MAXIMUM TERMS OF COMPOSITE ENTIRE FUNCTIONS AND THEIR
COMPARATIVE GROWTH RATES
SANJIB KUMAR DATTA
1*, ARUP RATAN DAS
2AND SAMTEN TAMANG
31&3
Department of Mathematics, University of Kalyani,
Kalyani, Nadia, PIN – 741235, West Bengal, India.
2
Pathardanga Osmania High Madrasah, P.O. – Panchgram,
Dist.- Murshidabad, PIN – 742184, West Bengal, India.
(Received on: 20-09-12; Revised & Accepted on: 21-12-12)
ABSTRACT
I
n the paper we investigate some comparative growth properties related to the maximum terms of composite entire functions.AMS Subject Classification (2000): 30D30, 30D35.
Keywords and phrases: Entire function, maximum term, composition, growth, order, lower order, hyper order, hyper
lower order, zero order, hyper zero order, hyper zero lower order.
1. INTRODUCTION, DEFINITIONS AND NOTATIONS.
Let f be an entire function defined in the open complex plane ℂ. The maximum term
µ
( )
r
,
f
of∑
∞
=
=
0
n n n
z
a
f
onr
z
=
is defined by( )
(
n)
n n
a
r
f
r
0
max
,
≥
=
µ
.To start our paper we just recall the following definitions:
Definition 1. The order
ρ
f and lower orderλ
f of an entire function f are defined as[ ]
( )
[ ]( )
,
log
,
log
inf
lim
and
log
,
log
sup
lim
2 2
r
f
r
M
r
f
r
M
r f r
f →∞
∞
→
=
=
λ
ρ
where
log
[ ]kx
=
log
(
log
[ ]k−1x
)
for k=1,2,3,… andlog
[ ]0x
=
x
.Definition 2. The hyper order
ρ
f hyper lower orderλ
f of an entire function f are defined as follows[ ]
( )
[ ]( )
.
log
,
log
inf
lim
and
log
,
log
sup
lim
3 3
r
f
r
M
r
f
r
M
r f r
f →∞
∞
→
=
=
λ
ρ
Since for
0
,
( )
,
( )
,
(
R
,
f
)
{
cf
.
[ ]
4
}
r
R
R
f
r
M
f
r
R
r
µ
µ
−
≤
≤
<
≤
it is easy to see that
[ ]
( )
[ ]( )
r
f
r
r
f
r
r f r
f
log
,
log
inf
lim
and
log
,
log
sup
lim
2
2
µ
λ
µ
ρ
∞ → ∞
→
=
=
and
[ ]
( )
[ ]( )
r
f
r
r
f
r
r f r
f
log
,
log
inf
lim
and
log
,
log
sup
lim
3
3
µ
λ
µ
ρ
∞ → ∞
→
=
=
.Definition 3. ([3]) Let f be an entire function of order zero. Then the quantities
ρ
*
f
,
λ
*
f
andρ
*
f
,
λ
*
f
are defined in the following way :
[ ]
( )
[ ][ ]
( )
[ ]r
r
f
M
r
f
r
M
r f r
f
2 2 *
2 2 *
log
,
log
inf
lim
,
log
,
log
sup
lim
∞ → ∞
→
=
=
λ
ρ
and
[ ]
( )
[ ][ ]
( )
[ ]
.
log
,
log
inf
lim
,
log
,
log
sup
lim
2 3 *
2 3 *
r
f
r
M
r
f
r
M
r f r
f
∞ → ∞
→
=
=
λ
ρ
Definition 4. The type
f
σ
of an entire function f is defined as( )
.
0
,
,
log
sup
lim
<
<
∞
=
∞
→ f
r
r
ff
r
M
f
ρ
σ
ρIn the paper we would like to establish some new results based on the comparative growth properties of maximum terms of composite entire functions. We do not explain the standard notations and definitions in the theory of entire functions as those are available in [5].
2. LEMMAS.
In this section we present some lemmas which will be needed in the sequel.
Lemma 1. ([1]) Let f and g be any two entire functions. Then for all sufficiently large values of r,
( )
0
,
(
,
)
(
( )
,
,
)
.
