TITLE: A LOT OF “COUNTEREXAMPLES” TO LIOUVILLE’S THEOREM.
AUTHOR: LUIS BERNAL-GONZ´
ALEZ.
AFFILIATION: DEPARTAMENTO DE AN´
ALISIS MATEM´
ATICO. FACUL-
TAD DE MATEM´
ATICAS. AVENIDA REINA MERCEDES. APARTADO
1160. 41080 SEVILLA, SPAIN. E-MAIL: lb[email p o ec ed].
FOOTNOTES TO THE TITLE: *This wo k is suppo ed in pa by DGICYT
g an PB93-0926.
A.M.S. Subjec Classi ica ion: P ima y 30D15. Seconda y 30E10, 44A05.
Key wo ds and ph ases: Liou ille’s heo em, en i e unc ions, dense linea mani old,
Radon ans o m, A akelian se , s ips and sec o s, g ow h index.
1
ABREVIATED TITLE: LIOUVILLE’S THEOREM.
NAME AND MAILING ADDRESS OF THE AUTHOR TO WHOM
PROOFS SHOULD BE SENT: LUIS BERNAL-GONZ´
ALEZ. DEPARTA-
MENTO DE AN´
ALISIS MATEM´
ATICO. FACULTAD DE MATEM´
ATICAS.
AVENIDA REINA MERCEDES. APARTADO 1160. 41080 SEVILLA, SPAIN.
E-MAIL: lb[email p o ec ed].
2
A LOT OF “COUNTEREXAMPLES” TO LIOUVILLE’S THEOREM
by
LUIS BERNAL–GONZ´
ALEZ*
Abs ac . We p o e in his pape ha , gi en α∈(0,1/2), he e
exis s a linea mani old Mo en i e unc ions sa is ying ha Mis dense
in he space o all en i e unc ions and, in addi ion, limz→∞ exp(|z|α)
(j)(z) = 0 on any plane s ip o e e y ∈Mand o e e y de i a ion
index j. Mo eo e , i is shown he exis ence o an en i e unc ion wi h
in ini e g ow h index sa is ying he la e p ope y.
1. INTRODUCTION AND NOTATION
One o he mos elemen a y, su p ising and beau i ul esul s in Complex Analy-
sis is Liou ille’s heo em: each bounded en i e unc ion is cons an . Ne e heless, i
boundedness condi ion is sligh ly weakened ( o ins ance, by allowing boundedness
*This wo k is suppo ed in pa by DGICYT g an PB93-0926.
1991 Ma hema ics Subjec Classi ica ion: P ima y 30D15. Seconda y 30E10,
44A05.
Key wo ds and ph ases: Liou ille’s heo em, en i e unc ions, dense linea ma-
ni old, Radon ans o m, A akelian se , s ips and sec o s, g ow h index.
3
on e e y line), hen noncons an en i e unc ions can be ob ained. Fo his, see
o ins ance [7, pp. 9-10], whe e i is e en shown a noncons an en i e unc ion
ha ing limi ze o on any line. This is achie ed by using a esul o A. Ro h abou
app oxima ion on closed se s.
Bu se e al sha pe esul s ha e been ob ained. Recen ly, D.H. A mi age [3]
has cons uc ed a noncons an en i e unc ion such ha each de i a i e (n)is
in eg able on e e y line lwi h espec o leng h measu e sand
∫l
(n)ds = 0
o e e y n∈N0, whe e N0=N∪ {0}and Nis he se o posi i e in ege s.
Mo eo e , sa is ies
lim
z→∞
z∈l
(n)(z) = 0
o e e y n∈N0and e e y line l( he e o e each de i a i e is bounded on each
line). This is ob ained by an elemen a y pole-pushing echnique. In he same pape
i is no ed ha i is a con inuous unc ion on he complex plane C, in eg able on
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Cwi h espec o plane Lebesgue measu e, and i
∫l
ds = 0 (1)
o e e y line l, hen is iden ically ze o (see [2] o ins ance). We ema k he e
ha he mapping ˆ
(l) = ∫l
ds (l∈L) is known as he Radon ans o m o . We
ha e deno ed by L he se o all s aigh lines in he plane. Thus, we a e dealing
wi h he injec i i y o he linea ans o m 7→ ˆ
in R2. The co esponding p ob-
lem o hype planes in Rnis ex ensi ely s udied in [8] o ce ain unc ion spaces.
