376
F U N C T I O N S OF B O U N D E D C H A R A C T E R I S T I C
A N D L I N D E L Ö F I A N MAPS
By M A U R I C E H E I N S
1. Introduction
The study of the boundary behavior of analytic functions in a modern
sense begins with the fundamental and justly celebrated memoir of
FatouE2]. An essential feature of that paper is the exploitation to the
fullest of the then novel ideas of Lebesgue. The half century which has
elapsed since Fatou's memoir has witnessed the flowering of a large and
important chapter of analysis whose concern is the boundary behavior
of analytic functions. Our attention will be centered on questions whose
parentage may be traced back to the results of Fatou.
Let us recall that the original theorem of Fatou states that a bounded
analytic function/in the open unit disk possesses a radial limit p.p. and
that the theorem of F. and M. Riesz[13] states that the radial limit function of/ vanishes only on a set of measure zero, if/ =£ 0. The functions
of bounded Nevanlinna characteristic in the open unit disk are precisely
the non-constant meromorphic functions which admit representation
as quotients of bounded analytic functions. Thanks to the original
Fatou theorem and the theorem of F. and M. Riesz, the conclusion of
the Fatou theorem persists for functions of bounded characteristic.
When we turn to the study of asymptotic values, wefindthat a marked
contrast appears. For a bounded non-constant analytic function in the
open unit disk, each asymptotic path tends to a point of the unit circumference and the associated asymptotic value is the Fatou radial
limit at this point. On the other hand, while it is still true for a function
of bounded characteristic that each asymptotic path tends to a point
of the unit circumference, there exist, as examples of LohwaterC9] and
Lehto[8] show, functions of bounded characteristic having more than
one asymptotic value associated with a given boundary point. A recently
announced theorem of GehringE4] asserts that an analytic (pole-free)
function of bounded characteristic in the open unit disk admits at most
two distinct finite asymptotic values associated with a given boundary
point. We shall see later that the situation changes radically for unrestricted functions of bounded characteristic. In fact, there exists a
quotient of two Blaschke products whose zeros cluster solely at z = 1
which has the property that the set of asymptotic values associated with
F U N C T I O N S OF B O U N D E D C H A R A C T E R I S T I C
377
z = 1 has the power of the continuum.f In the construction which we
give of such a function, a decisive role is played by Valiron's example[15]
of a meromorphic function of finite order whose set of asymptotic values
has the power of the continuum.
However, the realm of pathology is small. In fact, for an arbitrary
non-constant meromorphic function/ in {\z\ < 1}, the set of points of
{|z| = 1} with which more than one asymptotic spot[5] is associated is
countable. Examples of functions of bounded characteristic where such
sets are infinite are readily constructed. We understand that a point TJ
of {|z| = 1} is associated with an asymptotic spot cr of / provided that
{^} = Ç]cr(ù)), a) € domain of cr.
The notion of a function of bounded characteristic may be generalized to
conformai maps of Riemann surfaces [6>12]. The Nevanlinna characteristic
function maybe adapted to conformai maps of Riemann surfaces in such
a manner that the first fundamental theorem and the basic theorems of
the Nevanlinna theory associated with it persist in the general theory
of conformai maps of Riemann surfaces. The notion of a map of bounded
characteristic becomes an important element of the extended theory.
In the present paper we shall confine our attention to Lindelöfian
conformai maps ( = maps of bounded characteristic) whose domain is
the open unit disk (meromorphic functions of bounded characteristic
are included) and shall apply the methods developed in our papers C5]
and [6] tö problems concerning the boundary behavior of such maps and
the relation between their boundary behavior and their 'covering
properties'4 By confining our attention to Lindelöfian maps with
domain the open unit disk we put at our disposal an extensive apparatus
from real function theory. This fact permits us to expect more precise
results than one could for arbitrary Lindelöfian maps.
