On The Growth Of An E -Ealued Meromorphic Function And Its Derivative
Veena L. Pujari
yReceived 18 July 2014
Abstract
In this article, the relative growth of an E-valued meromorphic function and its derivative is studied and we obtain the bound for T(r;fT(r;f)0) for an E-valued meromorphic function of …nite order. We also extend the related results of S.
K. Singh and H. S. Gopalakrishna [4] toE-valued meromorphic function. Our results are signi…cant and much stronger than the result obtained by Z. Wu and Y. Chen [5].
1 Introduction
In 1982, H. J. W. Ziegler [6] successively extended the classical Nevanlinna theory of meromorphic functions to vector valued meromorphic functions in …nite dimensional spaces. Later in 1996, C. G. Hu and C. C. Yang [3] established the Nevanlinna’s theory in an in…nite dimensional Hilbert space. C. G. Hu [2] assumed,E is an in…nite dimen- sional Banach space with a Schauder basis fejg,j = 1;2; : : :and was able to present the statement of …rst and second fundamental theorem of Nevanlinna and Nevanlinna’s de…ciency relation in E. In 2006, C. G. Hu and Q. J. Hu [1] successively proved the generalized Poisson-Jensen-Nevanlinna formula, …rst and second fundamental theorem of Nevanlinna forE-valued meromorphic functions.
2 Basic Notions of Nevanlinna Theory in In…nite Di- mensional Banach Space
Assume that E is a in…nite dimensional complex Banach space with a Schauder ba- sis fejg1j=1 and C is a complex plane. Let D = Cr = fz:jzj< rg. An E-valued meromorphic functionf(z)in a domainD Ccan be written as
f(z) = X1 j=1
fj(z)ej = (f1(z); f2(z); : : : ; fj(z); : : :);
Mathematics Sub ject Classi…cations: 30D35, 30D30.
yPost-Graduate Department of Mathematics, Vijaya College, Basavanagudi, Bangalore-04. India
137
where eachfj(z)is a complex-valued meromorphic functions inD:We now introduce the generalized quantities of the Nevanlinna theory (see [1]): For any a2 E[ f1g, n(r; a; f) =n(r; a)denotes the number ofa-points off injzj r; counted with multi- plicities and n(r;1; f) =n(r; f)denote the number of poles of f in jzj r. Then we have the counting function of …nite or in…nitea-points as
N(r; a) N(r; a; f) =n(0; a) logr+ Z r
0
n(t; a) n(0; a)
t dt;
N(r; f) N(r;1; f) =n(0; f) logr+ Z r
0
n(t; f) n(0; f)
t dt;
m(r; f) m(r;1; f) = 1 2
Z 2 0
log+ f(rei ) d ; m(r; a) m(r; a; f) = 1
2 Z 2
0
log+ 1
kf(rei ) akd ;(a6=1);
and
T(r; f) =m(r; f) +N(r; f);
where log+x= maxflogx;0g. The volume function associated with E-valued mero- morphic functionf is given by
V(r; a; f) = 1 2
Z
Cr
log r
logkf( ) akd ^d ; a2E and the curvature function is given by
V(r;0; f0) =G(r; f) = Z r
0
dt 2 t
Z
Ct
logkf0( )kd ^d : The order of anE-valued meromorphic functionf is de…ned by
= lim sup
r!1
logT(r; f) logr and the lower order off is de…ned by
= lim inf
r!1
logT(r; f) logr :
We now de…ne the following de…ciencies as in [2]: For any a2E[ f1g, the number (a) = (a; f) = lim inf
r!+1
m(r; a)
T(r; f) = 1 lim sup
r!+1
V(r; a) +N(r; a) T(r; f)
is called the de…ciency of the point a, a pointawith (a)>0is called de…cient.
The quantity
(a) = (a; f) = lim inf
r!+1
N(r; a) N(r; a) T(r; f)
is called the index of multiplicity of a, and (a) = (a; f) = lim inf
r!+1
m(r; a) +N(r; a) N(r; a) T(r; f)
= 1 lim sup
r!+1
V(r; a) +N(r; a) T(r; f) : In particular, we have
(1) = lim inf
r!+1
m(r; f)
T(r; f) = 1 lim sup
r!+1
N(r; f)
T(r; f) sinceV(r;1) = 0;
(1) = lim inf
r!+1
N(r; f) N(r; f) T(r; f) ; (1) = 1 lim sup
r!+1
N(r; f) T(r; f): The quantity
G= G(f) = lim inf
r!+1
G(r; f) T(r; f) is called the Ricci Index of f(z).