L. Zalcman [16] in 1982 had indica ed ha i ac ually suffices o (1) o hold only
o almos e e y line belonging o a dense se o di ec ions in o de ha ≡0. In
he same pape , he cons uc ed –by using a esul (Theo em 1 below) due o A ake-
lian conce ning angen ial app oxima ion (see [1, p. 1189]and [6, pp. 160-162])–
a noncons an en i e unc ion sa is ying (1) on e e y line l. Nonnull en i e unc-
ions ending o ze o on e e y line had al eady been cons uc ed by Mi ag-Leffle
[10, pp. 290-294], and D.J. Newman [11] p o ided an explici noncons an en i e
unc ion which is bounded on e e y line h ough he o igin. Zalcman also poin ed
ou ha , i nea ly no hing is assumed o , hen he e exis nonnull unc ions o
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which (1) holds: ake =χE, whe e Eis Sie pinski’s nonmeasu able se wi h he
p ope y ha any line in e sec s Ein a mos wo poin s (see [15]). All hese esul s
can be ca ied in o he se ing o ha monic unc ions on Rn(n≥2) (see [4] and
[5]).
In his pape we imp o e all abo e asse ions abou en i e unc ions and p o ide
a dense linea mani old o en i e unc ions “ iola ing” Liou ille’s heo em. All
nonnull unc ions in his mani old ha e in ini e g ow h o de . We can ge e en o
a leas one o hese unc ions o ha e ex emely as g ow h.
We will need a bi o no a ion and a p elimina y esul . I > 0, B deno es
he closed ball {z:|z| ≤ }. A s ip is he plane egion lying be ween wo pa allel
s aigh lines. I β∈(0,2π), he sec o sβis he se sβ={z: 0 ≤a g z≤β}(he e
a g zis e iden ly being allowed o be ou side he “p incipal in e al” (−π, π] used
by some au ho s) . Σ will s and o he amily consis ing o all s ips in Cand all
sec o s sβ(β∈(0,2π)). H(C) is he space o all en i e unc ions, endowed wi h he
compac -open opology. The ex ended plane C∞is he one-poin compac i ica ion
o C. I F⊂Cis a closed se , hen A(F) is he space o all con inuous unc ions
6
on Fwhich a e holomo phic in he in e io o F. A closed subse F⊂Cis said o
be an A akelian se [13] whene e C∞ Fis bo h connec ed and locally connec ed
a in ini y.
We will use he ollowing (abo e men ioned) heo em due o A akelian.
THEOREM 1. Assume ha F⊂Cis an A akelian se and ha ε( )is con-
inuous and posi i e o ≥0. In addi ion, suppose ha
∫∞
1
−3/2log ε( )d > −∞.(2)
Then o e e y g∈A(F) he e exis s an en i e unc ion such ha
| (z)−g(z)|< ε(|z|)∀z∈F.
The s a emen does no emain alid o e e y Fi (2) is iola ed.
No e ha , o ins ance, ε( ) = exp(− 1/2) does no sa is y (2), bu ε( ) =
exp(− α) does o α < 1/2.
Finally, le be an en i e unc ion. I > 0, we de ine exp1 = exp ,
expk+1 = exp(expk ) (k∈N), and µ( ) = max{| (z)|:|z|= }. Fo > 0
7
la ge enough, we deno e log1 = log , logk+1 = log(logk ) (k∈N). The ollow-
ing de ini ions can be ound in [12] and [14]. The g ow h k-o de ρk=ρk( ) o
is
ρk= lim sup
→∞
logk+1 µ( )
log .
I k= 1, hen ρkis called he o de o , and we shall deno e i by ρ( ). The g ow h
index i( ) o is i( ) = min{k∈N:ρk( )<∞}, whe e we se i( ) = ∞when
ρk( ) = ∞ o all k.
2. RESULTS
Recall ha H(C) is sepa able: he amily {Pn}∞
1o holomo phic polyno-
mials ha ing coefficien s wi h a ional eal and imagina y pa s is an example o a
coun able dense subse o H(C). Wi h his in mind, we a e now eady o s a e he
ollowing heo em.
THEOREM 2. Assume ha α∈(0,1/2). Then he e is a linea mani old
M⊂H(C)which is dense in H(C)and sa is ying
lim
z→∞
z∈S
exp(|z|α) (j)(z) = 0 (3)
8
∀S∈Σ,∀ ∈Mand ∀j∈N0.
