Advances in Pure Mathematics, 2013, 3, 586-589
http://dx.doi.org/10.4236/apm.2013.36075 Published Online September 2013 (http://www.scirp.org/journal/apm)
Estimates for Holomorphic Functions
with Values in


0,1
P. V. Dovbush
Institute of Mathematics and Computer Science, Academy of Sciences of Moldova, Kishinev, Republic of Moldova
Received June 29, 2013; revised July 28, 2013, accepted August 24, 2013
Copyright © 2013 P. V. Dovbush. This is an open access article distributed under the Creative Commons Attribution License, which
permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Abstract
Extension of classical Mandelbrojt’s criterion for normality to several complex variables is given. Some inequalities for
holomorphic functions which omit values 0 and 1 are obtained.
Keywords: Complex Space; Holomorphic Functions
1. Introduction
In 1929, Mandelbrojt [1] has asserted his criterion for
normality of a family of holomorphic zero-free functions
of one complex variables.
In [2], the author has proved a generalization of Man-
delbrojt’s criterion to several complex variables. In order
to state this criterion precisely, we introduce some nota-
tions.
Let be a family of zero-free holomorphic func-
tions in a domain and be a subdomain in
such that
n
D
D. So that the quantities


 

,
,
ln ,if1 for all ;
sup ln
, if1 for some ,
zw D
mfD
fz fww D
fw
f
ww







D
 


,
,, sup
zw D
f
z
mfD
f
w








,min ,,,LfDmfDm fD

,
are well defined for each function . f
Theorem 1. (See [2].) Let be a family of holo-
morphic functions in a domain with values in
Then is normal in if and only if for
each point 0 there exists a ball




0, 1
z

00
,Bz r
such that the the set of quantities




00
,,LfBzr ,
is bounded.
,f
It is well known that a family of functions holo-
morphic on a domain


all of which omits the values 0
and 1 is normal, so by the Theorem




00
,,LfBzr for some 0 and all r.f

 But
for this case we may obtain a more plain inequalities:
Proposition 2. Let X
K
be the Kobayashi distance on
a connected complex space Let be the family
of all holomorphic functions on
.X
X
with values in


0,1. Then, for all ,
x
yX and all ,f
 


 
log
exp ,log
exp, ,
X
X
cfx
Kxy cfy
K
xy




(1)
where
2
1
44.3768796 .
4
c





Furthermore, if there exists continuous log
f
on
X
then
 




log
exp ,log
exp,.
X
X
cfx
Kxycfy
Kxy




(2)
In the proof of this proposition, we combine the result
of Lai [3] with the definition of the Kobayashi metric and
obtain a very elementary proof of Proposition 3 in [4].
2. The Proof of Mandelbrojt’s Criterion
Proof of Theorem 1.  Fix a point in  and con-
0
z
C
opyright © 2013 SciRes. APM

P. V. DOVBUSH 587
sider a ball


0,Bz r
LfB
. Suppose that  is normal
in but the set for some 
 
00
,,zr

,f,
0,rr is unbounded. Then there exists a sequence


j
f suchthat
 
00
, ,LfBzr

forall .jj  (3)
By hypothesis is normal, and therefore, the fol-
lowing two cases exhaust all the possibilities for se-
quence


:
j
f
1) The sequence


j
f
has a subsequence


k
j
f
which converges uniformly on

00
,Bzr to a holomor-
phic function ;
f
2) The sequence


j
f
has a subsequence


k
j
f
which converges uniformly on

00
,Bzr to Since
is a family of zero-free holomorphic functions in a do-
main by Hurwit’s theorem


f
is either nowhere
zero or identically equal to zero.
Therefore the following three cases exhaust all the
possibilities for sequence


:
j
f
a) The sequence


j
f
has a subsequence


k
j
f
which converges uniformly on

00
,Bzr to the holo-
morphic function
0;f
b) The sequence


j
f
has a subsequence


k
j
f
which converges uniformly on

00
,Bzr to a holomor-
phic function
f
which is zero-free on

00
,Bzr;
c) The sequence


j
f
has a subsequence


k
j
f
which converges uniformly on

00
,Bzr to
.

