f
z
f
z
f
z
f
z
f
z
f
ϕ
ϕ
ϕ
ϕ
=
=
=
=
=
... ... ...
f z
n( )
=
f
( ( ( ...( ( )
φ
f
φ
f z
orϕ
(
z
)
according asn
is odd or even)...)))
=
f
(
ϕ
n−1(
z
))
=
f
(
ϕ
(
f
n−2(
z
)))
Dibyendu Banerjee*, Biswajit Mandal / On The Existence uf Generalised Fix Points / IJMA- 7(1), Jan.-2016.
and so
)))
(
(
(
))
(
(
)
(
))
(
(
))
(
(
)
(
)
(
)
(
1 2
3
1 2
1
z
f
z
f
z
z
f
z
f
z
z
z
ϕ
ϕ
ϕ
ϕ
ϕ
ϕ
ϕ
ϕ
ϕ
=
=
=
=
=
... ... ...
ϕ
n(
z
)
=
ϕ
(
f
n−1(
z
))
=
ϕ
(
f
(
ϕ
n−2(
z
)))
.If
f
(
z
)
andϕ
(
z
)
are functions in class II, then so aref
n(
z
)
andϕ
n(
z
)
.A point
α
is called a fix point off
(
z
)
of ordern
with respect toϕ
(
z
)
iff
n(
α =
)
α
and a fix point of exactorder
n
iff
n(
α =
)
α
butf
k(
α
)
≠
α
,
k
=
1
,
2
,...,
n
−
1
. Such pointα
is also called relative fix point.Theorem: C [7] If
f
(
z
)
andϕ
(
z
)
belong to class II, thenf
(
z
)
has an infinity of relative fix points of exact ordern
for every positive integern
, provided)
,
(
)
,
(
n n
f
r
T
r
T
ϕ
is bounded.
Recently, Banerjee and Mandal [2] introduced the concept of relative fix point of exact factor order and using this concept proved analogous theorem of Lahiri and Banerjee [7].
A point
α
is called a relative fix point off
(
z
)
with respect toϕ
(
z
)
of exact factor ordern
iff
n(
α =
)
α
butα
α ≠
)
(
k
f
andϕ
k(
α
)
≠
α
for all divisorsk
(
k
<
n
)
ofn
.Theorem: D [2] If
f
(
z
)
andϕ
(
z
)
belong to class II, thenf
(
z
)
has an infinity of relative fix points of exact factororder
n
for every positive integern
provided)
,
(
)
,
(
1n n
f
r
T
f
r
T
−is bounded.
In [3], Banerjee and Mondal introduced another type of iteration called generalised iteration which runs as follows.
Let
f
(
z
)
andϕ
(
z
)
be two entire functions andα
∈
(
0
,
1
]
be any number. Then the generalised iteration off
(
z
)
with respect toϕ
(
z
)
is defined as follows.
))
(
(
)
(
)
1
(
)
(
))
(
(
)
(
)
1
(
)
(
)
(
)
1
(
)
(
2 2
3
1 1
2 1
z
f
z
z
f
z
f
z
z
f
z
f
z
z
f
ϕ
α
ϕ
α
ϕ
α
ϕ
α
α
α
+
−
=
+
−
=
+
−
=
... ... ...
f
n(
z
)
=
(
1
−
α
)
ϕ
n−1(
z
)
+
α
f
(
ϕ
n−1(
z
))
and
( )
))
(
(
)
(
)
1
(
)
(
))
(
(
)
(
)
1
(
)
(
)
(
)
1
(
2 2
3
1 1
2 1
z
f
z
f
z
z
f
z
f
z
z
z
z
αϕ
α
ϕ
αϕ
α
ϕ
αϕ
α
ϕ
+
−
=
+
−
=
+
−
=
... ... ...