,
2
8
1
f
g
r
M
M
g
f
r
M
f
g
g
r
M
M
≤
≤
−
Lemma 2. ([2]) Let
f
be an entire function of finite lower order. If there exist entire functions(
i
=
n
n
≤
∞
)
i
a
1
,
2
,...,
;
satisfying
( )
o
{
T
( )
r
f
}
i
a
r
T
,
=
,
and( )
,
1
1
;
∑
=
=
n
i
f
i
a
δ
the( )
( )
,
1
.
log
,
lim
π
=
∞→
M
r
f
f
r
T
r3. THEOREMS.
In this section we present the main results of the paper.
Theorem 1. Let
f
,
g
andk
be any three entire functions with( )
i
0
< < < ∞
λ ρ
k k,
( )
ii
0
< < ∞
ρ
g,
( )
iii
λ
*f<
ρ
*f< ∞
and
( )
iv
0
<
σ
g< ∞
.
Also let there exist entire functionsa
i(
i
=
1
,
2
,...,
n
;
n
≤
∞
)
satisfying( )
o
{
T
( )
r
g
}
i
a
r
T
,
=
,
and∑
( )
=
=
n
i
g
i
a
1
.
1
;
δ
Then[ ]
(
)
[ ]
(
( )
)
.
.
,
exp
log
,
log
sup
lim
.
4
1
* 2
2
*
g f
k g r
g f
k g
g
f
r
g
k
r
g g
λ
λ
ρ
πσ
µ
µ
ρ
ρ
λ
πσ
ρ ρ
≤
≤
∞
→
Proof. Putting
R
=
2
r
in the inequality( )
,
( )
,
(
R
,
f
)
{
cf
.
[ ]
4
}
r
R
R
f
r
M
f
r
µ
µ
−
≤
≤
we get that
( )
r
,
f
M
( )
r
,
f
2
µ
(
2
r
,
f
)
.
µ
≤
≤
Now in view of the first part of Lemma 1 and the inequality
M
( )
r
,
f
≤
2
µ
(
2
r
,
f
)
, we obtain for all sufficiently large values of r,[ ]
(
)
[ ]
≥
M
r
k
g
g
k
r
,
2
2
1
log
,
log
2µ
2[ ]
,
,
( )
1
4
16
1
log
2M
M
r
g
k
+
O
≥
(
)
,
.
4
log
−
≥
λ
kε
M
r
g
(1)Again in view of the second part of Lemma 1 and for all sufficiently large values of
r
,[ ]
(
( )
)
[ ](
( )
)
g
f
r
M
g
f
r
g
g
,
exp
log
,
exp
log
2µ
ρ≤
2 ρ
≤
log
[ ]2M
(
M
(
exp
( )
r
ρg,
g
)
,
f
)
(
*f)
.
(
g)
.
r
g.
ρ
ε
ρ
ε
ρ
+
+
≤
(2)Now using Lemma 2 and from (1) and (2) we get for all sufficiently large values of
r
,[ ]
(
)
[ ]
(
( )
)
(
)
(
)
(
)
gg
r
g
r
M
g
f
r
g
k
r
g f
k
ρ
ρ
ρ
ε
ρ
ε
ε
λ
µ
µ
.
.
,
4
log
,
exp
log
,
log
* 2
2
+
+
−
≥
(
)
(
)
(
)
g
g
r
g
r
T
g
r
T
g
r
M
g f
k
ρ ρ
ε
ρ
ε
ρ
ε
λ
+
+
−
≥
4
1
.
4
,
4
.
,
4
,
4
log
.
.
* .
Since
ε
( )
>
0
is arbitrary,[ ]
(
)
[ ]
(
( )
)
g f
k g r
g
g
g
f
r
g
k
r
ρ
ρ
λ
πσ
µ
µ
ρρ
.
4
1
,
exp
log
,
log
sup
lim
2 *2
≥
∞
→
. (3)Again by the second part of Lemma 1 and the inequality
µ
( )
r
,
f
≤
M
( )
r
,
f
, we get for all sufficiently large values ofr,
[ ]
(
)
[ ](
)
g
k
r
M
g
k
r
,
log
,
log
2µ
≤
2
≤
log
[ ]2M
(
M
( )
r
,
g
,
k
)
Also in view of the first part of Lemma 1 and for all sufficiently large values of
r
we obtain that[ ]
(
( )
)
[ ]( )
≥
M
r
f
g
g
f
r
g g
,
2
exp
2
1
log
,
exp
log
2 2ρ ρ
µ
[ ]
( )
≥
M
M
r
g
f
g
,
,
4
exp
16
1
log
2ρ
(
)
[ ]( )
−
≥
M
r
g
g
f
,
4
exp
log
2*
ρ
ε
λ
(
f)
(
g)
r
gρ
ε
λ
ε
λ
*−
−
.