Since it follows readily from (3) that
 
00
,, jBzr for all Lf j (4)
In case a) (respectively in case c)) we have

12
k
j
fz (respectively

2
k
j
fz for all

00
,zBzr and all sufficiently large. Hence k

ln k
j
f
z


is a negative (respectively positive) pluri-
harmonic function in Pluriharmonic functions
form a subclass of the class of harmonic functions in
(obviously proper for ). So by Harnack’s
inequality there exists some constant

0
Bz

,r

0,Bz r1n

 00 0
,,, ,CCBzrBzr 

1,,C
that




00
,B
ln for all and ,
ln
k
k
j
j
fz Czwzr
fw


and hence


00
,,
k
j
mf BzrC

for all suffi-
ciently large.
k
In case b), we have




lim k
j
k
f
zfz
 

for all

0,.zBzr It follows






00
lim uniformly for and ,
k
k
j
kj
fz fz zwBzr
fw
fw
 
The function




f
zfw is holomorphic on

00 00
,Bz rBz r,, it follows that




00
,,mfBzr
 is
bounded.
Since



00
,,
k
j
Lf Bzr

is the minimum of


00
,,
k
j
mf Bzr and


00
,,
k
j
mf Bzr

the set of quantities



00
,,,Lf Bzrk,
k
j is bounded,
which is a contradiction to (4).
Fix a point in
0
z

and define the families
and by






0
,1ffz  ,



0
,1ffz.


It will be shown that is normal in



Ω and
that is normal in 


,.CΩC
To prove that the family



0
,1ffz is
normal, it is sufficient to show that each sequence


j
f contains a subsequence converging locally
uniformly in


00
,Bzr to a holomorphic function or to

. The following two cases exhaust all the possibilities:
a) There exists a subsequence k

j
f
such that for any
k

 the function ln k
j
does not vanish in


00
,;Bz r
f
b) For each j

 there exists such
that

00
,zr
j
zB


ln 0.
jj
fz

In case a), we have that 1
k
j in for all
elements of the sequence. Such a subsequence is normal
in

00
,Bzrf


00
,Bzr by Montel’s theorem and hence we are
done in casea).
In case b), we have for all
 
00
,,
j
mf Bzr
.j


Therefore, according to the hypothesis,




00
,,
j
mfBzr C


for all and some constant j
0.C It follows that j
f
C in for all

00
,Bzr

,j


which means that


j
f
is a normal family in


00
,Bzr
f and hence finishes the proof in caseb). If

, then 1
f
is holomorphic on

because
f
never vanishes. Also1
f
never vanishes and


0
1fz 1.

Hence reasoning similar to that in the
above proof shows that

1:ff



is also
normal in




00
,Bz r. So if


j
f
is a sequence in
 there is a subsequence

k
j
f
and an analytic func-
tion on h


00
,Bzr such that


1k
j
f
converges in




00
,Bzr to . By the generalized Hurwitz’s
Theorem, either
h
h0

or never vanishes. If h0h

it is easy to see that k
j uniformly on compact
subsets of
f


00
.,rBz If never vanishes then h1h is

P. V. DOVBUSH 588
analytic and it follows that




1
k
j
f
zhz
uniformly
on compact subsets of .

00
,Bzr

It follows that and are normal at 0 so that
the union is normal in
z



00
,rBz Since normality is
a local property, is a normal family in 

This
completes the proof of the theorem.
Remark 1. It should be pointed out that the above
theorem is not true if the condition “for each point
there exists a ball
0Ωz


00
,r
 
,,
.Bz
00
r
such that the
the set of quantities
 
,LfBz ,
f
F ised”
is replaced by the condition “the corresponding family of
bound
functions given by






,
f
zwfzf w
 is locally
bounded on ΩΩ (cf. [5, Theorem 2.2.8]).
, consTo see thisider the family

:j
1
j
Fz

 of holo-
morphic functions. If we take

::121,Az z 
then A
F
is a set of bounded (b-
morphunctions in
y 1) zero-free holo
ic f
A
so Montel’s theorem guaran-
tees that
F
is normal. It is plain by inspection that the
family