ϕ
n(
z
)
=
(
1
−
α
)
f
n−1(
z
)
+
αϕ
(
f
n−1(
z
))
.Note-1: When
α
=
1
, generalised iteration reduces to relative iteration.Definition: 1 A point
β
is called a generalised fix point off
(
z
)
of ordern
iff
n(
β =
)
β
and a generalised fix point of exact ordern
iff
n(
β =
)
β
butf
k(
β ≠
)
β
,k
=
1
,
2
,
3
,...,
n
−
1
.β
is called a generalised fix point of)
(
z
f
of exact factor ordern
iff
n(
β =
)
β
butf
k(
β ≠
)
β
andϕ
k(
β
)
≠
β
for all divisorsk
(
k
<
n
)
ofn
.Example: 1 Let
f
(
z
)
=
2
z
+
1
,ϕ
(
z
)
=
2
z
−
1
andα
∈
(
0
,
1
]
. Thenα
α
+
=
2
z
is generalised fix point of exactorder 2 of
f
(
z
)
and3
3
1
22
+
+
+
+
−
=
α
α
α
α
z
is generalised fix point of exact factor order 3 off
(
z
)
.We normalise the functions in class II by taking
a
=
0
,b
=
∞
and throughout this paper we consider such type of functions and their generalised iteration unless otherwise stated.Let
f
(
z
)
be meromorphic inr
0≤
z
<
∞
,r
0>
0
. Following notations given in {[5], pp.88}, the first fundamental theorem takes the form)
(log
)
,
(
)
,
,
(
)
,
,
(
r
a
f
N
r
a
f
T
r
f
O
r
m
+
=
+
(1)where the region is always
r
0≤
z
<
∞
,r
0>
0
.Further if
f
(
z
)
is non-constant anda
1,
a
2,...,
a
q;
q
≥
2
,
be distinct finite complex numbers,δ
>
0
withδ
ν µ
−
a
≥
a
for1
≤
µ
≤
ν
≤
q
, then)
(
)
(
)
,
(
2
)
,
,
(
)
,
(
11
r
S
r
N
f
r
T
f
a
r
m
f
r
m
q
+
−
≤
+
∑
=
ν ν
(2)
where
N r
1( )
N r
,
1
2 ( , )
N r f
N r f
( ,
)
f
′
=
+
−
′
and
1
( )
,
,
(log )
q
f
f
S r
m r
m r
O
r
f
ν=f
a
ν
′
′
=
+
+
−
∑
.Adding
∑
=
+
qN
r
a
f
f
r
N
1
)
,
,
(
)
,
(
ν ν
to both sides of (2) and using (1), we obtain
∑
=
−
−
−
≥
q
r
S
f
r
N
f
r
T
q
f
a
r
N
1
1
(
)
)
,
(
)
,
(
)
1
(
)
,
,
(
ν ν
(3)
where
S
1(
r
)
=
O
(log
T
(
r
,
f
))
andN
corresponds to distinct roots.Also since
f
n has an essential singularity at∞
, we have {[5], pp.90},0
)
,
(
log
→
n
f
r
T
r
as
r
→
∞
.The object of this paper is to extend Theorem C and Theorem D to functions in class II, with generalised iteration.
2. LEMMAS
The following lemmas will be needed in the sequel.
Lemma: 1 If
f
andϕ
are functions in class II, then for anyr
0>
0
andM
, a positive constantM
f
r
T
f
r
T
>
)
,
(
))
(
,
(
ϕ
,
for all large
r
, except a set ofr
intervals of total finite length.Dibyendu Banerjee*, Biswajit Mandal / On The Existence uf Generalised Fix Points / IJMA- 7(1), Jan.-2016.
Lemma: 2 If
n
is any positive integer andf
(
z
)
andϕ
(
z
)
are functions in class II, then for anyr
0>
0
andM
1, a positive constant 1)
,
(
)
,
(
M
f
r
T
f
r
T
n p n+>
or 1
)
,
(
)
,
(
M
f
r
T
r
T
n p n+>
ϕ
according as
p
is even or odd, for all larger
except a set ofr
intervals of total finite length.Proof: For
j
=
1
,
2
,...,
n
and for all larger
, by using Lemma 1 we get)
1
(
))
(
,
(
)
)
1
(
,
(
)
,
(
r
f
1T
r
T
r
f
O
T
j+≤
−
α
ϕ
j+
α
ϕ
j+
≤
T
(
r
,
ϕ
j)
+
T
(
r
,
f
(
ϕ
j))
+
O
(
1
)
( , ( )) 1
( ,
)
(1)
( , ( ))
( , ( ))
j j
j j
T r
O
T r f
T r f
T r f
φ
φ
φ
φ
=
+
+
=
(
1
+
O
(
1
))
T
(
r
,
f
(
ϕ
j)).
