The star function for meromorphic functions of several complex variables

0.14 MB·

by  Faruk Abi-Khuzam

THE STAR FUNCTION FOR MEROMORPHIC FUNCTIONS OF
SEVERAL COMPLEX VARIABLES
FARUKABI-KHUZAM,FLORIANBERTRANDANDGIUSEPPEDELLASALA
7
1
0 ABSTRACT. We define an analogueof the Baernsteinstar functionfor a meromorphicfunctionf
2 in several complex variables. This function is subharmonic on the upper half-plane and encodes

n someofthemainfunctionalsattachedtof. Wethencharacterizemeromorphicfunctionsadmitting
a aharmonicstarfunction.
J

1
3

INTRODUCTION
]
V
One aspect of the classical theory of meromorphic functions of finite order, is the search for
C
sharpasymptoticinequalitiesbetweencertainfunctionalsassociatedwithagivenfunctionf. Such
.
h
functionals include, among others, counting functions for a-values, the Nevanlinna characteristic
t
a orthemaximummodulus,denotedrespectivelyby
m
[ N(r,a;f),T(r,f),M(r;f).

1 Thereisavastbodyofliteratureon thoseinequalitiesnotablyforfunctionsoforderlessthanone.
v
Wenoteinparticularaunified approach to someofthoseinequaltiesthathas been presented byJ.
3
9 Rossi and A. Weitsmann in [14] using the theory of the Phragme´n-Lindelo¨f indicator along with
0 theBaernstein star functionof f. The star function,denoted by T (reiθ,f),was introduced by A.
∗
9
Baernstein [3, 4], and used successfully by him in severalproblems beginningwith thesettlement
0
. of Edrei’s spread conjecture [6]. The crowning achievement in the use of the star function by A.
1
0 Baernstein, was the proof that the Koebe function is extremal for the Lp norms of all functions in
7 thestandardclassS. AkeyingredientinBaernsteinproofswasthefactthatwhilethestarfunction
1
of a typical meromorphic function f is always subharmonic in the upper half-plane, that of the
:
v
extremalfunctionisharmonic.
i
X The problems and techniques above have been considered and extended for subharmonic func-
r tions in Rn by many authors (see for instance [5, 13, 14, 10]). In particular, A. Baernstein and B.
a
A. Taylor[5] introduced an analogueof thestar function in higherdimension. However,although
such approach is rather natural for the study of subharmonicor δ-subharmonicfunctions in Rn, it
does not seem that the star function introduced in [5] is well adapted to the distribution theory of
entire, meromorphic or plurisubharmonic functions in several complex variables. In this respect,
the first author had already suggested at least two possible definitions for a general star function
[2]inseveralcomplexvariables. Inthepresentwork,wefollowoneofthoseapproachesandintro-
ducethestarfunctionofameromorphicfunctionF inCn byaveragingovertheunitspherethestar
functions T (.,F ) of its “slices” F : C → C defined by F (z) = F(zζ). Our first concern is to
∗ ζ ζ ζ
studythecontinuousdependenceofT (.,F )ontheparameterζ (Theorem1). Inanalogywith[1],
∗ ζ
where the first author characterized all meromorphicfunctions admittinga harmonicstar function
2010MathematicsSubjectClassification. 32A20,32A22,32A60,32A30,30D35.
1
2 FARUKABI-KHUZAM,FLORIANBERTRANDANDGIUSEPPEDELLASALA
in one variable (see also [9]), we provide a similar characterization in several complex variables
(Theorem 2). As might be expected, new elements enter the picture in the several variables case.
In particular, it connects with the problem of determining a meromorphic function F : Cn → C
from the knowledge of zero sets of its ”slice” functions. We hope that our approach will allow to
extendtoseveralcomplexvariablessomeoftheknowninequalitiesinCandcarry overaprogram
similartotheonevariablecase. Thiswillbethefocusofforthcomingwork.