The functionf is called admissible if TS(r(r ;f)) !0 for a sequencer !+1: THEOREM 1 ([1]). (E-valued Nevanlinna’s …rst fundamental theorem) Let f(z) be anE-valued meromorphic mapping in CR. Then for0< r < R,a2E,f(z)6=a;
T(r; f) =V(r; a) +N(r; a) +m(r; a) + logkcq(a)k+ (r; a):
Here (r; a)is a function such thatj (r; a)j log+kak+log 2, (r;0) 0;andcq(a)2E is the co-e¢ cient of the …rst term in the Laurent series at the pointa.
THEOREM 2 ([1]). (E-valued Nevanlinna’s second fundamental theorem) Letf(z) be a non-constant E-valued meromorphic mapping of compact projection in CR and a[k] 2E[ f1g(k= 1;2; : : : ; q)beq 3 distinct …nite or in…nite points. Then
Xq
k=1
m r; a[k] +G(r; f) T(r; f) N1(r) +S(r);
where N1(r) =N(r;0; f0) + 2N(r; f) N(r; f0)and G(r; f) =
Z r 0
dt 2 t
Z
Ct
logkf0( )kd ^d :
If R= +1, thenS(r) satis…esS(r) =OflogT(r; f)g+O(logr)as r!+1without exception iff(z)has …nite order and otherwise asr!+1outside a setJof exceptional intervals of …nite measure R
Jdr <+1:If0< R <+1, then S(r) =O log+T(r; f) +O log 1
R r
holds as r!R without exception iff has …nite order
= lim sup
r!R
logT(r; f) log(1=R r);
and otherwise as r!R outside of a set J exceptional intervals such that R
JdR r1 <
+1:In all cases, the exceptional setJ is independent of the choice of the …nite points a[k] 2E and of their number.
THEOREM 3 ([2]). (E-valued Nevanlinna de…ciency relation) Let f(z)be an E- valued meromorphic function and admissible with the property of compact projection.
Then the setfa2E[ f1g: (a)>0gis at most countable and summing over all such
points X
a
[ (a) + (a)] + G
X
a
(a) + G 2:
THEOREM 4 (Lemma 3.1(A) of [1]) Letf(z)be anE-valued meromorphic function with the property of compact projection, and let
S1(r) = 1 2
Z 2 0
log+ f0(rei )
kf(rei )kd + 1 2
Z 2 0
log+ F(rei ) f(rei ) d +plog+2p
logkc0l0k: Then
G(r) + Xp+1
k=1
m(r; a[k]) +N1(r) 2T(r; f) +S1(r);
where N1(r) =N(r;0; f0) + 2N(r; f) N(r; f0)is the generalized counting function of multiple points,a[ ]= (a[ ]1 ; : : : ; a[ ]j ; : : :)(p 2)2E are distinct …nite points, and
F(z) = Xp
=1
1 f(z) a[ ] :
3 Main Results
S. K. Singh and H. S. Gopalkrishna [4] proved the following result:
THEOREM 5. Iff is a non-constant meromorphic function of order then lim inf
r!1
T(r; f0) T(r; f)
X
a2C
(a; f)
wherer! 1without restriction if is …nite andr! 1outside an exceptional set of
…nite measure if = +1:
In [5], Z. Wu and Y. Chen proved the following result.
THEOREM 6. Letf(z)be an admissibleE-valued meromorphic function of com- pact projection inCof …nite order and assumeP
a (a) = 2:Then
r!lim+1
T(r; f0)
T(r; f) = 2 (1):
Now in this article, we obtain a THEOREM 5 forE-valued meromorphic function f(z) in modi…ed form and also extend the related results of S. K. Singh and H. S.
Gopalakrishna [4]. THEOREM 6 is also proved as a consequence of our main result.
We prove the following main results.
THEOREM 7. Letf(z)be an admissible and non-constantE-valued meromorphic function of …nite order with compact projection then
X
a2E
(a; f) + G lim inf
r!+1
T(r; f0) T(r; f);
where r !+1without restriction if is …nite andr! +1 outside an exceptional set of …nite measure if = +1:
To prove THEOREM 7, we …rst prove the following Lemma, which plays an promi- nent role in the proof of the THEOREM 7.
