on nevanlinna’s second theorem and extensions
TRANSCRIPT
346
ON N E V A N L I N N A ' S SECOND THEOREM AND
by W. K. H A Y M A N (Exeter, England)
EXTENSIONS
INTRODUCTION.
1) Suppose that f ( z ) is meromorphic in [z] ~ R < oo. We denote by
n (5 f ) the number of poles of f (z) in J z [ < r counted with correct multiplicity,
and by n (r, f ) the corresponding number with multiple poles counted only once. We also write
-{ d t (1.1) m (5 f ) = n ( t , f ) t '
We use (1.1) instead of the more usual (i)
N (r, f ) = In (t , i) - - n (o, f)] t +
( r , / ) = n (t, D c1 t. t
n (o, f) log r,
to avoid possible infinities if f (z) has a pole very near the origin. The radii of
all circles occurring will be bounded, so that our results will not be affected.
We also define as usual
and
1 (2~ ,,, (~, i ) = W~; .to log+ I i (r /~ a 0
(1.2) T (r, f ) = m (r, f ) -+- N (r, f).
We shall be interested in the behaviour of f (z) in circles with variable cen-
ires in the second section. We accordingly define
(1.3) m (z o, r, f (z)) = m (r,f (z o --~ z))
ON NEYANLINNA'S SECOND THEOREM AND EXTENSIONS 347
with similar definitions for N, N, T etc. The expressions 1 / ( f - - a ) are well
defined when a is finite. When a is co we define them formally by putting f
instead of 1 / ( f - - co). Thus
(1.4) m (r, 1 / ( f - c~)) - - m (r, f) .
W e shall assume throughout, that the circle [ z - - Zo I < r moves in a fixed
boundel domain Do in which f (z) is meromorphic and that r remains greater than a
positive constant, so that zo lies well inside Do. By O (1) w e denote any term
which remains bounded as Zo, r vary subject to such conditions. The first and
second fundamental Theorems of Nevanlinna may now be written as follows
Theorem A(~). For any fixed finite a we have
(1.5) T (Zo, r, 1 / ( f - - a ) ) --- T (Zo, r, f ) - ~ - 0 (1).
Theorem B~3;. If at, a2,..., aq are q >~ 3 distinct complex numbers, (possibly infinite), then
(1.6) (q - - 2) T(zo, r, f ) ~ ~ N Zo, r , - - - -~-S (Zo, r, f) , V~I f ~ a v
where
(1.7) S (zo, r , f ) = m (zo, r, ~ ) -[- m (Zo, r, ~ ) -~-O (1),
and �9 (z) ---- II ( f (z) - - av), the product being taken over those v for which
a~ is finite. If now instead of [ z -- Zo I <~ r we take a sequence of circles] z - z. [ < r . ,
where r. increases in such a way that
T (z., r,, f ) -~ co,
then Theorem A shows that for every a
(1.8) m (z . , r., 1 ~ ( f - - a ) ) q_ N (z. , r., l / ( f - - a)) -~ 1. T (z., r., f ) T (z., r., f )
We also write following Nevanlinna(4)
- - N (Z,~, r,. 1 / ( f - a)) (1.9) 0 (a) - - 1 - . lim
T (z., r . , f )
348 w. r H A Y M A N
Then Theorem B together with (1.8), (1.9) gives the defect relation
q
(1.1o) o (a) _< 2,
provided that
(1.11) s (z,, r~,r 0. T (z,,, r~,f)
3) We shall obtain in this paper some new conditions under which (1.11)
and hence (1.10) holds. Our results will be based on a new estimate for
m(Zo, r,f--'f),(Theorem l), which allows t h e c i r c l e l z - - Z o [ < r t o m o v e r i g h t
up to the boundary of the domain of Do in which f ( z ) is meromorphic in certain cases. From this we obtain for instance that the defect relation holds for functions of unbounded characteristic in I z t < 1, which are meromorphic and of finite order in a larger domain, bounded by a finite number of analytic arcs which lie except possibly for their endpoints in [ z l > 1, (Theorem II). In particular we obtain a Theorem of Picard type for such functions.
By mapping the unit circle onto an angle we obtain
Theorem IlL Suppose that f (z) is meromorphic of finite order in the plane.
Let 1 ~_ P ~ oo and let z~ (a) - - r~ e i o, be the roots of the equation f (z) - - a, 2
lying in the angle I arg z [ ~ --.~z Then either 2p
(0 f (z) has bounded characteristic in I arg z [ < ~ In which case 2p
E COS p 0,, (2.1) r" p
converges for every a; or
(i i) f (z) has unbounded characteristic in [ arg z I < ~ in which case the 2p'
series (2.1) diverges for every a with at most 2 exceptions. In particular we see that if (2.1) diverges for a single value of a then f ( z )
takes every value with at most two exceptions infinitely often in the angle.
