Internat. J. Math. & Math. Sci.
Vol. 8 No. 4 (1985) 635-640
635
ABOUT THE DEFECTS OF CURVES HOLOMORPHIC
IN THE HALF PLANE
MOQBUL HOSSAIN
Department of Mathematics
Jahangirnagar University
Savar, Dhaka, Bangladesh
(Received March 14, 1983)
ABSTRACT.
A study is made on the defects of curves holomorphic in the half plane.
Several results are proved.
WORDS AND PHRAS,’S.
/EY
|.
Menomorphzc function, entire functions.
AMS SUBJECT CLtSSIFICATION CODE.
/90
INTRODUCTION.
Let
f(z)
be a meromorphic function in the half plane
not equal to a constant identically.
of
32A70.
f(z)
N(r,f)
f
r
n(t, f) dt
where
pge
oe
arcsin
m(r,a)
T(r,f)
m(r,f) + N(r,f)
{Izl
f(z)
where
r
< r <
=.
We put
[1,2,3]
_I)
r
in+l f(re iO sin0)l
O}
I, Imz
dO
rsin20
does not contain neither zeros nor poles of
a, we get the following equality [I]:
N(r, {
arcsin
and f(z) is
1, the number of poles
f(z).
r
m(r,a,f)
When the half circle
n
r
f
arcsin
2--# I arcsln--r
O}
N(r a)
m(r,f)
the function
I}
(sog0g
i g eD(r,l)
are the poles of the function
_I_)
l](r, f-a
[z[
<
in
I
t
n(r,f),r
We denote by
z:[z-
lying inside the set
{Imz
in
-)
N(r,f)
If( reie sin0)
a
dO
+
$(r,l,f
a)
(1.1)
rsin20
(r,l,f
n
a)
f
arcsinr
arcsin-r
{inlf(eiOsinO) al
sine)}
r2
dO
636
M. HOSSAIN
0T(f)
The order
0T(f)
Similarly the lower order
in the sense of TSUJI is called the number
f(z)
of the funcitou
inT(r, f)
lira
kT(f)
f(z)
of
AT(f)
inr
is defined by the quantity
In T(r, f)
li--2m
lnr
r-m
The quantity
lira m_(
r-o T(r,f)
6T(a)
T(a,f)
f(z)
is called the deficiency of
6T(a)
which
of order
{a:
ET(f)
li---
N(r’a)-_
r
T(r,f)
at the point
and lower order
O}
T(a,f)
a
The value
{Imz
in the half z-plane
%T
ET(f)
The set
of
f(z)
for
>/
0}
we put
is called the set of deficient values of
f(z) in the sense of TSUJI.
2.
a
For a meromorphic function of
f(z)
is called a deficiency of
0
0T
1-
-
RESULTS ANALOCOUS TO TSUJI’S WORK.
We will derive analogue of TSUJI’S work for curves holomorphic in the half plane.
p
Let C
be a p-dimensional complex unitary space and
vector
(z)
gp(Z)}
{gl (z), g2(z)
is called p-dimensional entire curve of the complex parameter
p
fgn (z) n=l
vectors from
z
C
p.
The
where the function
are linearly independent entire functions without common zeroes.
The
scalar product:
((z).a)
P
I
k=l
gk(z)a k
is called an ordlnary entire function of the parameter
>. p)
The finite
Definition:
z.
or infinite system
A
of vectors is called ad-
missible, if any p-vectors of the system are linearly independent.
For
T*
teristics of
and
The quantity
tion
In this case the charac-
2, we shall get ordinary meromorphic functions.
n
((z).a)
coincide.
n*(r,a,)
in the region
ir
Iz
{z:
system of vectors.
We put
N*(r,,)
,,
We define the function
m* (r
{r n*(t,a,)
m
-
will denote the number of zeroes of the holomorphic func-
-
(r,a,)
dt
and
arcsinr
arcsln-
r
, Izl
r
<
< r <
T*(r,)
in+
> I},
where
aA,
the admissible
=.
by the quantities:
I I( reieie sine)l l.I [%11
((re
sin
) )l
de
rsin2O
DEFECTS OF CURVES HOLOMORPHIC IN THE HALF PLANE
arcsin
l-*(r,’)
2--
r
f
arcsin
de
(re i@ sinS)ll
II
In
637
rsin2e
r
where
l(z)ll
:/i=l l(z)l
For all admissable system
A
of vectors we get the equality:
m* (r,,,)
T*(r,G)
when
p
C
N* (r,,G)
+
+ O(I),
r
The order
and the lower order
OT
of the entire curve
%T
G(z)
may be defined
as before.