The paper is organized as follows. In Section 1, we study the continuity on the unit sphere of
T (.,F ) with respect to ζ which allows us, in particular, to define an analogue of the Baernstein
∗ ζ
starfunctionformeromorphicfunctionsinseveralcomplexvariables. InSection2,wecharacterize
meromorphicfunctionsadmittingaharmonicstarfunction.
1. STAR FUNCTION FOR MEROMORPHIC FUNCTION OF SEVERAL VARIABLES
We denote by ∆ = {z ∈ C | |z| < r} the disc in C centered atthe origin and of radius r > 0.
r
Wedenotetheupperhalf-planebyH = {z ∈ C | ℑmz > 0} andby S2n 1 theunitspherein Cn.
−
ConsiderameromorphicfunctionF : Cn → CsuchthatF(0) = 1. RecallthatF canbewritten
G
as F = where G and H are two coprime entire functions (see for instance Theorem 6.5.11 in
H
[11]). Definefor ζ ∈ S2n 1, thetraceofF onthecomplexline{zζ | z ∈ C}, F : C → C by
− ζ
F (z) = F(zζ).
ζ
For t > 0 and a ∈ C∪{∞}, let n(t,a;F ) be the numberofa-points of F in the closed disc∆ .
ζ ζ t
Fora ∈ {0,∞}and r ≥ 0,thecountingfunctionofF isdefined by
ζ
r n(t,a;F )
ζ
N(r,a;F ) = dt.
ζ
t
Z0
Notethataccording toJensen’sformula,onehas
1 π
(1.1) N(r,0;F )−N(r,∞;F ) = log|F(reiθζ)|dθ.
ζ ζ
2π
Z π
−
Forζ ∈ S2n 1, weconsidertheBaernstein starfunctionassociatedto F : C → C (see[3, 4])
− ζ
1
T (reiθ,F ) = sup log|F(reixζ)|dx+N(r,∞;F )
∗ ζ ζ
2π
E ZE
where reiθ ∈ H\{0} and where the sup is taken over all sets E ⊂ [−π,π] of Lebesgue measure
|E| = 2θ. We willwrite
1
F (reiθ) = sup log|F(reixζ)|dx.
ζ∗ 2π
E ZE
Notethat
T (r,F ) = N(r,∞;F )
∗ ζ ζ
and thatJensen’sformula(1.1)implies
T (−r,F ) = N(r,0;F ).
∗ ζ ζ
THESTARFUNCTIONFORMEROMORPHICFUNCTIONSOFSEVERALCOMPLEXVARIABLES 3
ThefundamentalresultofA. Baernstein states thatT (.,F ) issubharmoniconH and continuous
∗ ζ
on H\{0} [3, 4]; moreover,undertheassumptionthatF (0) = 1, T (.,F ) extendscontinuously
ζ ∗ ζ
on H. Our first main result is that for a fixed reiθ, r > 0, θ ∈ [0,π) the map ζ 7→ T (reiθ,F ) is
∗ ζ
continuousa.e. on thesphereS2n 1:
−
G
Theorem 1. Let F = : Cn → C bea meromorphicfunctionsatisfyingF(0) = 1,whereG and
H
H aretwo coprimeentirefunctions. Definethefollowingset
X = {ζ ∈ S2n 1 |G 1(0)∩H 1(0) 6= ∅}.
− −ζ ζ−
Then
i. Theset X hasLebesguemeasurezeroonS2n 1.
−
ii. Fora fixed reiθ,r > 0, θ ∈ [0,π),thefunctionζ 7→ F (reiθ) iscontinuousonS2n 1 \X.
ζ∗ −
iii. Fora fixed r > 0thefunctionζ 7→ N(r,∞;F ) iscontinuouson S2n 1 \X.
ζ −
InordertoproveTheorem1wefirstestablishtwolemmaswhichmaybeofindependentinterest.