(4)Again,
(
(
))
1
(
)
1(
)
1
z
z
f
z
f
j jϕ
jα
ϕ
α−α+
−
=
and so for larger
,)
1
(
)
,
(
)
,
(
))
(
,
(
r
f
T
r
f
1T
r
O
T
ϕ
j≤
j++
ϕ
j+
.Therefore
)
1
(
)
,
(
))
(
,
(
)
,
(
r
f
1T
r
f
T
r
O
T
j+≥
ϕ
j−
ϕ
j+
( , ( )) 1
( ,
)
(1)
( , ( ))
( , ( ))
j j
j j
T r
O
T r f
T r f
T r f
φ
φ
φ
φ
=
−
+
=
(
1
+
O
(
1
))
T
(
r
,
f
(
ϕ
j)).
(5)From (4) and (5), for all large
r
)).
(
,
(
))
1
(
1
(
)
,
(
r
f
j 1O
T
r
f
jT
+=
+
ϕ
(6)Similarly for all large
r
, we have)).
(
,
(
))
1
(
1
(
)
,
(
r
j 1O
T
r
f
jT
ϕ
+=
+
ϕ
(7)In particular,
)
,
(
))
1
(
1
(
)
,
(
r
f
1o
T
r
f
T
=
+
andT
(
r
,
ϕ
1)
=
(
1
+
o
(
1
))
T
(
r
,
ϕ
)
). Now we consider the following two cases.Case-(i): When
p
is even.For all large
r
except a set ofr
intervals of total finite length, we have from (6) and (7), by using Lemma 1)
,
(
))
(
,
(
))
1
(
1
(
)
,
(
)
,
(
1 n p n n p nf
r
T
f
r
T
O
f
r
T
f
r
T
+ + −+
=
ϕ
)
,
(
)
,
(
)
,
(
))
(
,
(
))
1
(
1
(
1 1 1 n p n p n p nf
r
T
r
T
r
T
f
r
T
O
+ −− + − +
+
=
ϕ
ϕ
ϕ
)
,
(
))
(
,
(
))
1
(
1
(
)
,
(
))
(
,
(
))
1
(
1
(
2 1 1 n p n p n p nf
r
T
f
r
T
O
r
T
f
r
T
O
+ −− + − +
+
+
=
ϕ
ϕ
ϕ
)
,
(
))
(
,
(
)
,
(
))
(
,
(
))
1
(
1
(
2 1 1 n p n p n p nf
r
T
f
r
T
r
T
f
r
T
O
+ −− + − +
+
=
ϕ
ϕ
ϕ
)
,
(
))
(
,
(
))
(
,
(
r
f
n p 1T
r
f
n p 2T
r
f
n p 2T
+ − + − + −+
... ... ... ... ... ...
1 2 3
1 2 3
( , (
))
( , (
))
( , (
)
( , (
))
(1
(1))
...
( ,
)
( ,
)
( ,
)
( ,
)
n p n p n p n
n p n p n p n
T r f
T r
f
T r f
T r
f
O
T r
T r f
T r
T r f
φ
φ
φ
φ
φ
φ
+ − + − + − + − + − + −
= +
>
(
1
+
O
(
1
))
M
p
=
M
1, saywhere
M
1=
(
1
+
O
(
1
))
M
p, a positive constant.i.e., 1
)
,
(
)
,
(
M
f
r
T
f
r
T
n p n>
+for all large
r
except a set ofr
intervals of total finite length.Case-(ii): When
p
is odd.For all large
r
except a set ofr
intervals of total finite length, we have from (6) and (7), by using Lemma 1)
,
(
))
(
,
(
))
1
(
1
(
)
,
(
)
,
(
1 n p n n p nf
r
T
f
r
T
O
f
r
T
r
T
+ + −+
=
ϕ
ϕ
)
,
(
)
,
(
)
,
(
))
(
,
(
))
1
(
1
(
1 1 1 n p n p n p nf
r
T
f
r
T
f
r
T
f
r
T
O
+ −− + − +
+
=
ϕ
)
,
(
))
(
,
(
))
1
(
1
(
)
,
(
))
(
,
(
))
1
(
1
(
2 1 1 n p n p n p nf
r
T
f
r
T
O
f
r
T
f
r
T
O
+ −− +
− +
+
+
=
ϕ
ϕ
)
,
(
))
(
,
(
)
,
(
))
(
,
(
))
1
(
1
(
2 1 1 n p n p n p nf
r
T
f
r
T
f
r
T
f
r
T
O
+ −− +
− +
+
=
ϕ
ϕ
)
,
(
)
,
(
)
,
(
))
(
,
(
)
,
(
))
(
,
(
))
1
(
1
(
2 2 2 1 1 n p n p n p n p n p nf
r
T
r
T
r
T
f
r
T
f
r
T
f
r
T
O
+ −− + − + − + − +
+
=
ϕ
ϕ
ϕ
ϕ
... ... ... ... ... ...1 2 3
1 2 3
( , (
))
( , (
))
( , (
)
( , (
))
(1
(1))
...