ON N E V A N L I N N A ' S SECOND THEOIIE/~I AND E X T E N S I O N S 349
7~
This result is not true for functions meromorphic merely in ] arg z F < - - , although 2,0
the condition that f ( z ) must be a meromorphic function of finite order could be relaxed considerably, it is also shown by an example that for any sequence z~ for which (2.1) converges and
E 1
for a finite k there exists a meromorphic function of finite order having f (z~) - - 0
and I f ( z ) I < 1 i n l a r g z [ < - - at least when p is an integer. 2p '
We next discuss the problem raised by BIoch (5) concerning the characteristic
of f (z) and f" (z). Some simple examples due to Littlewood (6) show that f (z ) may be bounded in ['z [ < 1 while f ' (z) has unbounded characteristic there. On
the other hand if f (z) is meromorphic of finite order in a domain Do satisfying the conditions of Theorem II, and has bounded characteristic in [ z [ < 1, then
all the derivatives of f ( z ) have bounded characteristic in I z i < 1 (Theorem IV).
Hence under the hypotheses of Theorem Ill, if (i) holds for f (z), then (i) holds
for all the derivatives of f (z) (Theorem V). The converse problem is left open.
3) In the second section of this paper we consider a function f ( z ) mero-
morphic of finite order in the unit circle. It is shown that if the circles
I z - - z ~ l <r,~ exhaust ] z [ < 1 properly i .e . so that
T (z,, r,, f ) (3.1) log (1/(1 - - r,~)) "~ oo,
then the defect relation (1.9) holds tl 'heorem VII). Two generalisations of Theorem
B due to Nevanlinna(:) and Milloux(s) are also discussed. If there exists a sequence of circles properly exhausting I z [ < 1, we say that the Riemann surface o f f ( z ) is properly exhaustible. This is shown to be the case (Theorem VI) whenever
(3.2) (1 - - r ) log M (r, f ) -~ c~, as r-~ 1,
where
(3.3) M (r , f ) : sup I f (z ) r. t z l = r
On the other hand to imply (3.1) with z , ~ : 0, the classical condition of 23 - R e n d , C i t e , M a t e m , P a l e r m o , - - s e r i e I I - t o m o I [ - a n n o I953
3 5 0 "W. IK, I t A Y M A N
Nevanlinna, we need the somewhat stronger condition
( 1 - - r) log 34 (r,f) ->- oo log (1/1 - - r)
on the maximum modulus. (3.2) cannot
f (z) -~- e 0+zV0-z) shows, for which I f (z)
Ahlfors(9) in a now classical paper
exhaustion from that of Nevanlinna. Let D
Riemann sphere. A region d contained with
be further weakened as the function
1 > 1 i n l z l < l . has developed a different theory of
be a simply connected domain on the
its frontier in I z - - Zo [ < r corresponds
to an island over D by f (z), if f (z) maps d p : 1 conformally onto D. We denote by
n (zo, r, D) the total number of such islands each counted with correct multiplicity p and
by n (zo, r, D) the corresponding number with each island counted once only.
Let DI, D~,..., Dq be nonoverlapping simply-connected domains on the Riemann
sphere. Then Ahlfors has proved the inequalities
Theorem A" Qo)
(3.4) n (Zo, r, Dr) < A (Zo, r) + C L (Zo, r);
Theorem B" (it)
(3.5) q
(q - 2) A (Zo, r) < ~ ~ (Zo, r, o , ) + c ' L (Zo, r);
analogous to Theorems A and B. Here ~ A (zo, r), L (Zo, r) are respectively the area and
length of the frontier of the image by f ( z ) of T z - - zr ! < r on the Riemann
sphere. The constants C, C' depend on the Dv only.
If a sequence of circlesJ z - - z , ] < r, exists exhaust ingJ z [ < 1 in such
a way that
(3.6) L (z., r.) *. O, A (zo, r.)
then the Riemann surface of f (z) is said to be regularly exhaustible (regular aussch6pJbar) by Ahlfors. in this case we can obtain the Ahlfors Defect Relation (t~)
q
(3.7) ~ o CO,) -< 2,
ON NEVANi,INNA'S SECOND THEOREM AND EXTENSIONS 3 5 ]
where
n-(z,, r., D,.) (3.8) O (O,) - - 1 - - lira
. * - - A (z . ,r . )
Suppose in particular that each island over Dv has multiplicity at least i~v.
Then (3.4), (3.0)and (3 .8)show that t-, (D,)_>_ (I P',I ) s o that (3.7)gives the
Scheibensatz (t8)
q( ,) (3.9) 1 - W -< 2.