FollowingA.Baernstein [4], weintroducethelevelsets
E(ζ,t) = {x ∈ [−π,π]| log|F(reixζ)| > t},
where ζ ∈ S2n 1, t ∈ R and r > 0. It follows from the proof of Proposition 1 in [4] that for any
−
ζ ∈ S2n 1 thereexistst(ζ) ∈ R suchthat
−
1
T (reiθ,F ) = log|F(reixζ)|dx+N(r,∞;F )
∗ ζ ζ
2π
ZE(ζ,t(ζ))
with|E(ζ,t(ζ))| = 2θ. Indeed,followingA.Baernstein’snotationsin[4],inourcasethedistribu-
tionfunctionλ(t) = |E(ζ,t)|ofF iscontinuoussinceeverylevelset ofF has measurezero and
ζ ζ
onecan takeE = A. It followsthat
1 π θt(ζ)
(1.2) T (reiθ,F ) = log+( F(reixζ)|−t(ζ) dx+ +N(r,∞;F ),
∗ ζ ζ
2π π
Z π
− (cid:12) (cid:1)
wherelog+(|F(reixζ)|−t(ζ)) = max{l(cid:12)og|F(reixζ)|−t(ζ),0}.
Lemma 1.1. The functionζ 7→ t(ζ)is continuousonS2n 1 \X.
−
G
Proof. Fix ζ ∈ S2n 1 \ X and ε > 0. Recall that F = where G and H are two coprime
0 −
H
entire functions. Denote by p = reixjζ ∈ Cn with x ∈ [−π,π], j = 1,··· ,N, the points such
j 0 j
that H(p ) = 0. Note that since ζ ∈ S2n 1 \ X then G(p ) 6= 0. There exists ε > 0 such that
j 0 − j ′
if Z ∈ ∪k B(p ,ε) then log|F(Z)| > t(ζ ) + 1. Here B(p ,ε) denotes the open ball centered
j=1 j ′ 0 j ′
at p and of radius ε. We then chose δ > 0 such that if |x − x | < δ for some j = 1,··· ,N
j ′ j
and kζ − ζ k < δ then reixζ ∈ B(p ,ε). Next we choose t large enough in such a way that if
0 j ′ ′
x ∈ E(ζ ,t)thenthereexists1 ≤ j ≤ N suchthat|x−x | < δ. Finallyweconsideracompactset
0 ′ j
K ⊂ [−π,π] avoiding the singularities of log|F(reixζ )| and containing E(ζ ,t(ζ )) \ E(ζ ,t).
0 0 0 0 ′
Thereexistsδ > 0 suchthat ifkζ −ζ k < δ then
′ 0 ′
sup|log|F(reixζ)|−log|F(reixζ )|| < ε.
0
x K
∈
4 FARUKABI-KHUZAM,FLORIANBERTRANDANDGIUSEPPEDELLASALA
Let x ∈ E(ζ ,t(ζ ))\E(ζ ,t). Thenlog|F(reixζ )| > t(ζ ) and so
0 0 0 ′ 0 0
(1.3) log|F(reixζ)| > log|F(reixζ )|−ε > t(ζ )−ε
0 0
wheneverkζ −ζ k < δ . Now let x ∈ E(ζ ,t). Then there is 1 ≤ j ≤ N such that |x−x | < δ.
0 ′ 0 ′ j
Ifkζ −ζ k < δ thenreixζ ∈ B(p ,ε)and therefore
0 j ′
(1.4) log|F(z)| > t(ζ )+1.
0
It followsfrom (1.3)and (1.4)that
E(ζ ,t(ζ )) ⊂ E(ζ,t(ζ )−ε).
0 0 0
whenever kζ − ζ k < min{δ,δ }. Since |E(ζ ,t(ζ ))| = |E(ζ,t(ζ))| = 2θ, this implies t(ζ) ≥
0 ′ 0 0
t(ζ )−ε. Bysymmetryweobtain|t(ζ)−t(ζ )| ≤ εifkζ−ζ k < min{δ,δ }. Thereforeζ 7→ t(ζ)
0 0 0 ′
iscontinuousonS2n 1 \X. (cid:3)
−
Lemma 1.2. Let H : Cn → C be an entire function and let r > 0. The function ζ 7→
π log|H(reixζ)|dxdefined onS2n 1 iscontinuous.