2. Non-negative harmonic functions
Parreau[11] has introduced the notions of a quasi-bounded non-negative
harmonic function and of a singular non-negative harmonic function.
f The author is indebted to Professor F . W. Gehring for pointing out the fact t h a t he
carried out a construction along the lines of the second paragraph of § 10 of the present
paper (F. W. Gehring, The asymptotic values for analytic functions with bounded
characteristic, Oxford Quart. J. Math. 1958). His construction shows the existence of
a function / of bounded characteristic in {\z\ < 1} having the property t h a t the set of
asymptotic values of/ associated with paths terminating at z = 1 has the power of the
continuum. However, he does not establish the more refined result to which the present
footnote refers.
J For the case of meromorphic functions of bounded characteristic, cf. Lehto [7»83.
I am indebted to Professor Pfluger for raising the question of extending the Fatou
theorem to Lindelöfian maps with domain the open unit disk.
378
MAURICE HEINS
These may be defined as follows. A non-negative harmonic function u
on a Riemann surface F is termed quasi-bounded provided t h a t it is the
limit of a monotone non-decreasing sequence of bounded non-negative
harmonic functions on F; u is termed singular provided t h a t the only
non-negative bounded harmonic function on F which is dominated by
u is zero.f A non-negative harmonic function on F admits a unique
representation as a sum of a quasi-bounded non-negative harmonic
function on F and a singular non-negative harmonic function on F.
A positive harmonic function u on F is termed minimal (Martin [10] )
provided t h a t the positive harmonic functions on F dominated by u are
proportional to u. If a minimal positive harmonic function is not
bounded, it is singular.
fi-maps. Let D denote a non-empty open subset of a Riemann surface
F which has the property t h a t each point of fr O is a point of a continuum contained in fr O. Given a non-negative harmonic function u
on O which vanishes continuously on fr Q., by /IQ(U) is meant the least
harmonic majorant of the subharmonic function on F which agrees
with u on CI and vanishes elsewhere on F, provided t h a t the subharmonic
functionin question admits a harmonic majorant; otherwise /iQ(u) = +00.
Let Qa denote the set of u for which /IQ(U) < + 00. The restriction of
fin to Qa is a univalent homogeneous additive map of Qa into the set of
non-negative harmonic functions on F. Further u (eQo) is quasibounded (singular) on O if and only if /ia (u) is quasi-bounded (singular)
on F. If O is a region, then u (€ Qa) is minimal on D if and only if
[içi(u) is minimal on F. By a /£-map we shall understand the restriction
of /IQ to Qa for some admitted Q.
3 . T h e Lindelöf p r i n c i p l e
Let 0 denote a conformai map (not necessarily univalent) of a Riemann
surface F into a Riemann surface G. Let n(p; <fi) denote the multiplicity
of <ß&tp € F. Let v^(q) denote the valence of (j) at q e G,i.e.H^p)=Qn(p; ç5).
Suppose now t h a t F and G are hyperbolic and t h a t ®F and &G are
their respective Green's functions. Let
S(p, q) = S ^ ^ n f o ci) &F(p, r).
The exact form of the Lindelöf principle asserts t h a t for each qeG,
®G^(P),q)
= S(p,q) + ua(p)
(peF),
(3.1)
where uq is a non-negative harmonic function on F. Obviously uq is
f These concepts are also pertinent to non-negative harmonic functions whose
domains are non-empty open subsets of a Riemann surface.
FUNCTIONS OF BOUNDED CHARACTERISTIC
379
unique. We note that uq is the greatest harmonic minorant (abbreviated
henceforth by 'G.H.M.') of the superharmonic function £> -> &a(^>(p), q).
Let vq denote the quasi-bounded component of uq and let wq denote the
singular component of uq. We have the alternatives: vq = 0 for all q e G,
or vq > 0 for all q e G. In the first case we say that ci is a map of type-Bl
of F into G, the designation 'Bl' being employed because of the close
relation of such maps with Blaschke products. If uq = 0 for all q e G,
we say that ci is a map of type-Blx of F into G. We observe that regardless
of whether ci is of type-Bl or not, wq = 0 save for an Fff of capacity
zero[5].