We shall show (Theorem VIII) that (3.0) holds for a sequence of circles
containing I z - z~ , [< r~ if 1
3.10) [(1 - - r;) 2 - - [ z;[~l --" A (z; , r;)--~ co.
The condition reduces to the one obtained by Ahlfors(14) if z , - - -0 . It is always
satisfied if the Riemann surface of f (z) is properly exhaustible in the sense of
3.1), so that the Scheibensatz holds in particular if (3.2) holds.
Our method also enables us to obtain good bounds for M (r, f ) in terms
, particularly when these tend to infinity slowly of the quantities N r, f - - a
as r-~-1. Thus Theorems Vl and VII imply for instance that if (3.2) holds, the
equations f ( z ) - - a have so many roots that
n r, d r - - - + c ~
except perhaps for 2 values of a. By a refinement of Ahlfors method(15) one
could even generalise the points a to domains D, using the Ahlfors notation.
Without going into this latter generalisation we shall prove the following rather precise theorem as an example of the method.
Theorem IX. Suppose that f (z) is meromorphic in I z ] < 1 and that p (p)
is the total number o f roots of the equations f (z) ~- av, v =- 1 to q, where the a,
3 5 2 w . x . H A ~ M A ~
are q ~ 3 distinct complex numbers one of which may be infinite. Then i f
(3.11) lira (1 - -r ) p (r) ~ b ,~ o%
we have
lira (1 - - r) log M (r, f) ~ 2 9,, ( 3 . 1 2 ) u -,~. I
log (1/1 - r )
where ~ : b/(q -- 1) or b/(q -- 2) according as the a~ are all finite or not. I f
f (z) is regular, lira may be replaced by lira in (3.12). Equality is possible in (3.12)
for every set of numbers a i to a~ and b < oo, for a regular function f ~z).
Using concentric circles (3.12) could only be proved with ~. -~ A replacing
~, where A is an absolute constant.(i6)
As a final application of our method we solve the following problem raised
by Littlewood in a group of research problems. Suppose that f(z), �9 (z) are
regular in I z ] < I and f (z), f (z) - �9 (z) have no zeros. Does
o [1) (3.13) log M (r, r - -
1 - - r always imply
(3.14) log MCr, f ) = 0 (1) ? l - - r
An affirmative answer was given in (4) Theorem V, which applied also to
meromorphic f (z) and �9 (z) but contained the assumption that
jl n (0, t, 1/O) d t < oo.
We can now eliminate this assumption and prove
Theorem X. Suppose c~ (z) ~ O, is regular of f inite order in I z i < 1 and
that f ( z ) is regular, and f ( z ) , f (z) - ~P (z) have no zeros there. Then we have
for 0 < r < p < 1,
log M (r, f ) < log M (p, O) -b
where K depends on f (z), ~P (z) but not on ~, r.
K p - - r '
ON NEVANLINNA S SECOND THEOREM AND EXTENSIONS 353
1 it is now clear that (3.13) implies (3.14) taking p ~ ~2- (l + r). Our method
would also enable us to prove a modified result if poles of f(z) and O(z)
and zeros of f (z), f (z) -- �9 (z) are allowed provided that their total number n (t) in I z l < t satisfies
i n (t) d t < oo.
For simplicity we restrict ourselves to the case quoted above.
S E C T I O N 1. 4) We start by proving the following fundamental result
Theorem L Suppose that f (z) is meromorphic in a bounded domain D containing
[ z I ~ R. Let dR (0) denote the distance of z ~ Re '~ from the boundary olD, and
nR (0) the total number of roots of the equations f (z) ~ 0,1, oo distant at least
1 - - d R (0)from the boundary of D. Then we have 2
( ) l/? ,o, , m R, ]'(')f (z)(Z) ._~ A (p) log+ f (Re '~ ) d 0 -l- I -l-- log+ -~ -1- 1 )
(4.1)
where
(4.2)
[ (7) ' ] ~_.A(p) log+ m (R , f ) q-- log+ m R, 1 -q--l--l-log+ ~ + 1
I - - I ( R ) 2 = J o log+ nn (o) + l o g + an(B) d0.
Here f(~> (z) is the p th derivative of f (z), and A (p) depends only on p. We need a series of lemmas.