π −
−
PRroof. Let ζ ∈ S2n 1 and let ε > 0. If E ⊂ [−π,π] is a set then, following[8] and using Lemma
0 −
IIIin [7], wehaveforany ζ ∈ S2n 1
−
1 1
|log|H(reixζ)||dx ≤ m(r;H ,E)+m r; ,E
ζ
2π H
ZE (cid:18) ζ (cid:19)
1 1
≤ c T(2r,H )+T 2r, |E| 1+log+
ζ
H |E|
(cid:18) (cid:18) ζ(cid:19)(cid:19) (cid:18) (cid:19)
1
≤ 3cT(2r;H )|E| 1+log+
ζ
|E|
(cid:18) (cid:19)
1
≤ 3clogM(2r;H )|E| 1+log+
ζ
|E|
(cid:18) (cid:19)
1
≤ c(r)|E| 1+log+ = Ψ(r,E)
′
|E|
(cid:18) (cid:19)
wherec > 0 is aconstant,c(r) > 0is aconstantdependingonlyonr, andwhere
′
1
m(r;H ,E) = log+H (reix)dx.
ζ ζ
2π
ZE
Considernowt < 0with−t largeenoughsuchthat2πΨ(r,[−π,π]\E(ζ ,t )) < ε. Thereexists
0 0 0 0
δ > 0 suchthat ifkζ −ζ k < δ then
0
sup log|H(reixζ)|−log|H(reixζ )| < ε.
0
x E(ζ0,t0)
∈ (cid:12) (cid:12)
Set (cid:12) (cid:12)
π
I = log|H(reixζ)|−log|H(reixζ )|dx .
0
(cid:12)Z π (cid:12)
(cid:12) − (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
THESTARFUNCTIONFORMEROMORPHICFUNCTIONSOFSEVERALCOMPLEXVARIABLES 5
Forζ ∈ S2n 1 such thatkζ −ζ k < δ, wehave
− 0
I ≤ log|H(reixζ)|−log|H(reixζ )|dx
0
(cid:12)ZE(ζ0,t0) (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
(cid:12)+ log|H(reixζ)|−log|H(re(cid:12)ixζ )|dx
0
(cid:12)Z[ π,π] E(ζ0,t0) (cid:12)
(cid:12) − \ (cid:12)
(cid:12) (cid:12)
≤ (cid:12) log|H(reixζ)|−log|H(reixζ )| dx+ (cid:12) log|H(reixζ)| dx
0
ZE(ζ0,t0) Z[ π,π] E(ζ0,t0)
(cid:12) (cid:12) − \ (cid:12) (cid:12)
(cid:12) (cid:12) (cid:12) (cid:12)
+ log|H(reixζ )| dx
0
Z[ π,π] E(ζ0,t0)
− \ (cid:12) (cid:12)
(cid:12) (cid:12)
≤ ε+4πΨ(r,[−π,π]\E(ζ ,t )).
0 0
Thisprovesthecontinuityofζ 7→ π log|H(reixζ)|dxon thesphereS2n 1. (cid:3)
π −
−
WenowproveTheorem1. R
Proofof Theorem 1. We provei. LetZ ⊂ Cn betheindeterminacyset ofF,that is
(1.5) Z = {Z ∈ Cn | G(Z) = H(Z) = 0}.