LEMMA 1. Letf(z) be a non-constant E-valued meromorphic function with the property of compact projection in Cr and
a[ ]= a[ ]1 ; a[ ]2 ; : : : ; a[ ]j ; : : : (p 2)2E are …nite or in…nite distinct points then
Xp
=1
m(r; a[ ]; f) +N r; 1
f0 +G(r; f) T(r; f0) +S(r; f);
where
S(r; f) = 1 2
Z 2 0
log+ F(rei ) f0i ) d log c0p +plog+2p and
F(z) = Xp
=1
1 f(z) a[ ] :
PROOF. Following the proof of Lemma 3.1(A) in [1], we obtain the required result.
PROOF OF THEOREM 7. Let a[ ] , = 1;2; : : : ;1 be an in…nite sequence of distinct elements of E;which includes everya2E for which (a; f)>0:Then
X1
=1
a[ ]; f =X
a2E
(a; f): (1)
We have
Xp
=1
m(r; a[ ]; f) +G(r; f) T(r; f0) N r; 1
f0 +S(r; f):
AddingPp
=1N r; a[ ]; f to both sides, we obtain Xp
=1
T(r; a[ ]; f) +G(r; f) T(r; f0) + Xp
=1
N r; a[ ]; f N r; 1
f0 +S(r; f)
= T(r; f0) + Xp
=1
N r; a[ ]; f N0 r; 1
f0 +S(r; f);
where N0 r;f10 is formed with the zeros off0 which are not zeros of any off a[ ] ( = 1;2; : : : ; p):SinceN0 r;f10 0; we have
Xp
=1
T(r; a[ ]; f) T(r; f0) + Xp
=1
N r; a[ ]; f G(r; f) +S(r; f):
By an E-valued Nevanlinna’s …rst fundamental theorem, we have T(r; a; f) =T(r; f) V(r; a; f) +O(1):
Using this in the above equation, we obtain Xp
=1
h
T(r; f) V(r; a[ ]; f) +O(1)i
T(r; f0) + Xp
=1
N r; a[ ]; f G(r; f) +S(r; f):
We further obtain
pT(r; f) T(r; f0) + Xp
=1
h
N r; a[ ]; f +V(r; a[ ]; f) i
G(r; f) +S(r; f):
Then
p lim inf
r!+1
T(r; f0) T(r; f) +
Xp
=1
lim sup
r!+1
N r; a[ ]; f +V(r; a[ ]; f)
T(r; f) lim inf
r!+1
G(r; f) T(r; f) + lim sup
r!+1
S(r; f) T(r; f):
It follows that
p lim inf
r!+1
T(r; f0) T(r; f) +
Xp
=1
h
1 (a[ ]; f) i
G(f):
So Xp
=1
(a[ ]; f) + G(f) lim inf
r!+1
T(r; f0) T(r; f): Lettingp! 1and using (1), we get
X
a2E
(a; f) + G(f) lim inf
r!+1
T(r; f0)
T(r; f): (2)
COROLLARY 1. Letf(z)be a admissibleE-valued meromorphic function of …nite order with the property of compact projection such that
X
a2E
(a; f) + G= 2; E=E[ f1g:
Then (i)
r!lim+1
T(r; f0)
T(r; f) = 2 (1; f):
(ii)
1 (a; f) + G lim inf
r!+1
V(r; a) +N(r; a)
T(r; f) lim sup
r!+1
V(r; a) +N(r; a) T(r; f)
= 1 (a; f):
PROOF. Given that X
a2E
(a; f) + G= 2;
we have X
a2E
(a; f) + (1; f) + G= 2:
It follows that X
a2E
(a; f) + G= 2 (1; f):
Using (2), we write lim inf
r!+1
T(r; f0) T(r; f)
X
a2E
(a; f) + G= 2 (1; f):
On the other hand, we know that
T(r; f0) = m(r; f0) +N(r; f0) =m(r;f0
f ) +m(r; f) +N(r; f0) T(r; f) +N(r; f) +S(r; f)
and
lim sup
r!+1
T(r; f0)
T(r; f) 1 + lim sup
r!+1
N(r; f) T(r; f): So
lim sup
r!+1
T(r; f0)
T(r; f) 2 (1; f):
Thus
r!lim+1
T(r; f0)
T(r; f) = 2 (1; f):
(ii) Let a 2E[ f1g and a[k] ; k = 1;2; : : :1 be an in…nite sequence of distinct elements ofE[f1gwhich includes everyb2E[f1gsuch thatb6=aand (b; f)6= 0:
Then X1
k=1
a[k]; f = X
b2E;b6=a
(b; f) = 2 (a; f): (3)
By E-valued Nevanlinna’s second fundamental theorem, we have
(q 2)T(r; f) +G(r; f)
q 1
X
k=1
h
V(r; a[k]; f) +N(r; a[k]; f) i
+ V(r; a; f) +N(r; a; f) +S(r; f);
(q 2)T(r; f)
q 1
X
k=1
h
V(r; a[k]; f) +N(r; a[k]; f) i
+ V(r; a; f) +N(r; a; f) G(r; f) +S(r; f);
(q 2)T(r; f)
q 1
X
k=1
V(r; a[k]; f) +N(r; a[k]; f)
T(r; f) + V(r; a; f) +N(r; a; f) T(r; f) G(r; f)
T(r; f) +S(r; f) T(r; f);
(q 2)
q 1
X
k=1
lim sup
r!+1
V(r; a[k]; f) +N(r; a[k]; f) T(r; f)
+ lim inf
r!+1
V(r; a; f) +N(r; a; f)
T(r; f) lim inf
r!+1
G(r; f)
T(r; f) + lim sup
r!+1
S(r; f) T(r; f);
(q 2) lim inf
r!+1
V(r; a; f) +N(r; a; f)
T(r; f) +
q 1
X
k=1
[1 a[k]; f ] G;
(q 2) + G lim inf
r!+1
V(r; a; f) +N(r; a; f)
T(r; f) + (q 1)
q 1
X
k=1
a[k]; f ;
G 1 lim inf
r!+1
V(r; a; f) +N(r; a; f) T(r; f)
q 1
X
k=1
a[k]; f :
So
lim inf
r!+1
V(r; a; f) +N(r; a; f) T(r; f)
q 1
X
k=1
a[k]; f + G 1:
Letq! 1 and using (3), we get lim inf
r!+1
V(r; a; f) +N(r; a; f) T(r; f)
X1 k=1
a[k]; f + G 1
= 2 (a; f) + G 1 = 1 (a; f) + G: On the other hand, by the de…nition of (a; f), we have
lim sup
r!+1
V(r; a; f) +N(r; a; f)
T(r; f) = 1 (a; f):
Thus
1 (a; f) + G lim inf
r!+1
V(r; a; f) +N(r; a; f) T(r; f) lim sup
r!+1
V(r; a; f) +N(r; a; f)
T(r; f) = 1 (a; f):
COROLLARY 2 Letf(z)be a admissibleE-valued meromorphic function of …nite order with the property of compact projection such that
X
a2E
(a; f) + G = 2:
Then
r!lim+1
T(r; f0)
T(r; f) = 2 (1; f):
PROOF. We know that (a; f) (a; f),8a2E[ f1g=Eand X (a; f) + G X
(a; f) + G 2:
Given P
(a; f) + G= 2:ThenP
(a; f) + G= 2. We observe that X (a; f) + G=X
(a; f) + G = 2:
Then X
a2E
(a; f) =X
a2E
(a; f):
So
(a; f) = (a; f) 8a2E:
By using Corollary 1(i), we have
r!lim+1
T(r; f0)
T(r; f) = 2 (1; f) = 2 (1; f):
So
r!lim+1
T(r; f0)
T(r; f) = 2 (1; f)
References
[1] C. G., Hu and Q. J., Hu, The Nevanlinna’s theorem for a class, Complex Var.
Elliptic Equ., 51(2006), 777–791.
[2] C. G., Hu, Nevanlinna’s theory in a Banach Space, Proceedings of the Fifth Inter- national Colloquium on Complex Analysis(1997), 109–115.
[3] C. G., Hu and C. C., Yang, Some remarks on Nevanlinna’s theory in a Hilbert space, Bulletin of the Hong-Kong Mathematical Society(1997), 267–272.
[4] S. K., Singh and H. S., Gopalakrishna, Exceptional values of entire and meromor- phic functions, Math. Ann., 191(1971), 121–142.
[5] Z., Wu and Y., Chen, E-valued Meromorphic functions with maximal de…ciency, Applied Mathematics E-Notes, 13(2013), 141–147.
[6] H. J. W., Ziegler, Vector Valued Nevanlinna Theory. Research Notes in Mathemat- ics, 73. Pitman (Advanced Publishing Program), Boston, Mass.-London, 1982.