≥
. (5)Now from (4) and (5) we get for all sufficiently large values of
r
,[ ]
(
)
[ ]
(
( )
)
(
)
( )
(
)
(
)
gg
r
g
r
M
g
f
r
g
k
r
g f
k
ρ
ρ
λ
ε
λ
ε
ε
ρ
µ
µ
.
,
log
,
exp
log
,
log
* 2
2
−
−
+
≤
.
As
ε
( )
>
0
is arbitrary,[ ]
(
)
[ ]
(
( )
)
g f
g k
r
r
f
g
g
k
r
g
λ
λ
σ
π
ρ
µ
µ
ρ
.
.
.
,
exp
log
,
log
sup
lim
2 *2
≤
∞→
. (6)
Thus the theorem follows from (3) and (6).
Theorem 2. Let
f
,
g
andk
be any three entire functions with( )
i
0
< < < ∞
λ ρ
k k,
( )
ii
0
<
ρ
g< ∞
,
and( )
* *f f
iii
λ
<
ρ
< ∞
( )
iv
0
<
σ
g< ∞
.
Also let there exist entire functionsa
i(
i
=
1
,
2
,...,
n
;
n
≤
∞
)
satisfying( )
o
{
T
( )
r
g
}
i
a
r
T
,
=
,
and∑
( )
=
=
n
i
g
i
a
1
.
1
;
δ
Then[ ]
(
)
[ ]
(
( )
)
.
.
,
exp
log
,
log
sup
lim
.
4
1
* 2
2
*
k f
k g r
k f
k g
k
f
r
g
k
r
g g
λ
λ
ρ
πσ
µ
µ
ρ
ρ
λ
πσ
ρ ρ
≤
≤
∞
→
Proof. In view of the following inequality and putting
R
=
2
r
( )
,
( )
,
(
R
,
f
)
{
cf
.
[ ]
4
}
r
R
R
f
r
M
f
r
µ
µ
−
≤
≤
we get that
( )
r
,
f
M
( )
r
,
f
2
µ
(
2
r
,
f
)
.
µ
≤
≤
Now in view of the first part of Lemma 1 and the inequality
M
( )
r
,
f
≤
2
µ
(
2
r
,
f
)
, we obtain for all sufficiently large values of r,[ ]
(
)
[ ]
≥
M
r
k
g
g
k
r
,
2
2
1
log
,
log
2µ
2[ ]
( )
1
,
,
4
16
1
log
2M
M
r
g
k
+
O
(
)
,
.
4
log
−
≥
λ
kε
M
r
g
(7)Again in view of the second part of Lemma 1 and for all sufficiently large values of
r
,[ ]
(
( )
)
[ ](
( )
)
k
f
r
M
k
f
r
g
g
,
exp
log
,
exp
log
2µ
ρ≤
2 ρ
≤
log
[ ]2M
(
M
(
exp
( )
r
ρg,
k
)
,
f
)
(
*)
.
(
)
.
r
g.
k f
ρ
ε
ρ
ε
ρ
+
+
≤
(8)Now using Lemma 2 and from (7) and (8) we get for all sufficiently large values of
r
,[ ]
(
)
[ ]
(
( )
)
(
)
(
)
(
)
gg
r
g
r
M
k
f
r
g
k
r
k f
k
ρ
ρ
ρ
ε
ρ
ε
ε
λ
µ
µ
.
.
,
4
log
,
exp
log
,
log
* 2
2
+
+
−
≥
(
)
(
)
(
)
g
g
r
g
r
T
g
r
T
g
r
M
k f
k
ρ ρ
ε
ρ
ε
ρ
ε
λ
+
+
−
≥
4
1
.
4
,
4
.
,
4
,
4
log
.
.
* .
Since
ε
( )
>
0
is arbitrary,[ ]
(
)
[ ]
(
( )
)
k f
k g r
g
g
k
f
r
g
k
r
ρ
ρ
λ
πσ
µ
µ
ρρ
.