BytheassumptionsonF,Z isacomplexanalyticsubvarietyofCn ofcomplexdimensionatmost
n − 2. Let τ : Cn → CPn 1 be the projection of Cn onto the projective space CPn 1. Note
− −
that the restriction τS2n−1 is a constant rank map S2n−1 → CPn−1; it is indeed a fibration - the
Hopf fibration - with| fiber S1. Therefore for any subset K ⊂ CPn 1 we have that the (2n − 2-
−
dimensional) Lebesgue measure of K vanishes if and only if the (2n−1-dimensional) Lebesgue
measure of the inverse imageτ 1 (K) ⊂ S2n 1 is zero. Since by definition X = τ 1 (τ(Z)),
−S2n−1 − −S2n−1
toprovei. it isenoughto show|that τ(Z) ⊂ CPn 1 has measure0. |
−
Since Z is a (n−2)-dimensional complex subvarietyof Cn, there exists a countable collection
{Zj}j N of locally closed, non-singular complex submanifolds of Cn, each one of dimension at
mostn∈−2,such thatZ = ∪j NZj. Fixedj ∈ N,considertherestrictionτ j : Zj → CPn−1. The
∈ |Z
map τ is smooth (and in fact analytic), and its rank at any point p of Z is less than n−1 since
j j
|Z
dimCZj ≤ n − 2, hence all p ∈ Zj are critical points of τ j. It follows by Sard’s theorem that
|Z
τ(Zj) has measurezero. Sinceτ(Z) ⊂ ∪j Nτ(Zj) weconcludethatτ(Z) hasmeasurezero.
∈
We now prove ii. We fix reiθ with r > 0 and θ ∈ [0,π). According to Equation (1.2) and
Lemma1.1 weonlyneed toshowthatthefunction
π
ζ 7→ log+ |F(reixζ)|−t(ζ) dx
Z π
−
defined on S2n \X iscontinuous. Letζ ∈ S(cid:0)2n \X and let ε >(cid:1)0. Set
0
π
J = log+ |F(reixζ)|−t(ζ) −log+ |F(reixζ )|−t(ζ ) dx .
0 0
(cid:12)Z π (cid:12)
(cid:12) − (cid:0) (cid:1) (cid:0) (cid:1) (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
6 FARUKABI-KHUZAM,FLORIANBERTRANDANDGIUSEPPEDELLASALA
Forζ ∈ S2n \X such thatkζ −ζ k < δ wehave
0
π
J ≤ log|F(reixζ)|−t(ζ) − log|F(reixζ )|−t(ζ ) dx
0 0
Z π
− (cid:12)(cid:0) (cid:1) (cid:0) (cid:1)(cid:12)
π (cid:12) (cid:12)
≤ log|G(reixζ)|−log|G(reixζ )| dx
0
Z π
− (cid:12) (cid:12)
(cid:12)π (cid:12) π
+ log|H(reixζ)|−log|H(reixζ )| dx+ |t(ζ)−t(ζ )|dx
0 0
Z π Z π
− (cid:12) (cid:12) −
(cid:12) (cid:12)
Thestatementii. nowfollowsfrom Lemma1.1 andLemma1.2.
Finallyiii. followsdirectlyfrom Lemma1.2 sincewhen ζ ∈ S2n \X wehave
N(r,∞;F ) = N(r,0;H )
ζ ζ
and byJensen formula(1.1)
1 π
N(r,0;H ) = log|H(reixζ)|dx.
ζ
2π
Z π
−
(cid:3)
NoticethatincasethesetX isempty,Theorem1impliesthat,forafixedreiθ,r > 0,θ ∈ [0,π),
the functions ζ 7→ F (reiθ) and ζ 7→ N(r,∞;F ) are continuous on S2n 1. This is in particular
ζ∗ ζ −
thecasewhenF : Cn → Cisentire,ormeromorphicwithoutzeros. However,notethatingeneral
thefunctionζ 7→ N(r,∞;F ) may notbecontinuouson S2n 1:
ζ −
Example1. Considerthemeromorphicfunctionon C2 defined by
z −1
1
F(z ,z ) = .