L e m m a l . Suppose f (z) - - a o -q- a~ z -~- . . . is regular and f (z) ~ 0 or 1
in l z l < d; then we have
l I iJa. (4.3) <A(n) 1 + log Oo ao gP i
Write
(4.4) g (z) - - log f (Z) ~-- s b. Z". o
354 w. r . HAvr~Ar~
We may without loss in generality take d : 2. Then i f l f ( 0 ) l . < 1 we have
by Schottky's Theorem I f ( z ) I < A in I z [ < 1, and if I f ( 0 ) l ~ 1 we have
I [ f (z)]-~ J < A in I z ] < 1. Thus we have either
o r
Rl g (z) < A
- - B l g (z ) < A
in I z l < 1. In either case we deduce(t7)
(4.5) ]b.l~2(IRtbol--I-A)---2 [lioglaol]+A]~A(lloglao[l-q-1),
We have next from (4.4) f ' (z) - - g ' (z) f (z) or
/ / a n Z ' ~ - i - - " ;7 bn z n - i a n z " .
1
Equating coefficients of z ~-i we deduce the recurrence relation
p--1
p a , : ~ ( p - - r) b,_~ a,, r ~ o
and using (4.5) we deduce
p - - I
}a, I<_A ( I log I ao ]l-q- 1) ~-" l ar l . ~ o
Now (4.3) follows by induction on p.
We have next
L e m m a 2. Suppose f (z) is meromorphic in [ z ] ~ t? and let 8n (0) be the radius
of the largest circle centre R e '~ in which f (z) is regular and unequal to 0 or 1.(t8) Then
m(.1"''z'~<_ t f~ i �9
ON NEVANI.INNA'S SECOND THEOREM AND EXTENSIONS 3 5 5
By applying lemma 1 to f (Re ~o + z) instead of f (z) we obtain
log+ f/~) (Re,0) < log+ A (p) 1 + log f ( R e '~ )
1 + log+ log <_A(p)+p log+ ~(0)
.--J--1 +1].
Lemma 2 now follows by integration�9
We need to estimate further the
quote the results.
two integrals occurring in lemma 2. We
L e m m a 3.(1"). We have
1 tn d ~ .< log+ m ( t ? , y O + l o g + m ( R , f ) + A.
L e m m a 4(~~ With the notation of Theorem ! we have
log+ 8\ (0-3 �9 O
( 1 ) aO<_A /+ tou+ ~ + t .
Theorem 1 now follows from lemmas 2, 3, 4.
5) Theorem I can be extended to the case when f (z) is not meromorphic
at every point of [ z ] ---R, provided only that the domain D contains almost
all points of I z ! - - R and the behaviour of 3' (z) in D is such that the integral
I (R) converges�9 We proceed next to obtain a general condition for this to occur.
We shall say that D properly contains a set of arcs z --- Re ~~ ~ < 0 < ~3,
i f these arcs lie in D and we have uniformly on all the arcs
(5.1)
where C~, C2 are positive constants. In effect this means that on each arc dn (0)
is not much smaller than the distance of z : t?e'O from the nearest endpoint
of the arc.
356 w. t~. H^VMAN
We shall also say that f ( z ) has finite order in D if for each complex a the
number n (d, a) of roots of the equation f (z) -- a which are distant at least d from
the boundary of D satisfies
(5.2) n (d, a) = O (d -c.~)
as d-t~ 0, where C.,~ is independent of d but may depend on a. This agrees
with the usual definition if D is a circle (,"t). With these definitions we can prove
Theorem I1. Suppose D is a bounded domain containing I z I ~ t? and pro-
perly containinff a set of arcs z = Re ~~ % < 0 < ~ , where
(5.3) S, ( ~ , - %) = 2 ~,
1 (5.4) ~ ('~- %)log ( L - ~,) < ~
Suppose further that f (z) is meromorphic of finite order in D. Then i.l S (r, f )
denotes the term in (1.7) with Z o - - 0 we have
S (r , f ) - - 0 flog T(r , f ) ] q- 0 (1),
as r-~ 1? from below. In particular if f ( z ) has unbounded characteristic in
z I < R, the defect relation holds as r -~ R from below.
We need two more lemmas
L e m m a 5. Under the hypotheses of Theorem 1I, let 0 ~ r < R and let
d, (0) denote the distance of z - - r e'O from the nearest frontier point of D. Then
we have uniformly in v and r
"t~v l~ ld, (0----) ( I ) , / d o - - O ( ~ - - %) log+ - + 1 .
We shall denote by C~, C:, .... constants independent of v and r. Let z ' = R'e'0'
be the nearest point to z -~ re ~o on the frontier of D. Suppose that ~v < 0 .< ~i v
and that
1 m i n l O - - % , ~ , , - - o (5.5) 10 - 0'~ < ~- ~.
O N N E V A N L I N N A ' S S E C O N D T H E O R E M A N D E X T E N S I O N S 357
Then i t follows that
1 (0 - - ~v ) ( , 8 v - - 0) (o' - % ) ( ~ , - o') > - g
so that (5.1) gives
I R ' e ~~ - R e ~ O ' l = R " - -
Thus in this case, since
we have
(5.0)
d, tO) - - j z ' - z [ > _ R ' - r>_R'--R,
d~ (0) > C4 [(~,, ..... O) (0 - - ~,,)]c.,.