( ,
)
( ,
)
( ,
)
( ,
)
n p n p n p n
n p n p n p n
T r
f
T r f
T r
f
T r
f
O
T r f
T r
T r f
T r f
φ
φ
φ
φ
φ
+ − + − + − + − + − + −
= +
>
(
1
+
O
(
1
))
M
p
=
M
1, saywhere
M
1=
(
1
+
O
(
1
))
M
p, a positive constant.i.e., 1
)
,
(
)
,
(
M
f
r
T
r
T
n p n>
+ϕ
for all large
r
except a set ofr
intervals of total finite length.Lemma: 3 If
n
is any positive integer andf
(
z
)
andϕ
(
z
)
are functions in class II, then for anyr
0>
0
andM
1, apositive constant 1
)
,
(
)
,
(
M
r
T
r
T
n p n>
+ϕ
ϕ
or 1
)
,
(
)
,
(
M
r
T
f
r
T
n p n>
+ϕ
according as
p
is even or odd, for all larger
except a set ofr
intervals of total finite length.3. THEOREMS
Dibyendu Banerjee*, Biswajit Mandal / On The Existence uf Generalised Fix Points / IJMA- 7(1), Jan.-2016.
Theorem: 1 If
f
(
z
)
andϕ
(
z
)
belong to class II, thenf
(
z
)
has an infinity of generalised fix points of exact ordern
for every positive integern
, provided)
,
(
)
,
(
n n
f
r
T
r
T
ϕ
is bounded.
Proof: We may assume that
α
≠
1
, because ifα
=
1
, the theorem coincides with Theorem C. We consider the function,
)
(
)
(
z
z
f
z
g
=
n<
<
∞
z
r
0then
T
(
r
,
g
)
=
T
(
r
,
f
n)
+
O
(log
r
).
(8) Assume thatf
(
z
)
has only a finite number of generalised fix points of exact ordern
. Now from (3) by takingq
=
2
,0
1=
a
,a
2=
1
, we obtain forg
,( )
r
g
S
g
r
N
g
r
N
g
r
N
g
r
T
(
,
)
≤
(
,
∞
,
)
+
(
,
0
,
)
+
(
,
1
,
)
+
1,
(9) whereS
1(
r
,
g
)
=
O
(log
T
(
r
,
g
))
outside a set ofr
intervals of finite length {[6], pp.47}.Since
f
n(
z
)
belongs to class II, it has a singularity atz
=
0
and an essential singularity atz
=
∞
and∞
≠
0
,
)
(
z
f
n inr
0<
z
<
∞
.Also the distinct roots of
g
(
z
)
=
0
inr
0<
z
≤
t
are the roots off
n(
z
)
=
0
inr
0<
z
≤
t
. Son
(
t
,
0
,
g
)
=
0
.Consequently
N
(
r
,
0
,
g
)
=
0
. By similar argumentN
(
r
,
∞
,
g
)
=
0
. Further ifg
(
z
)
=
1
, thenf
n(
z
)
=
z
.So,
N
(
r
,
1
,
g
)
=
N
(
r
,
0
,
f
n−
z
)
(
,
0
,
)
(log
)
,
11
∑
− =+
−
≤
nj
j
z
O
r
f
r
N
the term
O
(log
r
)
arises due to the assumption thatf
(
z
)
has only a finite number of generalised fix points of exact ordern
.Now from (9), we have
))
,
(
(log
)
(log
)
,
0
,
(
)
,
(
1
1
g
r
T
O
r
O
z
f
r
N
g
r
T
n
j
n
−
+
+
≤
∑
−=
(
,
)
(log
(
,
))
(log
)
11
r
O
g
r
T
O
f
r
T
n
j
j
+
+
≤
∑
−=
1 2
1 2
( ,
)
( ,
)
( ,
)
( ,
)
( ,
)
( ,
)
( ,
)