For a meromorphic function in the half plane the following assertions are true:
THEOREI A. [2]
Let
0
M
and
0
be an arbitrary, not more than countable,
set of points in the extended complex plane.
[2]
condition
{Im z
where
>
O}
0
min
THEORY! C. [2]
the half plane
THEOREM D
k}
ET(f)
of dificient
be an arbitrary sequence of different complex
such that
%
hum-
[%] + I)
(q
dT(ak ,f)
>
Kl(O,el)n.3
T(ak,f)
>
K1(O,el)n
36(0+1)’
For any
O}
{Im z
converges for each
for which the set
T
there exists a meromorphic function in the half plane
,l
of lower order
KI(O’OI
0
Then there exists a meromorphic function
be a sequence of positive number which satisfies the
{a
and let
For any
ber.
{n.o}
Let
=!Z n
of order
M.
values coincides with
THEOREM B.
O}
(Im z
in the half plane
for
p
18(+I) sinO- e
0 <
%
g
for
there exists a meromorphic function in
of finite lower order
%
such that the series
Z
(ak,f)_
a <
[3] Let
{w
be an arbitrary sequence of complex number and
be a sequence of positive number
(N -< oo)
the relationship
satisfying
Then tnere exists a meromorphic function in the half plane
{Im z
>
O}
Z
=I
{}
I
<
of finite order
such that
0
6T(W
f)
v’
w
e{w
and
T
(w f)
O
.{v
if
MAIN RESULTS.
In this article we shall consider the following assertions:
We denote by A(W) the fixed admissible system A of vectors, if the coordinates
of the vector
of the system depend on a complex parameter
the vectors of the admissible system
THEOREM i.
Let
f(z)
A(w)
of vectors by
w
We shall denote the
a(w)
be an arbitrary meromorphic function in the half plane.
It
M. HOSSAIN
638
T* (r,G)
ST((w),)
Let
COROLLARY 1.
l)(r,f)+
(p-
0
and
0T
2)
(z)
curve
be an arbitrary, not more than countable,
M
Then there exists a p-dimensional
ET(f)
For any
COROLLARY 2.
G(z)
curve
q
0
{Im z >
A(w
a(wk)
k
KI(0,I)
T(ak,G)
KI(0,OI)
K i(p,Bl
rain
%
holomorphic in the half plane
A(w)
for
there exists a p-dimensional curve
2
p
{Ira z > 0}
I,
%
t:)3
18+I) sinP- 18
p3
36;+I)
<
0
for
and an ad-
of finite lower order
of vectors such that the series
6
/l
aeA(w)
a<
and a ad-
of lower order
,2
k
T(ak,C)
For any
COROLLARY 3.
missible system
there exists a p-dimensional
[%] + I)
where
G%(z)
whose
0T
of vectors containing the sequence of distinct vectors
A(w)
a
such that
p > 2
and
%
0
%
of order
M
coincides with
holoorphc in the half plane
missible syste
{Im z > 0}
holomorphic in the half plane
set of deficient values
0(I)
AT((w),,) AT(W,f)
T(w,f),
set of vectors of the admissible system of vectors
(p
and an admissible
0(I)
l)(r, w, f) +
(r,a(w ,G)= (p-
m
2)
(r > I)
of vectors such that
A(w)
system
>
G(z) (p
is possible to find a p-dimensional holomorphic curve
Te (,G%)
diverges for
3
Let
COROLLARY 4
be a sequence of positive number
for any
%
0
p
%
in the half plane
{Ira z >.
PROOF OF THEOREM 1.
Let
{8
be an arbitrary sequence of complex number and
{w
> 2,
0}
(N < =)
there exists a p-dimensional curve
such that
f(z)
gl (z)
N
r. 6
=1
satisfying the condition
r((w),) 6
6T((w) .) 0
G%(z)
}N
<
Then
holomorphic
and
if w
{w
(2.])
g2(z)
be a meromorphic function in the half plane
{Im z >..0
where
g1(z)
and
g2(z)
are
entire functions having no zeroes in common.