Suppose that F and G are now unrestricted. We say that ci is of type-Bl
(Blt) at q e G provided that there exists a simply connected Jordan
region ù, qe Q, c: G, such that çi-1(£2) 4= 0 and the restriction of ci to
each component o) of ^ _ 1 (0) is a map of type-Bl (Bl^ of a> into O.
4. Lindelöfian maps
A conformai map (p of a hyperbolic Riemann surface J7 into a hyperbolic Riemann surface G satisfies
S(p,q) <+co
tf(p)*q).
(4.1)
Interest therefore attaches to the study of conformai maps ci: F ->G,
where F is hyperbolic but G is unrestricted, which satisfy (4.1). (It is
to be noted that in the definition of S(p, q) we need not assume that G is
hyperbolic.) Because of the genesis of this condition we term such maps
Lindelöfian. In terms of the extension of the Nevanlinna theory to
conformai maps of Riemann surfaces the Lindelöfian maps are precisely
the conformai maps of bounded characteristic.
5. Given distinct points a,b eG, there exists a function u on G which is
harmonic save at a and b, which has a normalized positive logarithmic
singularity at a and a normalized negative logarithmic singularity at b,
and which is bounded in the complement of some relatively compact
neighborhood of {a, b}. (If G is parabolic, then u is determined up to an
additive constant.) Given a Lindelöfian map <fi: F->G, it may be
concluded[6] from the extended form of Nevanlinna's first fundamental
theorem that uocj) admits a representation of the form
uo<f>(p) = S(p,a)-S(p,b)
+ H(p),
(5.1)
where H is the difference of two non-negative harmonic functions on F.
We shall see that this representation is the basis of the boundary
380
MAURICE HEINS
behavior theorems of the present paper. It is to be observed that H
admits a unique representation of the form P — N, where P and N are nonnegative harmonic functions on F satisfying G.H.M. min {P,iV} = 0.
There is an intimate connection between the harmonic function P and
the way in which ci covers a neighborhood of a.
In fact, let O0 = {u > 0}, let (oQ = çi_1(O0), and let çiWo denote the
restriction of ci to o)Q. If co0 4= 0, then, thanks to (5-1) and simple facts
concerning greatest harmonic minorants of superharmonic functions,
we are led to
/n TT -MT JL \ • T>
/* o\
(5.2)
Äo (G.H.M. u o & J = P.
If o)0 = 0, by convention we take the left-hand side of (5.2) to be zero;
it is readily verified that in this case we also have P = 0. It is now easy
to conclude that, if ß is an arbitrary Jordan region of G which contains a,
a) = çi-^O), and ® a is the Green's function of D, then
^{G.H.M.® Q (çi w ,a)}-P = 0(1).
(5.3)
Here ^ refers to the restriction of ci to a) and the obvious convention
prevails when o) = 0. From (5.3) we conclude that the singular component
of the first member of the left side of (5.3) is independent of £1 and is simply
the singular component of P. We denote it by Wa.
A situation in which (5.2) may be exploited advantageously is the
following. Suppose that O0 is connected and that the restriction of u
to Q0 is the Green's function of Q0 with pole at a. (This is certainly the
case if G is parabolic.) Suppose further that o)0 4= 0. Then P is singular if
and only if the restriction of ci to each component of o)Q is a map of typeBl of that component into O0. If P is singular, then by (5.2) G.H.M.
u o^Wo is singular, and ^ is readily seen to have the stated property.
The converse is also readily established on noting that a /*-map
carries a singular non-negative harmonic function into a singular nonnegative harmonic function.