( ,
)
( ,
)
( ,
)
( ,
)
( ,
)
( ,
)
( ,
)
( ,
)
(log )
log
( ,
) 1
( ,
)
(log )
( ,
)
(
p
q i
i i
n n n
j
j j n
n
n n n n
n
n
n
T r f
T r f
T r f
T r f
T r f
T r f
T r f
T r f
T r f
T r
T r f
T r
T r
T r
T r f
O
r
O
T r f
T r f
O
r
T r f
T
φ
φ
φ
φ
+
+ +
=
+
+
+ +
+
+
+
,
,
n)
r f
where
i i
1, ,...,
2i
p andj j
1,
2,...,
j
q are(
n
−
1
)
distinct index belong to the set{
1, 2, 3,...,
n
−
1
}
such that(
n
−
i
p)
's are even and(
n
−
j
q)
's are odd
]
4
1
4
1
)[
,
(
n
n
n
n
f
r
T
n−
+
+
<
, for all larger
, by Lemma 2 and since)
,
(
)
,
(
n n
f
r
T
r
T
ϕ
is bounded
(
,
)
2
1
n
f
r
T
=
.Therefore,
(
,
)
2
1
)
,
(
r
g
T
r
f
nT
<
for all larger
. This contradicts (8). Hencef
(
z
)
has infinitely many generalisedfix points of exact order
n
.This proves the theorem.
Theorem: 2 If
f
(
z
)
andϕ
(
z
)
belong to class II, thenf
(
z
)
has an infinity of generalised fix points of exact factororder
n
for every positive integern
, provided)
,
(
)
,
(
n n
f
r
T
r
T
ϕ
is bounded.
Proof: As in Theorem 1, we assume that
f
(
z
)
has only a finite number of generalised fix points of exact factororder
n
. Considering the function(
)
(
)
,
z
z
f
z
h
=
n<
<
∞
z
r
0 we have).
(log
)
,
(
)
,
(
r
h
T
r
f
O
r
T
=
n+
(10)Here also
N
(
r
,
0
,
h
)
=
0
andN
(
r
,
∞
,
h
)
=
0
.To calculate
N
(
r
,
1
,
h
)
we consider two cases separately.Case-(i): When
n
is even.Now,
N
(
r
,
1
,
h
)
=
N
(
r
,
0
,
f
n−
z
)
∑
−=
+
−
+
−
≤
21 ,
)
(log
)]
,
0
,
(
)
,
0
,
(
[
n
j n j
j
j
z
N
r
z
O
r
f
r
N
ϕ
∑
−=
+
+
≤
21 ,
)
(log
)
,
(
)
,
(
[
n
j n j
j
j
T
r
O
r
f
r
T
ϕ
2 3 2 1
2 4
( ,
)
1( ,
)
( ,
)
( ,
)
( ,
)
( ,
)
( ,
)
( ,
)
( ,
)
( ,
)
( ,
)
( ,
)
( ,
)
q p
j j j
j j j
n
n n n n n n
T r f
T r
T r
T r f
T r f
T r
T r f
T r f
T r f
T r f
T r f
T r f
T r f
φ
φ
φ
−
=
+
+ +
+
+
+ +
2 1 3
1
2
2 4
( ,
)
( ,
)
( ,
)
( ,
)
( ,
)
( ,
)
( ,
)
(log )
( ,
)
( ,
)
( ,
)
( ,
)
( ,
)
( ,
)
p
q j j
j
n n n
n
j
j j
n n n
T r f
T r f
T r f
T r
T r
T r
T r
O
r
T r
T r
T r
T r
T r
T r
φ
φ
φ
φ
φ
φ
φ
φ
φ
φ
−
+
+ +
+
+
+
+
+ +
where
j j
1,
3,...,
j
2p−1 are distinct odd divisors ofn
andj
2,
j
4,...,
j
2q are distinct even divisors ofn
and strictlyless than
n
)
(log
)
,
(
4
1
)
,
(
4
1
r
O
r
T
n
n
f
r
T
n
n
n
n
+
+
+
−
<
ϕ
,Dibyendu Banerjee*, Biswajit Mandal / On The Existence uf Generalised Fix Points / IJMA- 7(1), Jan.-2016.