1 2
z −1
2
Thenforanyr > 0,wehaveN(r,∞;F ) = 0forζ = 1 , 1 . Nowforζ = (r ,s ) ∈ S2n 1
ζ0 0 √2 √2 k k k −
convergingtoζ0 wehave (cid:16) (cid:17)
2 n(t,∞;F )
N(2,∞;F ) = ζk dt = log2+logs
ζk t k
Z0
sincen(t,∞;F ) equals 0 for 0 < t < 1/s and 1 for t ≥ 1/s . It is interestingto noticethat the
ζk k k
functionζ 7→ F (2eiθ) iscontinuousatζ . Indeed itcan checked thatifr > s > 0we have
ζ∗ 0 k k
1 θ r eix −1
F (2eiθ) = log k dx
ζ∗k 2π s eix −1
Z θ (cid:12) k (cid:12)
− (cid:12) (cid:12)
and ifs > r > 0 then
k k (cid:12) (cid:12)
(cid:12) (cid:12)
1 π+θ r eix −1
F (2eiθ) = log k dx.
ζ∗k 2π s eix −1
Zπ θ (cid:12) k (cid:12)
− (cid:12) (cid:12)
In both cases Fζ∗k(2eiθ) → 1 as ζk → ζ0. However not(cid:12)(cid:12)e that the s(cid:12)(cid:12)et E(ζ) realizing the supremum
inF (2eiθ)does notdepend continuouslyon ζ.
ζ∗
THESTARFUNCTIONFORMEROMORPHICFUNCTIONSOFSEVERALCOMPLEXVARIABLES 7
For a fixed reiθ, r > 0, θ ∈ [0,π), since ζ 7→ T (reiθ,F ) is bounded, Theorem 1 shows in
∗ ζ
particular integrabilityof T (reiθ,F ) on the unit sphere and therefore allows us to define an ana-
∗ ζ
logue of the Baernstein star function associated to a meromorphic function F of several complex
variables.
Definition 1.1. Let F : Cn → C be a meromorphic function satisfying F(0) = 1. The star
functionofF is defined by
1
(1.6) T (reiθ,F) = T (reiθ,F )dσ(ζ),
∗ ∗ ζ
σ
2n 1 S2n−1
− Z
where reiθ ∈ H\{0} and dσ denotes the Lebesgue surface area measure of the S2n 1 and σ
− 2n 1
−
itsarea.
Remark1. InordertoshowtheintegrabilityofthecountingfunctionN,itisnotstrictlynecessary
to show its continuity. Indeed in case H : Cn → C is an entire function then according to
Jensen’s formula, for a fixed r > 0, the positive function ζ 7→ N(r,0,H ) is plurisubharmonic
ζ
(see Proposition I.14 in [12]) and therefore L1 on the unit sphere S2n 1 ⊂ Cn. Now, with respect
−
to the notations of Theorem 1, we have N(r,∞,F ) = N(r,0,H ) for ζ ∈ S2n 1 \ X and so
ζ ζ −
ζ 7→ N(r,∞,F ) isL1 onS2n 1.
ζ −
Fora ∈ {0,∞}weset
1
N(r,a;F) = N(r,a;F )dσ(ζ).
ζ
σ
2n 1 S2n−1
− Z
ThefunctionN(r,a;F) can alsobeexpressedas
r n(t,a;F)
(1.7) N(r,a;F) = dt.
t
Z0
1
where n(t,a;F) = n(t,a;F )dσ(ζ) is, for a = 0, the Lelong number of the zero set
ζ
σ
2n 1 S2n−1
ofF (see [12]forinstan−ce)Z. NoticethatsinceF(0) = 1
T (r,F) = N(r,∞;F),
∗
and
T (−r,F) = N(r,0;F).
∗
In the next proposition, we extend to several variables the main property of the star function
(1.6):
Proposition 1.1. Let F : Cn → C be a meromorphic function satisfying F(0) = 1. Then the
functionT (.,F)issubharmonicon H.