To give a specific application of this result, we consider a function/of
bounded characteristic in {\z\ < 1} whose Fatou radial limits are p.p.
of modulus one. If/is bounded, a known theorem of Frostman[3] asserts
that for all a, |a| < 1, save for an Fa of zero capacity,
is a Blaschke product up to a constant factor of modulus one.f What
we shall now see is that, if neither / nor l//is bounded, then (5.4) is a
f The property of wq cited in § 3 constitutes a generalization of Frostman's theorem.
FUNCTIONS OF BOUNDED CHARACTERISTIC
381
quotient of two Blaschke products up to a constant factor of modulus one,
for alia, |a| < 1, save for an F^ of zero capacity. We remark that the case
where l//is bounded is readily reduced to the case where/is bounded.
To establish our assertion we note that with
a = 0, b = oo, u(z) = —log \z\,
P and N are both singular by virtue of the hypotheses on /. Hence the
restriction of / to a component of / - 1 {M < *} ( o r /""KM > 1}) *s °^
type-Bl relative to {|^| < 1} (or {|^| > 1} respectively). It now follows
from the property of wq cited in § 3 and (5.2) that save for an exceptional
set of oc of the described type, the P and N associated with
log
1-5/
/-*
vanish. The assertion is readily established.
This result admits generalizations to conformai maps of Riemann
surfaces. We shall not pursue this question further here.
6. The Fatou theorem
We shall now see that the Fatou theorem persists for Lindelöfian
maps <fi: F -> G, F = {\z\ < ï}, in the sense that for almost all £ on the
unit circumference, either ci tends to a point of G as z tends to £ sectorially,
or else ci tends to the ideal boundary of G as z tends to £ sectorially.
We consider u o <p of § 5 having fixed distinct points a,b eG. There
exists a meromorphic function / i n {| z | < 1} satisfying
log\f\=uo<f>.
(6.1)
By (5.1)/is of bounded characteristic, so that the Fatou theorem holds
for/. Suppose that / h a s a sectorial limit at TJ, \ TJ | = 1. Suppose that for
some S, 0 < S < \TT, it is not the case that as z tends to 7] in
^ = {|arg(l-^)|<£},
<f)(z) tends to a point of G or to the ideal boundary. There would then
exist a point qeG,a, relatively compact open disk A of G containing q,
and sequences (z'n) and (z"n) whose members he in Ss and which satisfy
not only
_. , .. „
h.mzn = iimzn = 7],
but also
and
çHOeA,
çi(4)€0-A
lim $(z'n) = q.
(aU n),
382
MAURICE HEINS
Let g denote a meromorphic function in A for which log |gr| is the
restriction of u to A. On considering / on the segments z'nz,rn we are led
to conclude that g is constant. This is of course impossible. The extended
Fatou theorem follows.
7. Criteria for sectorial and quasi-sectorial limits
We now turn to the examination of sufficient conditions for a point
G to be a sectorial (or a quasi-sectorialf ) limit of <fi.
(a) Suppose that (j) has a logarithmic ramification over a eG. Then a is
a sectorial limit of <fi.
More formally, we suppose that the following conditions are fulfilled:
(1) for each simply connected Jordan region Q, of G which contains a,
^~1(D) 4= 0, (2) there exists a function cr whose domain consists of the
set of such O and which satisfies
(i) or(O) is a component of çi _1 (n),
(ii) Ox c Q2 implies cr^-J <= <r(Q2)>
(iii) for Q, sufficiently small, (ß^Q), the restriction of ci to cr(£l), is a
universal covering of Q — {a}.
Thanks to these conditions, it follows that, for Ù small, ©^(Cì^Q), a)
is a minimal positive harmonic function on o"(Q). From (5.3) it may be
concluded that
/V(Q)[@W?W «)] < + oo.
(7.1)
For Q. sufficiently small, the left side of (7.1) is a minimal positive harmonic function in {j z | < 1} independent of £1, say
c«
[J=i] <°>°.M-i>.