Case-(ii): When
n
is odd.)
,
0
,
(
)
,
1
,
(
r
h
N
r
f
z
N
=
n−
∑
−
=
+
−
+
−
≤
21 ,
)
(log
)]
,
0
,
(
)
,
0
,
(
[
n
j n j
j
j
z
N
r
z
O
r
f
r
N
ϕ
∑
−
=
+
+
≤
21 ,
)
(log
)
,
(
)
,
(
[
n
j n j
j
j
T
r
O
r
f
r
T
ϕ
∑
∑
−
= −
=
+
+
=
21 , 2
1 ,
)
(log
)
,
(
)
,
(
)
,
(
)
,
(
)
,
(
)
,
(
n
j n
j n
j n
n
j n
j n
j
n
O
r
r
T
r
T
r
T
f
r
T
f
r
T
f
r
T
ϕ
ϕ
ϕ
(
,
)
(log
)
4
1
)
,
(
4
1
r
O
r
T
n
n
f
r
T
n
n
n
n
+
+
+
−
<
ϕ
,for all large
r
, by Lemma 2 and Lemma 3.Thus in any case,
)
(log
)
,
(
4
1
)
,
(
4
1
)
,
1
,
(
T
r
O
r
n
n
f
r
T
n
n
h
r
N
<
−
n+
+
ϕ
n+
.So,
)
(
)
,
1
,
(
)
,
(
r
h
N
r
h
S
1r
T
≤
+
(
,
)
(log
)
(log
(
,
))
4
1
)
,
(
4
1
h
r
T
O
r
O
r
T
n
n
f
r
T
n
n
n
n
+
+
+
+
−
<
ϕ
( ,
)
1
1
( ,
)
(log )
(log ( , ))
4
4
( ,
)
( ,
)
( ,
)
n n
n n n
T r
n
n
O
r
O
T r h
T r f
n
n T r f
T r f
T r f
φ
−
+
=
+
+
+
( ,
)
1
1
(log )
(log( ( ,
)
(log )))
4
4
( ,
)
( ,
)
n n
n n
O
T r f
O
r
n
n
O
r
T r f
n
n
T r f
T r f
−
+
+
≤
+
+
+
,Since
)
,
(
)
,
(
n n
f
r
T
r
T
ϕ
is bounded
(log )
log
( ,
) 1
( ,
)
1
(log )
( ,
)
2
( ,
)
( ,
)
n
n
n
n n
O
r
O
T r f
T r f
O
r
T r f
T r f
T r f
+
=
+
+
)
,
(
2
1
n
f
r
T
=
, for all larger
.Therefore,
(
,
)
2
1
)
,
(
r
h
T
r
f
nT
<
for all larger
. This contradicts (10). Hencef
(
z
)
has infinitely many generalisedfix points of exact factor order
n
.This proves the theorem.
REFERENCES
1. Baker, I. N., The existence of fix points of entire functions, Math. Zeit. 73 (1960), pp.280-284.
2. Banerjee, D. and Mandal, B., On the existence of relative fix points of a certain class of complex functions, İstanbul Univ. Sci. Fac. J. Math. Phys. Astr. Vol.5 (2014), pp.9-16.
3. Banerjee, D. and Mondal, N., Maximum modulus and maximum term of generalised iterated entire functions, The Allahabad Mathematical Society, Vol. 27, Part I, 2012, pp.117-131.
6. Hayman, W. K., Meromorphic functions, The Oxford University Press (1964).
7. Lahiri, B. K. and Banerjee, D., On the existence of relative fix points, Istanbul Univ. Fen Fak. Mat. Dergisi 55-56 (1996-1997), pp.283-292.
Source of support: Nil, Conflict of interest: None Declared