∗
8 FARUKABI-KHUZAM,FLORIANBERTRANDANDGIUSEPPEDELLASALA
Proof. Let z ∈ Hand letr > 0suchthatthecloseddisccenteredat z and radiusr isincludedin
0 0
H. Wehave
1 π 1 π 1
T (z +reiθ,F)dθ = T (z +reiθ,F )dσ(ζ)dθ
∗ 0 ∗ 0 ζ
2π 2π σ
Z−π Z−π 2n−1 ZS2n−1
1 1 π
= T (z +reiθ,F )dθdσ(ζ)
∗ 0 ζ
σ 2π
2n−1 ZS2n−1 Z−π
1
≥ T (z ,F )dσ(ζ)
∗ 0 ζ
σ
2n 1 S2n−1
− Z
= T (z ,F)
∗ 0
where the second equality follows from Theorem 1 and the inequality from the fact that the usual
Baernstein starfunctionissubharmonic. ThereforeT (.,F) issubharmoniconH. (cid:3)
∗
It is important to notice that the proof shows that T (.,F) is harmonic on H if and only if
∗
T (.,F ) isharmonicfora.e. ζ ∈ S2n 1. Thisfact willbeused intheproofofTheorem 2.
∗ ζ −
Now,Theorem 1and thecontinuityoftheusual Baernstein starfunctionimpliesdirectlythat:
Proposition 1.2. Let F : Cn → C be a meromorphic function satisfying F(0) = 1. Then the
functionT (.,F)iscontinuouson H.
∗
Notethat thecontinuityon {reiθ ∈ C | θ = π} andon {reiθ ∈ C| θ = 0} followsfrom (1.7).
2. ENTIRE FUNCTIONS OF SEVERAL VARIABLES WITH HARMONIC STAR FUNCTION
In the case of complex dimension one, as pointed out by A. Baernstein in [3], meromorphic
functionsofthekind
z z
f(z) = 1+ / 1−
r s
m (cid:18) m(cid:19) m (cid:18) m(cid:19)
Y Y
where r ,s > 0 for all integer m > 0 with 1 + 1 < ∞, admit a harmonic star
m m m rm m sm
function. In [1], the first author characterized all meromorphic functions with a harmonic star
P P
function;seealsotheworkofM.Esse´nandD.F.Sheain[9]forthecaseofmeromorphicfunctions
of zero genus. More precisely, it was proved in [1] that if f : C → C is a meromorphic function
satisfyingf(0) = 1andsuchthatitsstarfunctionisharmonicthenf canbewrittenf(z) = P(eiθz)
with
z z
(2.1) P(z) = eγz 1+ / 1− ,
r s
m (cid:18) m(cid:19) m (cid:18) m(cid:19)
Y Y
where θ ∈ R, γ ≥ 0 and r ,s > 0 for all m with 1 + 1 < ∞. From a geometric
m m m rm m sm
viewpoint, if the star function of f is harmonic then the zeros of f are distributed on one ray and
P P
itspoleson theoppositeray.
In this section, we characterise meromorphic functions F : Cn → C of several complex vari-
ables admittingaharmonicstarfunction.
THESTARFUNCTIONFORMEROMORPHICFUNCTIONSOFSEVERALCOMPLEXVARIABLES 9
Theorem 2. Let F be a meromorphic function on Cn with F(0) = 1. The star function T (.,F)
∗
is harmonic on H if and only if there exist a meromorphic function P : C → C of the form (2.1)
and a vector η = (η ,...,η ) ∈ Cn such that F(Z) = P(Z ·η) for all Z ∈ Cn, where we denote
1 n
Z ·η = z η +...+z η . In particular if T (.,F) is harmonicon H, then the indeterminacyset
1 1 n n ∗
ofF asdefined in (1.5)is emptyand forallζ ∈ S2n 1 thestarfunctionT (.,F )is harmonic.
− ∗ ζ
Remark 2. When F is nonconstant the function P is given by a (rescaled) restriction of F to the
complexline{z∂F(0) |z ∈ C},where ∂F(0) = ∂F (0),··· , ∂F (0) .