It follows from the Julia-Carathéodory theorem that for given S,
0 < S < \TT, and for each Q, the points of the sector Ss which are sufficiently close to 7] he in cr(O). We infer that a is the sectorial limit of ci at TJ.
The example
f(z) = exp Jexp \rz^j\
(N < X)>
shows that there exists an analytic function in {\z\ < 1} which possesses
a sectorial limit save at 1 and has infinitely many logarithmic ramifications over 0 and yet does not possess the sectorial limit 0. This shows
that the Lindelöfian character of ci is pertinent to the question considered.
f This notion will be denned below in the present section.
FUNCTIONS OF BOUNDED CHARACTERISTIC
383
Quasi-sectorial limit. We shall say that $ has the quasi-sectorial limit
a ( e G) at TJ provided that
lim4>[r](l-rei0)] = a (\6\ < In),
(7.2)
save for a set of 6 of zero capacity. We have:
(b)V
ITa(2)>c8t[|±£],
where c is a positive constant, then ci possesses the quasi-sectorial limit aatrj.
In fact, it follows from (5.1) that
«0(3» > c^^-{S{z,b)
+ N{z)).
Now N does not dominate a positive multiple of ?H[(T] + Z)I(TJ — Z)] in
{\z\ < 1}. It follows from the theorem of Ahlfors and Heins[1] that
]miu[(^(T] — T]reie)] = +oo,
|0| < \TT,
r->0
save for a set of 6 of zero capacity. Hence (j) has the quasi-sectorial limit
a at T).
Examples of this phenomenon are readily constructed. We cite the
example given by LehtoC8]
/(Z) = exp{i±|J6(Z) (H<1),
(7.3)
where 6 is a convergent Blaschke product whose zeros are real and
cluster solely at z = 1.
It is to be observed that the existence of a quasi-sectorial limit of a
Lindelöfian map which is not actually a sectorial limit imposes very
severe restrictions. In fact, if (7.2) holds even p.p. for \6\ < \n, but a is
not the sectorial limit of ci at TJ, then G is conformally equivalent to the
extended plane, and for each b ( 4= a) e G, for some sector
Sd = {T,(l-reM)\\d\
<£<|7r},
^_1({6}) n Ss clusters at TJ.
8. What information can be drawn from the non-vanishing of Waì We
shall see:
(i) / / G is compact, Wa > 0, and v^(b) < +oo for some b(e G) 4= a,
then a is a radial limit of (j).
384
MAURICE
HEINS
(ii) / / G is not compact and Wa > 0, then a is a radial limit of çi.f
A special instance of (i) is to be found in Lehto[8]. We shall revert to
this result of Lehto below.
Apart from the use of auxiliary harmonic functions on G with assigned
singularities and boundary behavior, and the study of the composition
of such functions with (j), the proof rests principally on the important
extension due to Saks[14] of the de la Vallée Poussin decomposition
theorem to the case of an unrestricted function of bounded variation of
one real variable.
Assertion (i) may be established as follows. We start with the representation (5.1) and note that our hypothesis on 6 renders S(z,b) innocuous near the unit circumference. Let
^jr*^]^
be a Poisson-Stieltjes representation for H(z), where ju, is a function of
bounded variation on each finite interval and satisfies
/t(0) = i M 0 + )+/t(0-)],
/i(d + 27T) = /i(6) + [fi(27r)-/i(0)]. •
We assert that fi'(d) = 4-00 for some 6. If this were not the case, then by
the de la Vallée Poussin-Saks decomposition theorem, we would be led
to conclude that H is of the form Q — TJ, where Q is a quasi-bounded nonnegative harmonic function and TJ is a non-negative harmonic function
in {\z\ < 1}. But Wa ^ P < Q so that Wa = 0. This is impossible. Hence
for some TJ, \TJ\ = 1, iimH(rT)) = +00. We conclude that limçi(n/) = a.
r->l
r->l
Part (ii) is similarly treated; the non-compactness of G serves as a
substitute for the finite valence hypothesis. The only change that is
called for is to replace u by a harmonic function onff-{a} which has
a normalized positive logarithmic singularity at a and is bounded above
outside of each neighborhood of a, and to note that (5.1) persists save
for the absence of — S(p, b).