PROOF. Conside he sequence {Pn}∞
1men ioned a he beginning o he
pa ag aph and ix α∈(0,1/2). Fix also a numbe β∈(α, 1/2). Fo e e y n∈N,
he unc ion
ε( ) = εn( )≡min{1/n, exp(− β)}
is posi i e and con inuous o ≥0 and sa is ies (2). Le Fn=Bn∪En, whe e
En={z∈C:|z| ≥ n+ 1 and dis (z, P )≥1}
and Pis he pa abolic cu e
P={x−ix1/2:x≥0}.
Then Fnis closed. In addi ion, a glance o e i s complemen e eals ha Fnis an
A akelian se . De ine he unc ion gn:Fn→Cby
gn(z) = {Pn(z) i z∈Bn
0 i z∈En.
Since Bn∩En=∅,gnis well-de ined. T i ially, gn∈A(En). By Theo em 1, he e
exis s an en i e unc ion nsuch ha
| n(z)−gn(z)|< ε(|z|)∀z∈Fn,
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hen he se
T(V) = { ∈H(C) : is bounded on V}
is o he i s ca ego y, because T(V) =
∞
∪
1
Cnwi h Cn={ ∈H(C) : | (z)| ≤
non V}and each Cnis closed and has emp y in e io ( o his, no e ha Cn∩
{noncons an polynomials}=∅and ha he second se in he in e sec ion is dense
in H(C)). Finally, obse e ha
AL ⊂∩{T( ) : is a ay om he o igin}.
So AL is, in his sense, e y small.
REFERENCES
1. N.V. ARAKELIAN, Uni o m app oxima ion on closed se s by en i e unc ions,
Iz . Akad. Nauk SSSR Se . Ma . 28 (1964), 1187-1206.
2. A.S. BESICOVITCH, A uniqueness heo em and a p oblem o in eg a ion,
J. London Ma h. Soc. 33 (1958), 82-84.
3. D.H. ARMITAGE, A non-cons an con inuous unc ion on he plane whose
in eg al on e e y line is ze o, Ame . Ma h. Mon hly 101 (1994), 892-894.
4. D.H. ARMITAGE and M. GOLDSTEIN, Be e han uni o m app oxima ion
16

on closed se s by ha monic unc ions wi h singula i ies, P oc. London Ma h.
Soc. 60 (1990), 319-343.
5. D.H. ARMITAGE and M. GOLDSTEIN, Nonuniqueness o he Radon ans-
o m, P oc. Ame . Ma h. Soc. 117 (1993), 175-178.
6. D. GAIER, Lec u es on complex app oxima ion, Bi kh¨ause , Bos on, 1987.
7. P.M. GAUTHIER, Uni o m app oxima ion, 1-37, in Complex Po en ial Theo y,
Kluwe Academic Publishe s, 1994.
8. S. HELGASON, The Radon ans o m, Bi kh¨ause , Bos on, 1980.
9. A.S.B. HOLLAND, In oduc ion o he heo y o en i e unc ions, Academic
P ess, New Yo k and London, 1973.
10. G. MITTAG-LEFFLER, Su la ep ´esen a ion analy ique d’une b anche uni-
o me d’une onc ion monog`ene (Sixi`eme no e), Ac a Ma h. 42 (1920), 285-308.
11. D.J. NEWMAN, An en i e unc ion bounded in e e y di ec ion, Ame . Ma h.
Mon hly 83 (1976), 192-193.
12. A.R. REDDY, On en i e Di ichle se ies o in ini e o de , Re . Ma . Hisp.-
Ame . 27 (1967), 120-131.
13. J.P. ROSAY and W. RUDIN, A akelian’s app oxima ion heo em, Ame . Ma h.
Mon hly 96 (1989), 432-434.
14. D. SATO, On he a e o g ow h o en i e unc ions o as g ow h, Bull. Ame .
Ma h. Soc. 69 (1963), 410-414.
15. W. SIERPINSKI, Su un p obl`eme conce nan les ensembles mesu ables supe -
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iciellemen , Fund. Ma h. 1(1920), 112-115.
16. L. ZALCMAN, Uniqueness and nonuniqueness o he Radon ans o m, Bull.
London Ma h. Soc. 14 (1982), 241-245.
DEPARTAMENTO DE AN´
ALISIS MATEM´
ATICO
FACULTAD DE MATEM´
ATICAS
AVENIDA REINA MERCEDES
APARTADO 1160
41080 SEVILLA, SPAIN
E-MAIL: lb[email p o ec ed]
18