4
1
,
exp
log
,
log
sup
lim
2 *2
≥
∞→
. (9)Again by the second part of Lemma 1 and the inequality
µ
( )
r
,
f
≤
M
( )
r
,
f
, we get for all sufficiently large values ofr,
[ ]
(
)
[ ](
)
g
k
r
M
g
k
r
,
log
,
log
2µ
≤
2
≤
log
[ ]2M
(
M
( )
r
,
g
,
k
)
≤
(
ρ
k+
ε
)
log
M
( )
r
,
g
. (10)Also in view of the first part of Lemma 1 and for all sufficiently large values of
r
we obtain that[ ]
(
( )
)
[ ]( )
≥
M
r
f
k
k
f
r
g g
,
2
exp
2
1
log
,
exp
log
2 2ρ ρ
µ
[ ]
( )
≥
M
M
r
k
f
g
,
,
4
exp
16
1
log
2ρ
(
)
[ ]( )
−
≥
M
r
k
g
f
,
4
exp
log
2*
ρ
ε
λ
≥
(
λ
*f−
ε
)
⋅
(
λ
k−
ε
)
⋅
r
ρg. (11)[ ]
(
)
[ ]
(
( )
)
(
)
( )
(
)
(
)
gg
r
g
r
M
k
f
r
g
k
r
k f
k
ρ
ρ
λ
ε
λ
ε
ε
ρ
µ
µ
.
,
log
,
exp
log
,
log
* 2
2
−
−
+
≤
.
As
ε
( )
>
0
is arbitrary,[ ]
(
)
[ ]
(
( )
)
k f
g k
r
r
f
k
g
k
r
g
λ
λ
σ
π
ρ
µ
µ
ρ
.
.
.
,
exp
log
,
log
sup
lim
2 *2
≤
∞→
. (12)
Thus the theorem follows from (9) and (12).
Theorem 3. Let
f
,
g
andk
be any three entire functions with( )
i
0
<
λ
k<
ρ
k< ∞
,
( )
ii
0
<
λ
g<
ρ
g< ∞
,
( )
* *f f
iii
λ
<
ρ
< ∞
and( )
<
<
∞
g
iv
0
σ
.Also let there exist entire functionsa
i(
i
=
1
,
2
,...,
n
;
n
≤
∞
)
satisfying
( )
o
{
T
( )
r
g
}
i
a
r
T
,
=
,
and∑
( )
=
=
n
i
g
i
a
1
.
1
;
δ
Then[ ]
(
)
[ ]
(
( )
)
.
.
,
exp
log
,
log
sup
lim
.
4
1
* 3
3
*
g f
k g r
g f
k g
g
f
r
g
k
r
g g
λ
λ
ρ
πσ
µ
µ
ρ
ρ
λ
πσ
ρ ρ
≤
≤
∞
→
Proof. Considering the following inequality and taking
R
=
2
r
( )
,
( )
,
(
R
,
f
)
{
cf
.
[ ]
4
}
r
R
R
f
r
M
f
r
µ
µ
−
≤
≤
we obtain that
( )
r
,
f
M
( )
r
,
f
2
µ
(
2
r
,
f
)
.
µ
≤
≤
Now in view of the first part of Lemma 1 and the inequality
M
( )
r
,
f
≤
2
µ
(
2
r
,
f
)
, we obtain for all sufficiently large values of r,[ ]
(
)
[ ]
≥
M
r
k
g
g
k
r
,
2
2
1
log
,
log
3µ
3[ ]
,
,
( )
1
4
16
1
log
3O
k
g
r
M
M
+
≥
(
)
,
.
4
log
−
≥
λ
kε
M
r
g
(13)Again in view of the second part of Lemma 1 and for all sufficiently large values of
r
,[ ]
(
( )
)
[ ](
( )
)
g
f
r
M
g
f
r
g
g
,
exp
log
,
exp
log
3µ
ρ≤
3 ρ
≤
log
[ ]3M
(
M
(
exp
( )
r
ρg,
g
)
,
f
)
(
)
.
(
)
.
.
*
g
r
gf
ρ
ε
ρ
ε
ρ
+
+
≤
(14)Now using Lemma 2 and from (13) and (14) we get for all sufficiently large values of
r
,[ ]
(
)
[ ]
(
( )
)
(
)
(
)
(
)
gg
r
g
r
M
g
f
r
g
k
r
kρ ρ
ε
ρ
ε
ρ
ε
λ
µ
µ
.