The theorem of Lehto to which reference has been made asserts that
a meromorphic function of bounded characteristic in {\z\ < 1}, whose
image is dense in the extended plane, omits at most one value outside
of the closure of the set of its radial limits. An examination of the proof
given by Lehto shows that it suffices, as far as Lehto's theorem is
f Can * radial ' be replaced by ' sectorial' ? The fact t h a t we are concerned with PoissonStieltjes integrals a t points where the generating function has an infinite derivative
renders the 'radial' easy; however, the argument used for the existence of a sectorial
limit of a Poisson-Stieltjes integral a t a point where the generating function has a
finite derivative does not appear to be available in the infinite case.
FUNCTIONS OF BOUNDED CHARACTERISTIC
385
concerned, to appeal to the original de la Vallée Poussin decomposition
for continuous functions of bounded variation rather than to the general
decomposition theorem.
The result of Lehto may be extended to Lindelöfian maps with domain
{\z\ < 1} and may be given $ slightly more refined formulation. Let R
denote the closure of the set of radial Kmits(€Ö) of ci. By(5.2)ifge G — R,
then either q belongs to the complement of the closure of the image
of (j) or else ci is of type-Bl at q. Let B denote the set of points in G — R
at which (j) is of type-Bl.
/ / G is not compact, then $ is of type-Bl± at each point of B. This is a
consequence of (ii) of the present section. Thus, in particular, on each
component of B, v^ is constant (finite or not).
If G is compact, two possibilities may occur. The first m that v$ is
identically infinite on G. There are examples of such ^ which are not of
type-Blx at several points of B (e.g./(z)//( — z) for/of (7.3); thisfunction
is not of type-Bli at 0 and oo which are both points of B). The second is
that for some point q0 e G, we have ^(g 0 ) <• + oo. Thanks to (i) of the
present section, it follows that if qQeG — B, then <p is of type-Blx at each
point of B; if q0e B, then $ is of type-Blx at each point of B — {q0} and in
this case if <f> is not of type-Bl± at q0, v^ is infinite at each point ofG — {q0).
For this latter phenomenon cf. (7.3).
9. From this point on we shall be concerned with applying the methods
described in this paper to problems concerning functions of bounded
characteristic.
We first consider a refinement of the theorem of Gehring which we
quoted earlier. We note that an asymptotic spot[5i of a function of bounded
characteristic in {\z\ < 1} has the property that precisely one point of
{\z\ = 1} adheres to all cr(o)), co e domain of or. Let A(TJ) denote the
number of distinct asymptotic spots cr off over finite points which have
the property that {TJ} = D or(a>); let B(TJ) similarly denote the number of
distinct asymptotic spots cr of / over oo which have the property that
{TJ} = PI cr(o)). The following theorem holds:
For an analytic function f of bounded characteristic in {\z\ < 1},
A(TJ) < 2 and B(TJ) < 3, \TJ\ = 1. There exists an f for which A(l) = 2 and
J5(l) = 3.
We note that, if A(TJ) ^ 3, then thanks to the classical results of
Lindelöf which are employed in the proof of the Denjoy-CarlemanAhlfors theorem and to (5.2), u(z) being taken as log \z\, we would infer
25
TP
386
MAURICE HEINS
the existence of two distinct asymptotic spots cr1 and cr2 over oo satisfying {TJ} = fi crk(o)), k = 1,2, and such that for some a) in their common
domain not only is it the case that o"x(w) n cr2(o)) == 0, but also
where ck> 0, ß = 1,2. This is impossible since /e-maps carry two admitted
positive harmonic functions with disjoint domains into two positive
harmonic functions the minimum of which has a vanishing greatest
harmonic minorant. Hence A(TJ) < 2. The proof that B(TJ) < 3 is similar;
it differs only in preliminary details.
