On Relative Order of Composite Function with
respect to a Composite Function
Dibyendu Banerjee1 And Mithun Adhikary2 Department of Mathematics, Visva-Bharati
Santiniketan- 731235, India
ABSTRACT
In this paper we prove some results on relative order of composite function with respect to a composite function formed with entire and meromorphic functions.
Keywords: Entire function, Meromorphic function, Relative order, Composite function. Mathematics Subject Classification (2010): 30D35.
I. INTRODUCTION AND DEFINITIONS
The Maximum modulus of an entire function f z( ) is defined by
( )
{| ( ) |:| |
}.
f
M
r
max
f z
z
r
If
f
is non-constant then Mf( )r is strictly increasing and continuous function of r and its inverse1
: (|
(0) |, )
(0, )
f
M
f
exists and is such that
lim
f( )
.
r
M
r
Definition 1.1The order of an entire function
f z
( )
is defined as[2]
log
lim
sup
( )
.
log
f fr
M
r
r
Definition 1.2([4])If
f z
( )
andg z
( )
are two entire functions then the relative order off z
( )
with respect to( )
g z
is defined as0
( )
inf{
0 :
( )
(
)
( )
0}
g
f
M
fr
M
gr
for all r
r
1
log
( )
sup
.
log
lim
g fr
M M
r
r
Definition 1.3 ([7]) The relative order of a meromorphic function
f z
( )
with respect to an entire functiong z
( )
is defined as0
( )
inf{
0 :
( )
(
)
( )
0
}
g
f
T r
fT r
gfor all r
r
1
log
( )
sup
.
log
l
im
g fr
T T r
r
Definition 1.4([1]) The relative order of a meromorphic function
f z
( )
with respect to another meromorphic functiong z
( )
is defined aslog
( )
( )
l m
sup
.
log
( )
i
fg
r h
T r
f
T r
II. LEMMAS
In this section we present some lemmas which will be needed in the sequel.
Lemma 2.1([6]) Let
g
be an entire function. Then for all large values ofr
( )
log
( )
3 (2 ).
g g g
T r
M r
T
r
Lemma 2.2([12]) Let
f
andg
be two entire functions. Then for a sequence of values ofr
tending to infinity1
1
( )
log
(
( )).
3
9
4
f g f g
r
T
r
M
M
Lemma 2.3([5]) Let
f
andg
be two entire functions. Then for all sufficiently large values ofr
1
(
( ) | (0) |)
( )
(
( )).
8
2
f g f g f g
r
M
M
g
M
r
M
M
r
Lemma 2.4([8]) Let
f
be a meromorphic function andg
be an entire function with0
g
and 0.f
Then for a sequence of values of
r
tending to infinity( )
(exp( ) ).
f g g
T
r
T
r
Lemma 2.5 ([3]) Let
f
be a meromorphic function andg
be an entire function with0
g
.
Then for a sequence of values ofr
tending to infinity( )
(exp( ) ).
f g f
T
r
T
r
Lemma 2.6 ([4]) Let
f
be a meromorphic function andg
be an entire function then for all large values ofr
( )
( ) {1 0(1)}
(
( )).
log
( )
g
f g f g
g
T r
T
r
T M
r
M
r
Lemma 2.7([9]) Let
f
andg
be two entire functions. IfM
g( )
r
2
| (0) |
g
ò
ò
for anyò
0,
then( )
(1
)
(
( )).
f g f g
T
r
ò
T M r
In particular if
g
(0)
0
then( )
(
( ))
f g f g
T
r
T M r
for all
r
0.
Lemma 2.8Let
f
be a meromorphic function andg
be an entire function with0
g
.
Then for all large values ofr
( )
{1 0(1)}
(exp( ) ).
f g f
T
r
T
r
Proof:From Lemma 2.6
( )
( ) {1 0(1)}
(
( )).
log
( )
g
f g f g
g
T r
T
r
T M r
M r
Now from the definition of order of
g
we have for anyò
0
and larger
log
( )
gg
M r
r
ò. ,
( )
exp
gexp
.
g g
i e M r
r
ò
r
when
(2.2) So using Lemma 2.1we have from (2.1) and (2.2)( )
{1
0(1)}
(exp( ) ).
f g f
T
r
T
r
III. MAIN THEOREMS
In this section we present the main results of the paper.