Suppose next that (5.5) is false. Then
z" - - z I - - t R " ei~ - - r e io I - - { R " ei(~176 - r l ~ R ' I sin (~' - - O) i --~> ~ [ O' - 0 I,
i f [ 0 - - 0 ' < - - . 7: Also i f - - = ~ 1 0 - - 0 ' [ ~ 7 : we have 2 2
[ z - z [ - - I R ' - - r e ~ c o - ~ ' ) [ _ > R ' - r c o s (0 - - 0') > R' _> R.
Thus in any ease
I z" - z t >- ~-to" - >- R_ rain (o - ~,, ;~, - (~) 2 n
t' (0 - =,)(}, -~)) . 4r~.
Thus in this case also (5.6) holds, possibly with different (74, C5. Thus (5.6) holds generally on all the arcs % < 0 < i3v. We have on integration
~v log+ l ~ d 0 < C:, log+ . . . . . 4- log+ �9 =~ d , ( 0 ) �9 = , L 0 - - % "r . . . . . . . .
dO
] . , a' V
= (}, - ~) [2 c~ log 1 ] ( ~ , - ~ , ) + 2 c ~ + c , .
3 5 8 W . K . H A Y M A N
This proves lemma 5. We have next
L e m m a 6, Suppose that D satisfies the conditions o[ Theorem II. Then
'/?[ '} l ( r ) : log+ n, (0) q-- log+ dO .2 .. a , io)
is uniformly bounaed for 0 .:_~,~ r < R. By condition (5.3) we have
Now
equations f (z) - - O, 1,
boundary of D we have
log 4- n, (0) = O ! log + , d , (0)
uniformly in r and e. Thus we obtain, using iemma 5,
i (r) : 2-~ : .. ~ d, (0)
since f ( z ) is of finite order and n, (0)is the number of roots of the
co within a distance not more than --1 d, (0) from the 2
+ 1!.
l ( r ) = O 1 -F log + d 0 ~V de
I ' )} = o ( ~ , - ~ , ) log , 1 = o ( t ) ,= , (~, - ~,v)
uniformly in 0 ~ . .-~ ,9 by (5.4). This proves lemma 6. We can now complete the proof of Theorem il. We have to show thai if
S (0, r , f ) is defined as in (1.7) then
(5.7) S (0, r, f ) - O 1 log T (r, f ) } --I- O t 1), as r -~ R.
To do this note that by Theorem 1 and lemma 6 we have as r-~./9
(5.8) m
ON NEVANLINNA'S SECOND THEOREM AND EXTENSIONS 3 5 9
Again
= < Z m r, -+-O(1), (5.9) m r, m r, = f - - a~ --- ~=l f Z a~
and the functions f ( z ) - a~ have finite order in D since f ( z ) does. We can thus apply (5.8) to f - av and obtain
(1,) _< log+ T ( r , f - - a,) -+- O (1) -~_ A log+ T(r , f ) -k- 0 (1). m r, f __ av
Thus (5.8), (5.9) yield
m + m A (q q- 1) [log+ T (r,)') -]- O (1)],
and now (5.7) follows from this and (1.7). It is also clear that if T ( r , f ) ~ a s r -~ R, then
S (0, r, f ) �9 .~ 0, T (r, y)
which is the relation (1.11), so that the defect relation (1.10) also holds. This completes the proof of Theorem 1I.
6) It may be worthwhile at this point to say a few words about the conditions on D in Theorem II. The condition (5.3) implies that D contains almost all points of [ z l = R. On the other hand (5.4) is not a metrical condition on the complement, since even the complement of a countable set need not satisfy (5.4). To see this, we first divide (0, 2 r~) into two equal arcs, then (=, 2 ~) into
( 2 2 equal arcs and more generally 2 ~ 2 n into 2 2~+1 2,~, 2~+i equal arcs, each
of length ~z/(2~"+l+"+~). Denoting these arcs by (~,, [3,) we have
1 _ _ rc 2 :~'~+~+'~+1
2'~+ ~ ~ log 2 __ - r c~.
On the other hand there exist Cantor sets of dimension 1 whose complementary arcs satisfy (5.4).