.
,
4
log
,
exp
log
,
log
* 3
3
+
+
−
≥
(
)
(
)
(
)
g
g
r
g
T
g
r
T
g
M
g f
k
ρ
ρ
ε
ρ
ε
ρ
ε
λ
+
+
−
≥
4
1
.
4
,
4
.
,
4
,
4
log
.
.
*
.
Since
ε
( )
>
0
is arbitrary,[ ]
(
)
[ ]
(
( )
)
g f
k g r
g
g
g
f
r
g
k
r
ρ
ρ
λ
πσ
µ
µ
ρρ
.
4
1
,
exp
log
,
log
sup
lim
3 *3
≥
∞
→
. (15)Again by the second part of Lemma 1 and the inequality
µ
( )
r
,
f
≤
M
( )
r
,
f
, we get for all sufficiently large values ofr,
[ ]
(
)
[ ](
)
g
k
r
M
g
k
r
,
log
,
log
3µ
≤
3
≤
log
[ ]3M
(
M
( )
r
,
g
,
k
)
≤
(
ρ
k+
ε
)
log
M
( )
r
,
g
. (16)Also in view of the first part of Lemma 1 and for all sufficiently large values of
r
we obtain that[ ]
(
( )
)
[ ]( )
≥
M
r
f
g
g
f
r
g g
,
2
exp
2
1
log
,
exp
log
3 3ρ ρ
µ
[ ]
( )
≥
M
M
r
g
f
g
,
,
4
exp
16
1
log
3ρ
(
)
[ ]( )
−
≥
M
r
g
g
f
,
4
exp
log
2*
ρ
ε
λ
≥
(
λ
*f−
ε
)
⋅
(
λ
g−
ε
)
⋅
r
ρg. (17)Now from (16) and (17) we get for all sufficiently large values of
r
,[ ]
(
)
[ ]
(
( )
)
(
)
( )
(
)
(
)
gg
r
g
r
M
g
f
r
g
k
r
g f
k
ρ ρ
ε
λ
ε
λ
ε
ρ
µ
µ
.
,
log
,
exp
log
,
log
* 3
3
−
−
+
≤
.
As
ε
( )
>
0
is arbitrary,[ ]
(
)
[ ]
(
( )
)
g f
g k
r
r
f
g
g
k
r
g
λ
λ
σ
π
ρ
µ
µ
ρ
.
.
.
,
exp
log
,
log
sup
lim
3 *3
≤
∞
→
. (18)
Thus the theorem follows from (15) and (18).
Theorem 4. Let
f
,
g
andk
be any three entire functions with( )
i
0
< < < ∞
λ ρ
k k,
( )
ii
0
< < < ∞
λ ρ
k k,
( )
iii
0
< < ∞
ρ
g,
( )
* *0
f fiv
<
λ
<
ρ
< ∞
and( )
v
0
<
σ
g<
∞
. Also let there exist entire functionsa
i(
i
=
1
,
2
,...,
n
;
n
≤
∞
)
satisfying
( )
o
{
T
( )
r
g
}
i
a
r
T
,
=
,
and∑
( )
=
=
n
i
g
i
a
1
.
1
;
[ ]
(
)
[ ]
(
( )
)
.
.
,
exp
log
,
log
sup
lim
.
4
1
* 3
3
*
k f
k g r
k f
k g
k
f
r
g
k
r
g g
λ
λ
ρ
πσ
µ
µ
ρ
ρ
λ
πσ
ρ ρ
≤
≤
∞
→
Proof. Putting
R
=
2
r
in the inequality( )
,
( )
,
(
R
,
f
)
{
cf
.
[ ]
4
}
r
R
R
f
r
M
f
r
µ
µ
−
≤
≤
We obtain that
( )
r
,
f
M
( )
r
,
f
2
µ
(
2
r
,
f
)
.
µ
≤
≤
Now in view of the first part of Lemma 1 and the inequality
M
( )
r
,
f
≤
2
µ
(
2
r
,
f
)
, we obtain for all sufficiently large values of
r
,[ ]
(
)
[ ]
≥
M
r
k
g
g
k
r
,
2
2
1
log
,
log
3µ
3[ ]
( )
1
,
,
4
16
1
log
3M
M
r
g
k
+
O
≥
(
)
,
.