Theorem 3.1Let
f f h
1,
2,
1 andh
2 be four entire functions of respective finite orders andg
be a polynomial of degreem
.
Then the relative order ofh
1
h
2 with respect tof
1
f
2
g
satisfies the inequality2 1 2
2
1 2
(
)
hf f g
f
h
h
m
and further when
|
h
2(0) | 0,
21 2
2
1 2
(
)
h.
f f g
f
h
h
m
Proof:We have by the definition of order for any
ò
0
there existsr
0( )
ò
0
such that1
1
( )
exp{
}
0( ).
f
f
M
r
r
òfor all r
r
ò
(3.1)1
1
1
1
[2]. ,
f( )
exp{
log
}.
f
i e M
r
r
ò
(3.2)Similarly
2
2
1
1
[2]( )
exp{
log
}.
f
f
M
r
r
ò
(3.3)
Again for arbitrary
ò
0
and for all large values ofr
1
1
( )
exp{
}
h
h
M
r
r
ò (3.4) and2
2
( )
exp{
}.
h
h
M
r
r
ò (3.5)Let
g z
( )
a
0
a z
1
a z
2 2
a z
m m.
Then for any
ò
0
there existsr
1( )
ò
0
such that1
1 1
1 1
( ) {
}
( ) {
} .
|
| (1
)
|
| (1
)
m m
g g
m m
r
r
M
r
and M
r
a
a
ò
ò
(3.7)Again from the Lemma 2.3we get
1 2
( )
1(
2(
( )))
f f g f f g
M
r
M
M
M
r
1 2 2 1
1 1 1 1
. ,
f f g( )
g(
f(
f( ))).
i e M
r
M
M
M
r
(3.8)Also by Lemma 2.3for two entire functions
h
1 andh
2 with|
h
2(0) | 0
we have1 2 1 2
1
( )
(
( )).
8
2
h h h h
r
M
r
M
M
(3.9)So
1 2 1 2
1 2
1
1 2
log
(
( ))
(
)
li
m u
s
p
log
f f g h h
f f g
r
M
M
r
h h
r
2 1 1 2
1 1 1
log
(
(
(
( ))))
(3.8)
log
g f f h h
r
M
M
M
M
r
limsup
from
r
2 1 1 2
1 1 1
1
log
(
(
(
(
( )))))
8
2
(3.9)
log
g f f h h
r
r
M
M
M
M
M
limsup
from
r
2
2 1 1
1 1 1
1
log
(
(
(
( exp( )
))))
8
2
(3.5)
log
h
g f f h
r
r
M
M
M
M
limsup
from
r
ò
2 1
2 1
1 1 1
1
log
(
(
(exp[ exp( )
]
)))
8
2
(3.4)
log
h h
g f f
r
r
M
M
M
limsup
from
r
ò ò
2 1
2
1
1 1
1
[2]1
log
(
(exp{
log (exp[ exp( )
]
)}))
8
2
(3.2)
log
h h
g f
f r
r
M
M
limsup
from
r
ò ò
ò
1 2
2
1
1 1
log
(
(exp{
( )
}))
(1)
2
log
h
h
g f
f r
r
M
M
O
limsup
r
ò
ò
ò
1 2
2 1
1
1
[2]log
(exp{
log (exp{
( )
}))
(1)
2
(3.3)
log
h
h g
f f
r
r
M
O
limsup
from
r
ò
ò
ò
ò
2
2
1
log
(exp{
log( )})
(1)
2
log
h gf r
r
M
O
limsup
r
ò
ò
2
2
1
log
(( )
)
(1)
2
log
h f
g r
r
M
O
limsup
r
2
2
1
( )
2
log{
}
(1)
|
| (1
)
(3.7)
log
h f
m m
r
r
O
a
limsup
from
r
ò ò
ò
2
2
log( )
log |
| (1
)
(1)
1
2
log
h f
m r
r
a
O
limsup
m
r
ò ò
ò
2
2
log( ) log |
| (1
)
(1)
2
1
log
hm f
r
r
a
O
limsup
m
r
ò
ò
ò
2
2
1
hf
m
ò
ò
2
2
1
,
0
.