3 6 0 w . x . R A ~ M A r ~
Consider next the condition that the arcs (~. ~) are properly contained in D. This is always satisfied if D is bounded by a finite number of analytic arcs lying except for
their endpoints in [ z I > R. Let yv be such an arc having endpoints (Re i%, Rei~). Since y~ is analytic and does not coincide with an arc of ] z [ - - R, y~ must
have contact of finite order n at most with [ z l - /? at these endpoints and
so we can find a region of the type
z - r d ~ < R + . (0 - %). ( ~ - 0)',, ~ < 0 < ~ ,
contained in D. Also since there are only a finite number of arcs we may take ,, n the same for all of them. Thus condition (5.1) is satisfied. In the case of infinitely many arcs (5.1) assures that D extends at a uniform rate in some sense
across those arcs of [ z ! : R contained in it.
P r o o f o f T h e o r e m ill 7) We proceed to prove Theorem Iii, quoted in the
introduction. Put
(7.1) w z P -t- i \1 - - w/
so that the angle l a r g z J < ,-c corresponds t o ] ~v [ < 1. Then 2p
(7.2) / (z) = / [z (w)] -- g (w)
is meromorphic in the w plane cut from - - 1 to - - co and + 1 to -[- co, along the
real axis. We take for D the part of this cut plane lying in ] w[ < 2. Then D
contains the whole of ] ~v ] ~ 1 except w - - + 1 and if w -- e/~ we have
2 d ( w ) ~ ] s in0 ] > - - r a i n t ] 0 ] , [ ~ - - 0 ] , I ~ + 0 l t , i n - - ~ < 0 < %
~z
so that D properly contains the arcs - - ~ < 0 < 0 and 0 < 0 <~z, and
(5.3), (5.4) are satisfied. Suppose next that n (d, a) is the number of roots of the equation g (w) - - a
for which w is distant at least d from the boundary of D. The corresponding
w must then satisfy l w - - l [ ~ d
and since I w l < 2 w e deduce from (7.1) that the corresponding z satisfies ]
ON NEVAm.~r~;S SF.CO~. THEOn~M ~Nt, EXa~SmNS 361
Now the number of roots of f (z) = a in the circle (7.3) is O [d -(k+')/p] if f (z) has
finite order k in the plane and since each z corresponds to at most l/p -l- 1 values
of w in D we obtain n (d, a) ~--- O [d-(k+s//P], so that g (w) has finite order in D.
Thus g(w) satisfies the conditions of Theorem II and we deduce from that
Theorem that if w, (a) are the roots of g (w) = a in [ w [ < 1 then
~ ( 1 - - I w,,(a) I)
either converges for every a, if g (w) has finite characteristic in [ w] ,< 1
diverges for every a with at most two exceptions.
Put now z = r r Then
o r
so that
I w I~ -- ] zP-l_zp@. 1 ~---- r ~ p - 2 r p cos pO-q- 1
r 2P + 2 r P c o s p 0 4 - 1
1 - I w I ~ = 4 rP cos p 0
r ~p -t- 2 rP cos p 0 -+- 1
Also w,, runs over the roots of g (w) = a in [ w I < 1, while z runs over the
' ~ The convergence of ~ (1 - - I w~ [) roots r,, e 0f, of f (z) =: a in I arg z I < 2--~o"
4 r~ cos p 0 is equivalent to that of ~ ( , 1 - - i w , = [ ~ ) i .e . ~ 1 + r: ~ + 2 r P c o s p O '
cos p 0 since r,, ~- co . ]h i s proves Theorem I11. and hence of ~ r~
We next show thai if p is a positive integer and the points z~ = r e~~
2-,~,are such that (2.1) converges, then there exists a function f ( z )
meromorphic in the plane and bounded in I arg z I < =- such that f (z,,) - - 0. 2 ~
Consider in fact
z : - z , ~ n=, z~ + z~ \ z n / "
362 w. l~.. I-I A Y ~r A N
It is easily seen that the product is convergent in the plane to a meromorphic
function of order k at most provided that (2.1) converges and ~ I z, I -k converges.
Also if all the z,, and z lie in } arg z ] < ~ each factor of the product satisfies 2 ~
z~ + z~ ~.
Thus I / (z) l < 1, i. e. f(z) takes no value of modulus greater than 1 in
I arg z I ~ ~ . Thus for a meromorphic function of finite order in the plane, the 2 p
condition that (2.1) taken over the zeros of f (z) in 1 arg z [ < -~-- should diverge, 2t~
is the weakest condition on the zeros which will insure that f ( z ) satisfies a
Picard Theorem in the angle.
C h a r a c t e r i s t i c of f (z) a n d f ' (z). 8) A problem mentioned by Nevanlinna
as due to Bloch is the following (~). Suppose f ( z ) has bounded characteristic
in I z l < 1, is the same always true of f" (z)? Without further restrictions the
answer is no. Consider in fact
f (z) ~ a . . . . " Z ,
where a is a positive integer. These functions have been investigated by Little-
wood (~) and he has shown that i f - 1 ( c < 0, then f(z) is continuous and
s o bounded and of bounded characteristic in I z l --< I. On the other hand.
f ' (z) - - ~ a (i+~ z ~'. t t = l
Here 1 - ] - c > 0 by hypothesis, and in this case Littlewood showed that if a is large enough there exists a sequence ~,, < ~,,§ "~ 1 such that
I f ' (z) I > A (a, c) (1 - - p,,~-<'+~ I z I = ~,.