4
log
−
≥
λ
kε
M
r
g
(19)Again in view of the second part of Lemma 1 and for all sufficiently large values of
r
,[ ]
(
( )
)
[ ](
( )
)
k
f
r
M
k
f
r
g g
log
exp
,
,
exp
log
3µ
ρ≤
3 ρ
≤
log
[ ]3M
(
M
(
exp
( )
r
ρg,
k
)
,
f
)
(
)
.
(
)
.
.
*
g
r
k f
ρ
ε
ρ
ε
ρ
+
+
≤
(20)Now using Lemma 2 and from (19) and (20) we get for all sufficiently large values of
r
,[ ]
(
)
[ ]
(
( )
)
(
)
(
)
(
)
gg
r
g
r
M
k
f
r
g
k
r
k f
k
ρ ρ
ε
ρ
ε
ρ
ε
λ
µ
µ
.
.
,
4
log
,
exp
log
,
log
* 3
3
+
+
−
≥
(
(
)
)
(
)
g
g
r
g
r
T
g
r
T
g
r
M
k f
k
ρ ρ
ε
ρ
ε
ρ
ε
λ
+
+
−
≥
4
1
.
4
,
4
.
,
4
,
4
log
.
.
* .
Since
ε
( )
>
0
is arbitrary,[ ]
(
)
[ ]
(
( )
)
k f
k g r
g
g
k
f
r
g
k
r
ρ
ρ
λ
πσ
µ
µ
ρρ
.
4
1
,
exp
log
,
log
sup
lim
3 *3
≥
∞
→
. (21)Again by the second part of Lemma 1 and the inequality
µ
( )
r
,
f
≤
M
( )
r
,
f
, we get for all sufficiently large values ofr,
[ ]
(
)
[ ](
)
g
k
r
M
g
k
r
,
log
,
log
3µ
≤
3
≤
log
[ ]3M
(
M
( )
r
,
g
,
k
)
Also in view of the first part of Lemma 1 and for all sufficiently large values of
r
we obtain that[ ]
(
( )
)
[ ]( )
≥
M
r
f
k
k
f
r
g g
,
2
exp
2
1
log
,
exp
log
3 3ρ ρ
µ
[ ]
( )
≥
M
M
r
k
f
g
,
,
4
exp
16
1
log
3ρ
(
)
[ ]( )
−
≥
M
r
k
g
f
,
4
exp
log
2* ρ
ε
λ
≥
(
λ
*f−
ε
)
⋅
(
λ
k−
ε
)
⋅
r
ρg. (23)Now from (22) and (23) we get for all sufficiently large values of
r
,[ ]
(
)
[ ]
(
( )
g)
(
(
)
)
(
( )
)
gr
g
r
M
k
f
r
g
k
r
k f
k
ρ ρ
ε
λ
ε
λ
ε
ρ
µ
µ
.
,
log
,
exp
log
,
log
* 3
3
−
−
+
≤
.
As
ε
( )
>
0
is arbitrary,[ ]
(
)
[ ]
(
( )
)
k f
g k
r
r
f
k
g
k
r
g
λ
λ
σ
π
ρ
µ
µ
ρ
.
.
.
,
exp
log
,
log
sup
lim
3 *3
≤
∞
→
. (24)
Thus the theorem follows from (21) and (24).
REFERENCES
[1] Clunie, J.: The composition of entire and meromorphic functions, Mathematical essays dedicated to A.J. Machintyre, Ohio University Press, 1970, pp. 75-92.
[2] Lin, Q. and Dai, C.: On a conjecture of Shah concerning small functions, Kexue Tong bao (English Ed.), Vol. 31 (1986), No. 4, pp. 220-224.
[3] Liao, L. and Yang, C.C.: On the growth of composite entire functions, Yokohama Math. J., Vol. 46 (1999), pp. 97-107.
[4] Singh, A.P. and Baloria, M.S.: On maximum modulus and maximum term of composition of entire functions,
Indian J. Pure Appl. Math., Vol. 22 (1991), No. 12, pp. 1019-1026.
[5] Valiron, G.: Lectures on the general theory of integral functions, Chelsea Publishing Company, 1949.