h f
since
is arbitrary
m
ò
From Lemma 2.3, for all sufficiently large values of
r
1 2 1 2 1 2
1
1
1
( )
(
( ))
(
(
(
( ))))
9
2
9
18
2
f f g f f g f f g
r
r
M
r
M
M
M
M
M
(3.10)
1 2 2 1
1 1 1 1
. ,
f f g( )
2
g(18
f(9
f( ))).
i e M
r
M
M
M
r
(3.11) Also for all sufficiently large values ofr
we get from Lemma 2.31 2
( )
1(
2( )).
h h h h
M
r
M
M
r
(3.12)Againfor sufficiently large values of
r
1 2
1
( )
exp{
}
2( )
exp{
}
h h
h h
M
r
r
òand M
r
r
ò (3.13) and1 2
1 2
1
1
[2] 11
[2]( )
exp{
log
}
( )
exp{
log
}.
f f
f f
M
r
r
and M
r
r
ò
ò
(3.14)Now
1 2 1 2
1 2
1
1 2
log
(
( ))
(
)
li
m u
s
p
log
f f g h h
f f g
r
M
M
r
h h
r
2 1 1 2
1 1 1
log(2
(18
(9
(
( )))))
(3.11)
log
g f f h h
r
M
M
M
M
r
limsup
from
r
2 1 1 2
1 1 1
log(2
(18
(9
(
(
( )))))
(3.12)
log
g f f h h
r
M
M
M
M
M
r
limsup
from
r
2
2 1 1
1 1 1
log(2
(18
(9
(
(exp{
})))))
(3.13)
log
h
g f f h
r
M
M
M
M
r
limsup
from
r
ò
2 1
2 1
1 1 1
log(2
(18
(9
(exp[exp{
}]
))))
(3.13)
log
h h
g f f
r
M
M
M
r
limsup
from
r
ò ò
2 1
2
1
1 1
1
[2]log(2
(18
(9 exp{
log (exp[exp{
}]
)})))
(3.14)
log
h h
g f
f r
M
M
r
limsup
from
r
ò ò
ò
1 2
2
1
1 1
log(2
(18
(9 exp{
})))
log
h
h
g f
f r
M
M
r
limsup
r
ò
ò
ò
1 2
2 1
1
1
[2]log(2
(18exp{
log {9 exp{
}}}))
(3.14)
log
h
h g
f f
r
M
r
limsup
from
r
ò
ò
ò
ò
1 2
2 1
1
1
log(2
(18exp{
log[
]}))
(1)
log
h
h g
f f
r
M
r
O
limsup
r
ò
ò
ò
ò
2
2
1
log(2
(18
))
(1)
log
h f
g r
M
r
O
limsup
r
ò ò
2
2 1
18
log{
}
(1)
|
| (1
)
(3.7)
log
h f
m m
r
r
O
a
limsup
from
r
ò ò
ò
2
2
1
log
(1)
log
h f
r
r
O
m
limsup
r
ò ò
2
2
1
,
0
.
h
f
since
is arbitrary
m
ò
Theorem 3.2Let
f f h
1,
2,
1 andh
2 be four entire functions of respective finite orders such that2
0
f
and1
,
2g g
be two polynomials of degreem m
1,
2 respectively such that|
g
2(0) | 0.