∂z1 ∂zn
(cid:16) (cid:17)
Wefirst establishthetwofollowinglemmas
Lemma 2.1. Let f : C → C bea meromorphicfunctionoftheform
eiθz eiθz
(2.2) f(z) = eγeiθz 1+ / 1−
r s
m (cid:18) m (cid:19) m (cid:18) m (cid:19)
Y Y
with θ ∈ R, γ ≥ 0, r ,s > 0for allmand
m m
1 1
+ < ∞.
r s
m m
m m
X X
Assumefurthermorethatf hasatleastonezeroor pole. Then
d
(2.3) f(k)(0) = d eikθ = k ·(f (0))k,
k dk ′
1
where d isreal forallk ≥ 0.
k
Proof. Forz smallenoughwehave
∞ (−1)k+1 eiθz k ∞ 1 eiθz k
logf(z) = γeiθz + +
k r k s
m k=1 (cid:18) m (cid:19) m k=1 (cid:18) m (cid:19)
XX XX
∞ (−1)k+1 1 eikθ
= γeiθz + + zk
rk sk k
k=1 m m m m!
X X X
∞ eikθ
= c(k) zk,
k
k=1
X
where
1 1
c(1) = γ + +
r s
m m
m m
X X
and fork ≥ 2
(−1)k+1 1
c(k) = + .
rk sk
m m m m
X X
Thistellsusthat fork ≥ 1
k!
Dklogf(z)| = c(k)eikθ
z=0
k
10 FARUKABI-KHUZAM,FLORIANBERTRANDANDGIUSEPPEDELLASALA
and sofork ≥ 0
f
Dk ′(0) = k!c(k +1)ei(k+1)θ.
f
1 1
Sincef(0) = 1andc(1) = γ+ + > 0,andsincewehaveatleastonezeroorpole,
r s
m m
m m
wehave X X
f (0) = c(1)eiθ 6= 0.
′
We set d = 1, d = c(1) ∈ R. We now proceed by induction. Having f(k)(0) = d eikθ with
0 1 k
d ∈ R for0 ≤ k ≤ m, wehave
k
m
m f
f(m+1)(0) = Dk ′(0)Dm kf(0)
−
k f
k=0(cid:18) (cid:19)
X
m
m
= k!c(k +1)ei(k+1)θd ei(m k)θ
m k −
k −
k=0(cid:18) (cid:19)
X
m
m
= k!c(k +1)d ei(m+1)θ,
m k
k − !
k=0(cid:18) (cid:19)
X
whichprovesthefirstequalityin (2.3). Thesecond equalityfollowsdirectly. (cid:3)
Lemma 2.2. Let F be a meromorphic function on Cn with F(0) = 1. Assume that its star func-
tion T (.,F) is harmonic on H. For any integer k > 0 let P be the polynomial giving the
∗ k
k-homogeneous part of the Taylor expansion of F at the point 0. Then there exists a sequence
{c } ofreal numberssuchthat
k k 2
≥
P = c (P )k
k k 1
forallk ≥ 2.
Proof. SinceT (.,F)isharmoniconH, thenbythedefinitionofT (.,F)andtheproofofPropo-
∗ ∗
sition 1.1, for a.e. ζ ∈ S2n 1, T (.,F ) is harmonic on H. Thus by Theorem 1 in [1], for a.e.
− ∗ ζ
ζ ∈ S2n 1,F has theform(2.2)inLemma2.1. So wehave
− ζ
F(k)(0) = d (ζ)eikθζ,
ζ k
with d (ζ) ∈ R for all k ≥ 0 for a.e. ζ ∈ S2n 1, and thus forall ζ ∈ S2n 1 by continuityof d (ζ).
k − − k
Forz ∈ C and α = (α ,··· ,α ) ∈ Cn \{0},define
1 n
F (z) = F(α z,··· ,α z).
α 1 n
Since Fα(z) = F α (kαkz), the function Fα has the form (2.2) of Lemma 2.1 and so Fα(k)(0) is a
kαk
real multipleof (F (0))k. In particular, notethat if α 7→ F (0) is identicallyequal to zero then F
α′ α′
mustbeidenticallyequal to1. ThehomogeneouspolynomialP is givenby
k
∂ F(0)
P (Z) = J ZJ,
k
J!
J =k
|X|