PAPER-I .
ON .B^QBPTIQK AL ..YALTO8 QP M&ROKQRPHIC FUNOTIOES. *
1*
Lot 4> (z)
r<»
z such that\ dz___
«*z
be an increasing function of
is convergent and let £ denote
4* (*)
the set of such functions
following s. lLShah
[S]. we define u to be an ezoeptionsl value (e.v.)B.
of a meromorphio
function f(B), if,
Lim inf__SiZl __
Y—>oo n(r, <*)#( r)
>0.
A value oc is slid to be an exceptional value (e.v)K
if# [lf?8-l 07~] ;
& (r, of )
= 5 (<* )= 1 - Ida supfil£j°U
y_^o°
>
0
T{rj
and e.v.v. (in tbe sense of Yallron) if
A(r. o< )
=
A (* )
=•
1- 11a infJ£i£ij*jL > 0.
T(r)
Also
let 0(o<)=r 1- lim sup K^r* °* )*
Y—VOO T^rj
and /*(r, o< ) = /*( <x ) « lim inf_
°1
I
I
y >oo
T(r)
where
f( a)
~n (r#o< )
•*«()
xn
rj»S£—)
denotes tbe number of roots of
|s|^r, each root being counted only
once*
.§s,.IS.tfE2y2.!;i?§.?2ll£?i5#-i'£§2£§51i__________________
*. it is a joint paper by tbe author ana his
Supervisor ir. s.K. Singh and is coununioated ibr publication.
- 2 -
Theorem 1,
Let
tl>at
where
alt
f( a) he a meromorphia function such
f
£ (»i )-+-!>< ai) = 1
aSt--.......................... - are any finite -
constant a different from each other, then, if oo
an e.v.B
is
for f( a), it is also an e.v.s.far f’(z)
and. o- on Tersely,
proof;- ’ oo is an e.v.a. for f(s).
so (5 ( oo ) = l, see 8.M . 8hah [2 ]
But, it is known that
llm sup
■ *.....1
y—+oo T( r,f)
See
2- 5(oo ) - m ( oo )
'
1 I
(2.1)
as
1................................
seTaniinna £l,104,]
Also it is known that ,
lia in£-IL£i£lL
y—>oo
^ £ 8{&i)
T(r , f)
1
-+- f M(ai)
' ‘
1........................ (2.2)
Seel littioh [6,2l]
from
( 2.1)
low
T(r, f»)
n(r ,f ) =r
*
and. (2.2)
it follows that
T(r, f)
n(r ,f)
\
2n(r, f)
n(r,f)
i
Henoe,
liLtlll—
^
1/2
n(r,f»)<J>(r)
__ Kirill v
n(r,f)<f(r) > LhlUJLtSl
n(r, f)^(r)
Since oo is an e.t.B for f(s), it follows
lim infJEi£i£:LY—►
oo
n (r, f')«£(r)
>
0
.
- 3 Hence
• 00 •
co is
an e. v.E
is en e.v.E.flor f*(a) .Sonversely ,if
S3or f»(B), w® prove that it is e, y.ii.
lor £(z) also*
Since
■But
<£(f, oo )
1.
N(r, f») =
H(r, f)
JL{J£m-Q.
i/2
0
f-vo° T<r#r,j
H(r, f) ^ 2N(r,f)
T(r,f)
lim .sup-fiLrjuf),
T(r#f)
So
But
r—>oo
is
«
Hence
© ( oo )
^
X
'
^
So----- £!£__ l£1__
n(r,f)<^(r)
oo
Bence oo is
>
<*,
S (r,aj)+ZKr.aj)
T(r, f»)
n(r~,f*)
since
i.
1
But, as before,
i in Uf .Hr.fll ..
-f—>0o T(y f)
Sow
__
known that £ 6,21 ] .
lim supJUS,^ 2 T—><» T(r, f)
Now
i(r, r*)
0(oq) =z l - lim aup-tLLr^f)^
t—voo i(r, f)
it
lim aupSLS^f'j-o
lim sup_M£tXL)
« i/2
11
f)
n(r, f).