We note that the entire functions:
k(z)
h
glP-k-l(z)
k
g2(z)
......
k
0,
are linearly independent in the field of complex number [5]
P-
(2.2)
DEFECTS OF CURVES HOLOMORPHIC IN THE HALF PLANE
639
We consider the curve
(z)
{h (z)
{Im z
holomorphic in the half plane
where for any complex
minant
O}
{a(w)}
A
then for the deter-
of the system we have
p(p-l)
p-i
C[-I-
A(w)
is an admissible system of vectors.
(z)
(w
ik
k=
the curve
A(w)
(-1) p-ln
k
p-different vectors from the system
(I)--2
Thus
A, A
and the system
(_l)k.p-l-"k
(1
{a(wk)} -_l_l
A
>
(2.3)
w
a(w)
If
h p-I (z)}
l(Z)
0
0
i)
We shall investigate the defects of
{Ira z
holomorphic in the half plane
w
k
> O}
for the vectors
a(w)
A(w).
contained in this admissible system
We have
(,(z)
p-I
a(w))
l
k=O
p-k-I
gl
For
z=re
iO
sln
C-I g2
k
(z) wk
]gl (z)12 (p-k- )I g2 (z)
k=O
k=0
k
g2(z)w)P-I g-I (z)(f(z)
(gl(z)
[gl(z>[P W
(z) (-I)
)p-1
2k
g2(z)IP-J/Pz
If(z)l 2k
f(z)l 2 (p-k-l)
2 (p-k-l)
k 0 If(z)[
{f(reiOsin 8) w{P-I
((reiOsin O)(w))
(2.5)
vk=O
0, we get
[(reiSsin 0)11
(2.4)
>
max(lf(.z){ p-I, I) (2.6)
f (rei8 sinO)
w{ p-
Then
(p-1) in +
< in +
{f(reiSsin O) w{
{l(w)llmax(1,lf(-)l p-I)
{f(reiOsinS) w{P-i
in +
Ip-I
{f(reiesln
e)
w
{((reiesin
e)
Z(w)){
(2.7)
From (2.6) and (2.7) we get
{J(re_iOsin O)l lJ (w) ll
((rei sin B)%(w))
For the curve
<_
_p_f(reiesin e){ + I) p-I,
{f(reiOsin e) wlP-I
(2.8)
holomorphic in the half plane defined by the relation (2.3) we get
(z)
by (2.7) and (2.8)
(p- l)m(r,w,f) s
m*(r,a(w),)
(p-
l)(r,w,f)+
C
(2.9)
consider the arguments of the poles, the characfor fixed value of
teristics n(r,f) may decrease, if we simply change its argument
we
get
its modules. Then from (2.4), (2.5), and (I.I),
Since the characteristics
(r,f)
M. HOSSAIN
640
(p- l)N(r,=,f)+ (p- l)N(r,w,f)- (p- l)N(r,=,f)
N*(r,a(w),G) -<
1)N(r,w,f)
From (2.9) and (2.10) we get
(r,)
(p
T * (r,)
m* (r,(w) )
g(z)
If in (2.9), f(z)
same way considering
+
gl(z)
G(z)
N* (r,a(w),G)
N*(r,a(w),G)
(2.10)
+ C.
l)(r,w,f) +
S
(p-
S
(p- l)T(r,f)+ C
C
+ (p
l)(r,w,f)
is an analytic function, then we shall proceed in the
g(z)
and
g2(z)
in
(2.1), (2.2)
and (2.3)
with
gp-I (z), gp-2 (z)
Corllary I, 2, 3 follow from Theorem I, and Theorem A, B, C, for meromorphic
functions.
fined in
Corollary 4 follows from Theorem
and Theorem D for analytic function de-
([3], 131-135).
REFERENCES
Distribution of values of Meromorphic Func-
I.
GOLDBERG, A. A. and OSTROVSKI, E. V.
tions, M. 1970
2.
FAINBERG, E.D.
3.
FAINBERG, E. D. About the Valiron’s defects of meromorphic functions in the half
plane, Ibid. Vol. 26(1977), 134-138.
LYBOVA, S. V. About the Valiron’s defects of meromorphic functions and holomorphic
curves in the half plane, Ibid. Vol. 27(1979) 48-53.
HOSSAIN, M. About the defects and deviation quantities of entire curves, Theory
of functions, functional analysis and their application, Vol. 20(1974) 161170.
4.
5.
About the defects of meromorphic functions in the half plane,
Theory of functions, functional analysis and their application, Vol. 25(1976)
120-131.