Then the relative order ofh h
1
2
g
2 with respect tof
1
f
2
g
1 satisfies the inequality2
1 2 1
2
2
1 2 2
1
(
)
hf f g
f
m
h
h
g
m
and2 1 2 1
2
2
1 2 2
1
(
)
h.
f f g
f
m
h h
g
m
Proof:Let 1 1 2
1
( )
0 1 2m m
g z
a
a z
a z
a z
and
2
2
2
2
( )
0 1 2m m
g z
b
b z
b z
b z
be two polynomials of degree
m m
1,
2 respectively. By the definition of relative order of an entire function with respect to another entire function we have1 2 1 1 2 2
1 2 1
1
1 2 2
log
(
( ))
(
)
sup
l g
im
o
l
f f g h h gf f g
r
M
M
r
h
h
g
r
1 2 1 1 2 2
1 1 1
log
(
(
(
( ))))
(3.8)
log
g f f h h g
r
M
M
M
M
r
limsup
from
r
1 2 1 1 2 2
1 1 1
1
1
log
(
(
(
(
(
( ))))))
9
18
2
(3.10)
log
g f f h h g
r
r
M
M
M
M
M
M
limsup
from
r
2
1 2 1 1 2
1 1 1
1
1
log
(
(
(
(
(
|
| (1
)( ) )))))
9
18
2
(3.6)
log
m
g f f h h m
r
r
M
M
M
M
M
b
limsup
from
r
ò
2 2
1 2 1 1
1 1 1
1
1
log
(
(
(
( exp(
|
| (1
)( ) ))
)))
9
18
2
(3.5)
log
h
m
g f f h m
r
r
M
M
M
M
b
limsup
from
r
ò
ò
2 2 1
1 2 1
1 1 1
1
1
log
(
(
(exp[ exp(
|
| (1
)( ) )
]
)))
9
18
2
(3.4)
log
h h
m
g f f m
r
r
M
M
M
b
limsup
from
r
ò ò
ò
1 2 2
1 2
1
1 1
1
1
log
(
(exp{
log( exp(
|
| (1
)( ) )
)}))
9
18
2
(3.2)
log
h
h m
g f m
f r
r
M
M
b
limsup
from
r
ò
ò
ò
ò
1 2 2
1 2
1
1 1
|
|
log
(
(exp{
(
)(1
)( ) }
))
(1)
18
2
log
h
h m m
g f
f r
b
r
M
M
O
limsup
r
ò
ò
ò
ò
1 2 2
1
2 1
1
1
[2]|
|
log
(exp(
log (exp{
(
)(1
)( ) }
)))
(1)
18
2
(3.3)
log
h
h m m
g
f f
r
b
r
M
O
limsup
from
r
ò
ò
ò
2 1 2
1
2 1
1
|
|
log
(exp{
log({
(
)(1
)( ) })})
(1)
18
2
log
h h m m
g
f f
r
b
r
M
O
limsup
r
ò
ò
ò
ò
ò
2 2
1
2
1
log
(exp{
log( ) })
(1)
2
log
h m
g
f r
r
M
O
limsup
r
ò
ò
2
2 2
1
1
log
[( ) ]
(1)
2
log
h f
m g r
r
M
O
limsup
r
ò ò
2
2 2
1
[( ) ]
1
2
log
(1)
|
| (1
)
(3.7)
log
h f
m
m r
r
O
m
a
limsup
from
r
ò ò
ò
2 2
2
2
2
1 2
1
1
log( )
(1)
2
log
h f
m
h r
f
r
O
m
m
limsup
r
m
ò ò
ò
ò
2
2
2 1
,
0
.
h f
m
since
is
arbitrary
m
ò
Using the same arguments as in Theorem 3.1we can show that
2 1 2 1
2
2
1 2 2
1
(
)
h.
f f g
f
m
h
h
g
m
Theorem 3.3Let
f
1,
f
2 andh
2 be three entire functions of respective positive orders andh
1 be meromorphic function of finite order1 h
andg
be a polynomial of degreem
.
Then the relative order ofh
1
h
2 with respect tof
1
f
2
g
satisfies the inequality2 2
1 2 1 2
2 2
1 2 1 2
1 1
( ) h ( ) h .
f f g f f g
f f
h h and h h
m m
Proof:For any
ò
0
and for all large values of r we get1 1
1
( )
1( )
.
f h
f h
T
r
r
òand T
r
r
ò (3.15)1 1
1 1
( )
ff
T
r
r
ò (3.16) for all large values of r.1
1
( )
.
h
h
T
r
r
ò (3.17) For three entire functionsf
1,
f
2 andh
2we use the Lemma 2.7we get1 2
( )
3
1(
2(
( ))).
f f g f f g
T
r
T
M
M
r
So for all large values of r we get
1 2 2 1
1 1 1 1
( )
(
(
( ))).