1
j
jj

is not locally bounded on ,
zw
A
A

while

1
ln ln
j
j
j
zw

 is a locally bounded family on
.
A
A.2.8 in [5] is not true. Hence Theorem 2
3. Estimates for Holomorphic Functions
Pr m of Landau
Which Omit the Values 0 and 1
oof of Proposition 2. The classical theore
may be stated in the form that if the function


f
z is
holomorphic in the unit disk

Δ:1zz nd

1, then
a
does not take the values 0 and

0f has a
bound depending only on


0f. In fact

0f has a
bound depending only on


0f
  


0 log0cf (5)
where
02ff

2
1
44.3768796
4
c





(see, for example, [4]).
Let

denote the Poincaré distance on i.e., the
die fi
,
stancunction defined by the Poincaré metr c

2
2
2
4.
1
dzdz
ds z 

For define
Δz






1
zwwz zw
  . Since
z
f

 ot take t (5) we
the following inequality
does nhe values 0 and 1, from
derive


  
 

0
20log0.
zz
zz
f
fcf



 


(6)
Let
2
10zf  

be a pair of points in
X
. Since X
K
,
x
y is an
inner pometric (see [6]), for ea 0 seud ch

 t exist
an integer 1,l

here

1,, Δ,, l
H
ol X

 and 1,,a


0, 1
l
a satng isfyi


10,
y






10 j
,
for
jj
a  
1, ,jl1

 and 


,
l
ax
l


and



1
0,, 2.
lxy
jX
j
aK




Set

.
jj
gf


 From inequality (6), we obtain


 

2
2 for 0,.
1
log
j
j
jj
gt ta
t
gtc gt




 (7)
Since for 0,
j
ta






 
 
log
log log
jj
j
jj
gt gt
gt t
g
tg
tt





t
from (7), we obtain


2
2
log .
1
j
gt
tt  (8)
If we integrate both sides from
su
logc

0t to , the re-
j
a
lt becomes



log
log0, .
log 0
jj
j
j
cga a
cg









Then


 
log
log, .
log X
cfx
Kxy
cfy



, we finally get


Letting 0


 

log fx
exp ,
log X
c
K
xy
cfy


so the second inequality in (1) is proved. Since
x
and
u
y play symmetric roles, it is evident that the first ine-
qality in (1) also holds.
For obtaining inequalities (2), let us notice that there
exists continuous


log log
j
g
on 0,
j
ta

 . Since
 
log logloglog
jj
g
tg
tt
t

we have


P. V. DOVBUSH 589
References

  


log log
log .
jjj
j
gt
g
tg
gt tt
cgt
t






t
From this inequality and inequality (8), we obtain
[1] S. Mandelbrojt, “Sur les Suites de Fonctions Holomorphes.
Les Suites Correspondantes des Fonctions Dérivées.
Fonctions Entières,” Journal de Mathématiques Pures et
Appliquées, Vol. 9, No. 8, 1929, pp. 173-196.
http://portail.mathdoc.fr/JMPA/afficher_notice.php?id=J
MPA_1929_9_8_A10_0


2
1
jj
tt 
Integrating both sides of this inequality as abov
ob
o
fu
or Robert B. Burckel for all
2
log log for 0,.cgt ta

 [2] P. V. Dovbush, “On a Normality Criterion of S. Man-
delbrojt,” 2013. http://arxiv.org/abs/1302.1695
e we [3] W. T. Lai, “The Exact Value of Heyman’s Constant in
Landau’s Theorem,”Scientia Sinica, Vol. 22, 1979, pp.
129-133.
tain the inequality (2).
The proof of the theorem is now complete. 
rphic [4] E. M. Chirka, “Harnack Inequalities, Kobayashi Distances
and Holomorphic Motions,” Proceedings of the Steklov
Institute of Mathematics, Vol. 279, No. 1, 2012, pp. 194-
206.
Remark 2. Proposition 2 holds also for holom
nctions defined on an infinite dimensional complex
Banach manifold with values in

0, 1, the same
proof works. So we give here mole proof of
Proposition 3 in [4].
4. Acknowledgements
re simp
doi:10.1134/S0081543812080135
[5] J. L. Shiff, “Normal Families,” Springer-Verlag, New
York, 1993. doi:10.1007/978-1-4612-0907-2
[6] S. Kobayashi, “Hyperbolic Complex Space,” Springer-
Verlag, New York, 1998.
doi:10.1007/978-3-662-03582-5
I would like to thank Profess
his help during my work on this paper.