ON N E V A N I , I N N A ' S SECOND T H E O R E M AND EXTENSIONS 303
Thus
, I f " ~ [ , T (p,,, [ (z)) "-- - - - log + I (P- e '~ I a o 2 ~ J o
> tl + c) log | m [~n . . . . . . -~ O (1), as n-)- o%
and so / ' (z) does not have bounded characteristic in I z I < 1. We can however prove
Theorem IV. Suppose that f (z) satisfies the hypotheses of Theorem I!
and has bounded characteristic in [ z [ ~ R. Then all the derivatives of f (z)
also have bounded characteristic in [ z I < R. We note that if f ( z ) has a pole of order q at a point, /(~)(z) has a pole
of order p -~-q ~ (p -Jr- 1) q. Thus
N (r, I<~' (z)) ~ (p -F 1) N (r, f (z) - - O (l), as r-~./?,
by hypothesis. Again
f(') (z) m(r,f<'>(z))_<m(r,f)+m r, f i - ~ ]"
Here the first term on the right hand side is bounded by hypothesis as r - ~ ,q
from below. Also for the second term we have by Theorem I
( ,,<z> I ( ' ) f m r,-7~zj .)~- .4 (p) log + m (r, f ) - ] - l o g + m r , ~ - ~ - / ( r ) - - ~ O ( l ) ,
and here the right hand side remains as bounded as r -~ R by our hypotheses and
lemma 6. This proves Theorem IV.
We can deduce a corresponding result under the hypotheses of Theorem 11I. We have
Theorem V. Suppose that f (z) is meromorphic of finite order in the plane
and has bounded characteristic in I arg z I < - - . Then the same is true of 2 ~
all the derivatives of f (z).
364 w. K.. HA'gMAN
We again make use of the substitutions (7.1) and (7.2). Then since f (z) has bounded characteristic in j z l < 1, g I w) has bounded characteristic in
w I "< 1, and satisfies the conditions of Theorem IV. Thus g" (w) has bounded
characteristic in ] w l < 1. Now 1 + l
f ' (z) = g ' [w (z)] a w = g, (w) P (l - w) ~ d z •
2 (1 + w) P
Since g" (w), (1 - w ) t / 0 + ~, (1-~-w)~-I/0 each have bounded characteristic in
] w l G 1, so has f ' [z (w)], and so [ ' (z) has bounded characteristic in
By repeating the argument the same is true of the higher deri- [arg z I < 2 ~"
vatives of f(z). This completes the proof of Theorem V.
S E C T I O N !I . 9 ) I n this section we consider functions [(z) mero- morphic and of finite order in 1 z S < 1, and obtain conditions under which the second fundamental Theorem holds effectively for f (z) as [ z ! < 1 is exhausted by a sequence of circles ] z - - z,~ ] < r,. In addition to (1.1) to (1.4) and (3.3) we define
(9.1) ['* (z) - - f (z) 2- ~ [ ~ "l - ~ z '
where bl , . . ,b~,r are the poles of J(z)lying in [ b ~ - - z 1 1 �9 ~ - - . An empty I 1 - - b ~ z [ 2
product is taken to be 1. This definition, which is convenient for our purposes here differs slightly from one given in M. M. p. 128, where [(z) is also divided by a corresponding product taken over the zeros.
Circles j z - - zo I < r will be supposed contained with their circumference in [ z l < 1 unless the contrary is stated, i. e.
(0.2) IZo I < I - r < 1.
A sequence of circles I z -- z, I < r,, will be said to exhaust I z I < 1 if
(0.3) r. -~ 1 as n -~ c~,
) O N N E V A N L I N N A S S E C O N D T H E O n E M A N D E X T E N S I O N S 365
and to exhaust ] z i < 1 properly (w. r. t. a function f (z)) if in addition
T (z., r., [)
1 (9.4) log 1 - - r .
if there exists a sequence of circles exhaus t ing [ z [ < 1 properly w. r. t. f (z) , we shall say that the Riemann surface of f ( z ) is properly exhaustible. We shall prove the following results.
Theorem VL Suppose that f (z) is meromorphic in l z I < 1 and that
(9.5) lim (I -- I) log M (r, f , (z)) - - + c~.