3
f f g g f f
r
T
r
M
M
T
(3.18) Again from Lemma 2.5we get for a sequence of values of r tending to infinity and
2
0
h1 2( ) 1(exp( ) ).
h h h
T r T r (3.19)
Let
g z
( )
a
0
a z
1
a z
2 2
a z
m m. Now1 2 1 2
1 2
1
1 2
log
(
( ))
(
)
li
m u
s
p
log
f f g h h f f g
r
T
T
r
h h
r
1 2
2 1
1 1 1
( )
log
(
(
(
)))
3
(3.18)
log
h h g f f r
T
r
M
M
T
limsup
from
r
1
2 1
1 1 1
(exp
)
log
(
(
(
)))
3
(3.19)
log
hg f f
r
T
r
M
M
T
limsup
from
r
1
2 1
2
1 1 1
log
(
(
({exp
}
)))
(3.17)
log
h
g f f
r
M
M
T
r
limsup
from
r
ò
1 1
2
1 2
1 1
log
(
([{exp
}
]
))
(3.16)
log
h f
g f
r
M
M
r
limsup
from
r
ò ò
1
1
2
2
1 1
log
(
([{exp
}]
))
log
h f
g f
r
M
M
r
limsup
r
ò ò
1
1
2
2
1
1
[2]log
(exp{
log ({exp
}
)})
(3.3)
log
h f
g
f r
M
r
limsup
from
r
ò ò
ò
1
2 1
1
1
2
log
(exp{
log(
)})
log
h g
f f
r
M
r
limsup
r
ò
ò
ò
1
2 2 1
1
1
2
log
(exp{
log
log
)})
log
h g
f f f
r
M
r
limsup
r
ò
2
1
log
(
)
(1)
log
f
g r
M
r
O
limsup
r
ò
2
log(
)
(1)
|
| (1
)
1
(3.7)
log
f
m r
r
O
a
limsup
from
m
r
ò
ò
2
1
f
m
ò
2
2
1
,
0
.
h f
since
is arbitrary
m
ò
We know from the definition of lower order for all large values of r
1
1
1 1 [2]
( ) exp{ log }. f
f
M r r
ò (3.20) Similarly, for all large values of r
2
2
1
1
[2]( )
exp{
log
}.
f
f
M
r
r
ò
(3.21)Also 1
1
( ,
)
h.
T r h
r
òfor all large values of r
Again by Lemma 2.2we get
1 2 1 2
1 1
( ) log ( ( ))
3 9 4
f f g f f g
r
T r M M
1 2
1 1 1
log ( ( ( ))) 3 f 9 f 10 g 8
r
M M M
1 2 2 1
1 1 1 1
. ,
f f g( )
8
g(10
f(9
f(exp 3 ))).
i e
T
r
M
M
M
r
(3.22) Also for a sequence of values ofr
tending to infinity we get from Lemma 2.81 2
( )
{1
0(1)}
1(exp( ) )
2.