(1-6 ) Tfr. fM
n(r,f»)
<f> (r)
is ane.v.i for f»(B), lim inuJElEtHL >0
T-—>oo n(r,
an e. v.E. of f(z),
Theorem
2:
If
• oo *
is an e, v.B. lor a meromorp-telc
X 5(
tion l(a) and
+ 0°
Sc ffc, 0)4- 6{&,
—
then §£
l
OO ) =r 2, for k
prooi:- Sinoa oo is an e. v.E, £(f# oo )
But lim sup_Hli£l_L < 2- £ ( oo )
Y*—*0° T(r,f)
~~
Abain
lim lnf__T(r* f<)
^
57£(o< )
*=
-
As
1,2-r —
1.
m ( oo )
/
-+- £(*(<* 5
Y—>oo T(r, f)
ao
func­
I
T(r,f*) r^> T(r, f).
in
theorem
1, sinoa oo is an e, v.E. for f»(z),
•8 {£'j o° ) =
Atosin# we
H(r.
have
*
£ 6,18,
1.
,
»(r.aj)
S(r)
so,
^
T(r#f')
<*
H(r# !/£*) -h m(r# l/£') - m(r, !/*•)+* Z m(r,aj)
-4-8(r)
^ T(r,f»)
*
X m(r,aj) -4- S(r) « m(r#l/f*) -f- o(l).
Mil-.) iS(t’.o)
Hence l im inf /X
BCr> aJ )
Y -r->oo l i
T(r, f»)
Bnzai so,
!
lim sup ^(r»^')
Y—>oo T(r, f)
| Sir. aj) ^
5(1". 0)
5
But lim mip._ I(r,f>)
■f1 -—oo
T(r, f)
Z 5 (r,aj)
Hence
^
1
=■
1
S (f • ,0)
So $(f«,0) -+- £(f\ oo)
.
5=1
1
«
2
Be pea ting the argument we get,
8(fk , <>)+£(** , oo )
-
and
2
Corollary:*
If oo is an e,v.:E. ter a meromorphio
oo 6(
* <xu ) =• i# then
function f(a) and if lUrtherZ
p , the order of the function. Bust he a positive
integer, Por an alternative proof of the corollary
see
S.lLShah and 8.K. Singh [5,3
Proof:-
By theorem 2,
0) = 1.
S(f,00) = 1.
Hence m W -0)+B(r, f■-*>)
Y—^oo
Hence
the order
of ,f'
(a)
=
Q_
must he a positive
integer because ter non-integral order,
h.
o
lim
>
r—> oo
T(r)
see Hevanllnna [ 1,51 ] .
fcr all a and
But $ the order of a meromorphio tenotion is the same
as
the order of its derivative.
corollary,
Theorem
a.
This proves
the -
'
If there is at least one
e, v.H. ter a
6
meromorpbio funotion f(s), then
0
mist "be an except­
ional value of f*(a) In the sense or Valiron.
Proof:- Suppose, if possible 0
is a
normal value (for
f'(B) In the sense of Valiron.
Then,
A(f\Q)
But, it; is known
=
0.
[ 4 ] that,
A(f\0) lim inf^irL£l5—. ^
r
—*oo T(r,f)
0
^ X S(ai).
So
2
*
<S (ai)
00
X1
Contradicting the hypothesis
Hence
0
is an
{
exceptional value of f*(s) in the
sense of Valiron,
********
Beferanoes.
[
B. Nevanlinna,
: Le theorems de PicardBorel et la theorie des
fonotlons meromorphes,
pari s, 1929.
[•]
S. 1C. Shah,
: on exceptional values of
entire functions.Comp.
lfeth.9.227-238(1951).
[»1
S. X. Shah,
: Exceptional values of
entire and. meromorphio
functions. Duke Math. J
19,586-594(1952).
H
S. M. Shah. and.
S.K.Singh,
.
a
On the derivatives of
meromorphic funotion
with maximum deiteot,
Ifeth. Zeitsohr.Bd.65, s.
171-174(1966).
8. It. Shah and
8.K. Singh,
; Msromorphio functions
vith maxi nun dsifcot sum.
1959. To ho leu Math. J. Vol.
11, pp. 447- 452.
H^Wittioh,
: Neure untersuohungen Ufcer
eindfiutiga anslytlflbhe funistionen, Berlin 1955,
■****#*•*