To construct an/with -4(1) = 2, B(l) = 3, we may proceed as follows.
Let F(z) =
t^siatdt,
gz > — 1. Now the positive and negative real
Jo
axes are asymptotic paths of F and the corresponding asymptotic values
are finite and distinct; the positive imaginary axis is an asymptotic
path of F and the associated asymptotic value is oo. The function
g{z) =
F
( ^
(\Z\<1)
is of bounded characteristic.
Next we observe that there exists a function of bounded characteristic
in {\z\ < 1}, say h, which has the following two properties: (1) B(l) = 2,
(2) h tends to zero as z -> 1 in A = {|z —2 -1 | < 2 -1 }. Such a function h
may be constructed as follows. We observe that a positive harmonic
function p in {\z\ < 1} which does not dominate c8i[(l + z)/(l —z)] for
any positive constant c may be approximated uniformly on the intersection of A with the open unit disk by a function of the form — log \b\,
where 6 is a Blaschke product whose zeros cluster solely at z = 1. If we
choose p tending to infinity as z -> 1 (in the open unit disk), let w denote
an analytic function satisfying log \w\ = \p, and let b be as above with
p 4-log |6| = 0(1) on the intersection of A with the open unit disk, then
wb is an admissible h.
It suffices now to t a k e / = g+wb.
10. The striking contrast between the analytic and meromorphic cases,
in so far as asymptotic values of functions of bounded characteristic are
concerned, is brought out by the following example of a function f of
bounded characteristic in {\z\ < 1} which is the quotient of two Blaschke
products whose zeros cluster solely at 1, and which has the property that the
FUNCTIONS OF BOUNDED CHARACTERISTIC
387
set of asymptotic values of f associated with z = 1 has the power of the
continuum.
The construction is based on Vahron's exampleC15] of an even meromorphic function g in the finite plane, which satisfies T(r; g) = 0(r), and
has the property that there is a non-countable subset (£ of [0, n] such
that, for 6 e @, lim g(reie) exists and is finite, and 6 -> Mm g(reie) is unir~»»oo
valent on @. Let a,ß e @, where a < ß < n and a condensation point
of (£ lies in (à, ß), and let h denote the restriction of g to {a < arg z < /?}.
Thanks to the fact that T(r; g) = 0(r), A is a Lindelöfian meromorphic
function.
There exists a finite point w distinct from iim g(reict) and limg(re^)
which is a condensation point of
{iimg(rei6)\a < d < ß, d e $}.
It follows that there exists an open circular disk A centered at w such that
one of the components £1 of A-^A) has the following properties: (1) every
finite frontier point of £1 Mes in {a < arg z < ß) and fr £l is regular analytic
at each such point, (2) the set of asymptotic values in A of h restricted
to £i is not countable (and consequently has the power of the continuum
since it is an analytic set). Let p denote the radius of A.
Let T denote the component of fr O (relative to the finite plane) which
separates 0 from O, and let Q* denote the component of the complement
of T with respect to the finite plane which contains £1. Then Ü* is simply
connected and its frontier (relative to the extended plane) is a closed
Jordan curve. Let $ denote a univalent conformai map of {|z| < 1}
onto O* which 'carries' 1 into oo. We introduce
fi = P~1\go(p-w]
and note that/! is of bounded characteristic and further has the property
that it possesses a limit of modulus one at each point of {|z| = 1} other
than 1. Further, the set of distinct asymptotic values of f± associated
with z = 1 has the power of the continuum. It follows from § 5 that there
exist complex numbers oc, e, \a\ < 1, |e| = 1 such that
is a quotient of Blaschke products. Clearly / satisfies all the imposed
requirements.
25-2
388
MAURICE HEINS
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