h h h h
T
r
T
r
where
(3.23) Now1 2 1 2
1 2
1
1 2
log
(
( ))
(
)
sup
lo
im
g
l
f f g h hf f g
r
T
T
r
h
h
r
2 1 1 2
1 1 1
log(8
(10
(9
(exp(3
( ))))))
(3.22)
log
g f f h h
r
M
M
M
T
r
limsup
from
r
2 1 1
1 1 1
log(8
(10
(9
(exp(3{1 0(1)}
(exp
))))))
(3.23)
log
g f f h
r
M
M
M
T
r
limsup
from
r
2 1
1
1 1
1
[2]log(8
(10
(9 exp{
log (exp(3
(exp
)))})))
(1)
log
g f h
f r
M
M
T
r
O
limsup
r
2 1
1
1 1
1
log(8
(10
(9
exp{
log(3
(exp
))})))
(1)
log
g f h
f r
M
M
T
r
O
limsup
r
ò
1
2 1
1
1 1
log(8
(10
(9[3
(exp
)]
)))
(1)
log
f
g f h
r
M
M
T
r
O
limsup
r
ò
1 1
2
1
1 1
log(8
(10
(9[3(exp
)
]
)))
(1)
(3.15)
log
h f
g f
r
M
M
r
O
limsup
from
r
ò ò
1 1
2
1
1
1
[2]log(8
(10 exp{
log (9[3(exp
)
]
)}))
(1)
(3.21)
log
h f
g
f r
M
r
O
limsup
from
r
ò ò
ò
1
2 1
1 1 2
log(8 (10 exp{ log( )})) (1)
log
h g
f f
r
M r O
limsup
r
ò
ò ò
1
2 2 1
1
1
2
log(8
(10 exp{
log
log
}))
(1)
log
h g
f f f
r
M
r
O
limsup
r
ò
ò
ò
ò
2
1
log(8
(10 exp{log
}))
(1)
log
f
g r
M
r
O
limsup
r
ò
2
1
log(8
(10
))
(1)
log
f
g r
M
r
O
limsup
r
ò
2
1
log(
(10
))
(1)
log
f
g r
M
r
O
limsup
r
ò
2
10
log(
)
(1)
|
| (1
)
1
(3.7)
log
f
m r
r
O
a
limsup
from
m
r
ò
ò
2
(log10
log
log |
| (1
))
(1)
1
log
m f
r
r
a
O
limsup
m
r
ò
ò
2
1
f
m
ò
2
2
1
,
0
.
h f
since
is arbitrary
m
Theorem 3.4Let
f
1 andh
1be two meromorphic functions of finite non-zero orders such that1
0
f
and2
f
andh
2 be two entire functions of finite non-zero orders such that2 0
f
and
g
be a polynomial of degreem
.
Then the relative order of h1h2 with respect to f1 f2g satisfies the inequality
1 1
1 2 1 2
1 2
1 2 1 2
( ) h ( ) h .
f f g f f g
f f
h h and h h
m
Proof:Let 2
0 1 2
( ) m
m
g z a a za z a z be a polynomial of degree
m
.
We have from Lemma 2.51 2
( )
1(exp( ) ).
h h h
T
r
T
r
(3.24) Again for all large values of r we get from Lemma 2.71 2
( )
1(
2( )
f f g f f g
T
r
T
M
r
1( 2( ( ))).
f f g
T M M r
(3.25) Now by the definition of relative order we get
1 2
1 2
1 2
1 2
l
lim
og
( )
(
)
sup
log
( )
h h
f f g
r f f g
T
r
h
h
T
r
1
1 2
log
(exp
)
sup
(3.24)
log
(
lim
)
h
r f f g
T
r
from
T
r
1
1 2
log (exp )
sup (3.25)
log ( ( ( ) m
) li
) h
r f f g
T r
from
T M M r
1
1 2
log (exp )
sup (3.6)
log ( (| | (1 ) l
))
im h m
r f f m
T r
from
T M a r
ò
1
2
1
log (exp ) sup
log (exp[| | (1 ) ] li
) m
f h
m r
f m
T r
T a r
ò
ò
1
2 1
log
(exp
)
sup
(3.15)
(
)
lim
[|
| (1
)
]
fh
m r
f m
T
r
from
a
r
òò
ò
1
2
1
(
)
sup
(
)[|
| (1
)
lim
]
fh
m r
f m
r
a
r
òò
ò
ò
1
2
1
0
.
h
f f
when
m
and
is
arbitrary
ò
Also we have from Lemma 2.6
1 2
( )
1(
2( )).
h h h h
T
r
T
M
r
(3.26)
T
f1 f2 g( )
r
T
f2 g(exp
r
))
for
f2g
2
log
M
f g(exp
r
)
2
1
exp
log
(
(
))
8
4
f g
r
M
M
. (3.27)Now by the definition of relative order we get
1 2
1 2
1 2
1 2
l
lim
og
( )
(
)
sup
log
( )
h h f f g
r f f g
T
r
h
h
T
r
1 2
1 2
log
(
( ))
sup
(3.26)
l
lim
og
( )
h h
r f f g
T M
r
from
T
r
1 2
2
[2]
log
(
( ))
sup
(3.27)
1
exp
log (
( (
(
))))
8
4
lim
h hr
f g
T M
r
from
r
M
M
1 2
2
[2]
log
(
( ))
sup
(3.6)
1
exp
log (
( |
| (1
)[
] ))
4
l
)
8
im
)
h h
r m
f m
T M
r
from
r
M
a
ò
2
1
2
[2]
log
(exp{
})
sup
1
exp
log
li
(
( |
| (1
] ))
4
m
)[
8
h
h
r m
f m
T
r
r
M
a
òò
2
1
2
(
) log[exp{
}]
sup
1
exp
(
) log( |
| (1
)(
) )
8
lim
4
hh
r m
f m
r
r
a
ò
ò
ò
ò
2
1
2
(
){
}
sup
(
)
(1)
lim
h
h
r f
r
mr
O
ò
ò
ò
1
2 2
.
h
h f
when
m
Theorem 3.5Let
f
1 andh
1 be two meromorphic functions of finite non-zero order andf h
2,
2andg
be three entire functions such that2
0.
f
Then1 2
(
1 2)
.
h h
f
f
g
Proof:From the definition of relative order of meromorphic function with respect to another meromorphic function we have
1 2
1 2
1 2
1 2
log
( )
(
)
sup
.
lo
l
g
( )
im
f f gh h
r h h
T
r
f
f
g
T
r
( ) exp{ g }. g
M r r ò (3.28) Now
2
1 2
1 2
[2]
1 2
1
exp
log
(
(
))
8
4
(
)
sup
(3.27)
log
( )
li
m
f g
h h
r h h
r
M
M
f
f
g
from
T
r
2
1 2
[ 2]
1
exp
log
(
(
))
8
4
sup
(3.26)
log
(
( ))
lim
f g
r h h
r
M
M
from
T
M
r
2
1 2
[ 2]
1
exp
log
(
exp(
)
)
8
4
s
li
up
log
(
( ))
m
g
f
r h h
r
M
T
M
r
ò
2
2
1
[ 2]
1
exp
log
(
exp(
)
)
8
4
sup
log
(exp(
)
lim
)
gh
f
r
h
r
M
T
r
ò ò
2
2
1
exp
(
)(
)
4
sup
(
im
)(
)
l
g
h
f
r
h
r
r
ò ò
ò
ò
Thus1 2
(
1 2)
.
h h
f
f
g
This completes the proof.
REFERNCES
[1] Banerjee. D, A note on relative order of meromorphic functions, Bull.Cal.Math.Soc., 98, 1(2006), 25-30. [2] Bergweiler. W, On the Nevanlinna characteristic of composite functions, Complex Variables, 1(1988), 225-236. [3] Bergweiler. W, On the growth rate of composite meromorphic functions, Complex Var. Elliptic Equ., 14(1990), 187-196. [4] Bernal. L, Orden relativo de crecimiento de funcionesenteras, Collectanea Mathematica, Vol 39(1988), 209-229.
[5] Clunie. J, The composition of entire and meromorphic functions, Mathematical essays dedicated to A. J. Macintyre, Ohio University Press (1970), 75-92.
[6] Hayman. W. K., Meromorphic functions, The Clarendon Press, Oxford(1964).
[7] Lahiri. B. K. and Banerjee. D., Relative order of an entire and meromorphic functions, Proc. Nat. Acad. Sci. A., 69(A), 3(1999), 339-354.
[8] Lahiri. I. and Sharma. D. K., Growth of composite entire and meromorphic functions, Indian J. Pure Appl. Math, 26(5)(1995), 451-458.
[9] Mondal. M., Growth properties of composition of two meromorphic functions, I.J.M.T.T, 57(2018),31-48. [10] Niino. K. and Smita. N., Growth of a composite function of entire functions, KodaiMath.J., 3(1980), 374-379.
[11] Niino. K. and Yang. C. C.,Some growth relationship on factors of two compositive entire func-tions, in: Factorization Theory of Meromorphic Functions and Related Topics, Marcel Dekker Inc., New York/Basel,(1982), 95-99.