Then T (Zo, r , f ) is unbounded as Zo varies subject to !Zo ! < 1 - r for every f ixed r in 0 < r < 1. Further i f
(9.6) lim (1 r) log M (r, f ) : - ~ - oo,
then either (9.5) holds, or
(9.7) T (r, f) . . . . . . . . . . . . . . - ~ r
1 ,Og 1 - r
, a s r ~ l .
In either case the Riemann surface of f (z) is properly exhaustible.
Note that if f (z) is regular (9.5) reduces to (9.6) with lira instead of lira. Also the conclusions of Theorem VI are not altered if f ( z ) is replaced by 1/( f ~ a ) for a finite a in (9.5) or (9.6), since by Theorem A the condition (9.4) is unaltered by such a transformation. [f the sequence of circles [ z -z,~ I < r,, e x h a u s t s [ z [ < 1 properly, the second fundamental Theo rem and its usual extensions become effective. We have.
Theorem VIA Suppose that f (z) is meromorphic and o f finite order in z : < 1 and that the circles [ z - - z . ] < r. exhaust [ z ] < 1 properly w. r. t. f (z). Then
(0 I f ai, a~, .. . , aq denote q ~_ 3 distinct complex numbers, finite or inlinite, and ~ > O, we have for n > nl
(9.8) ( q - - 2) T(z. ,r , , , f ) < (l 4- ~) ~ N z . , r . , - -
~4 " R e n d . C i r c . ; ' ) l a t c )B . P a l $ r ) ) l o , - - s e r i e I I . t o m o I I - a n n o 1953
1
I - a,)
3 6 0 w . ~ . H A Y M A ~
(i i) I f q - - 3 in (i) we may replace the av by distinct meromorphic functions
av (z) having bounded characteristic in [ z [ < 1.
(i i i) Suppose that I is a positive intes av (z), v - - 0 to l are meromorphic
functions having bounded characteristic in [ z [ < I and
(9.9) ~b (z) = ao (z) f (z) -}- ai (z) f ' (z) -}-- �9 �9 �9 --}- aL (z) fc,) (z).
Then if n > n2 we have
(9.10)
T ( z . , r . , f ) < (l q-r [ N ( z . , r , . f ) + N ( z . , )) ( ' ) ] r., + N z., ,r., b ( z ~ l .
Here (i) is just Theorem B. in the case of concentr ic circles, the extension
(i i) is due to Nevanlinna(7) and (i i i) as well as s o m e more general results
is due to Milloux(S).
P r o o f o f T h e o r e m Vi. 10) W e base our proof of Theorem VI on a
number of lemmas. r - - p 1
L e m m a 7. Suppose that 0 < ~ < r < 1 a n d - - _~_ -- . Then 1 - - o r 2
log i _ - . ~ r < A o 1_+_ ~ log ~, r ~ p l ~ , a r
where Ao - - 3 log 2 - - 1.039 .... Asymptotic equality holds 2
r - - p 1 while . . . . . . . . . . = - - .
1 - - ~ r 2
W e put
and
so that
1 - - 0 1 - - x 1 + ~ ' l + x
r - - ~ 1 - - h
1 - - p r 1 f h
1 - - r 1-~-~
1-+- r 1 - - ~
1 - - h x
l + h x
as ~,r tend to 1,
ON NEVANLINNA S SECOND THEOREM AND EXTENSIONS 3o7
We also put
~ (p, r):
log 1 ~ p r log 1 4-____hh 1 - - 9 r - - p - - x 1 - - h
1 4- p log _1 log 1 -[- h x r 1 - - h x
1 h ~ 4 - 1 h4_~ - .. ~ + ~ - ~- .
1 1 1 4- ~- (h x)~ + -~ (h x)' + . . .
so that
(10.1) 1 h e j v 1 h4 1 l o g - - - - c D ( p , r ) < 1 -~ -~ - -5 4- . . . . . 2 h
l + h 1 - - h
It is clear that right hand side of (10.1) increases with h and by the hypotheses
of lemma 7 we have 1 - - h ~ 1 1 - - - - so that h ~__ -- . Thus (10.1) gives (b (p, r) < ~ + k 2 3
3 -2- log 2, which is the desired inequality. Also equality holds in (10.1) asymptotically
l __3 log 2 if p, r tend to 1 while h - - - as x - ~ 0 i. e. ~-~ 1 and s o ~ (p, r)-~ 2 3
i . e . r - - ~ _ 1 This completes the proof of lemma 7. 1 - - p r 2
We have next
L e m m a 8 . Suppose that O ~ l z o [ ~ 1 - - r < l , [ Z - Z o l ~ r , l a [ < l. Let
r 2 - ( z - z o ) ( a - - z o ) 1 - - a z (10.2) g (Zo, r, z, a) - - log+ -- log+ 2 (z a) " r (a - - z)
Then we have
00.3) g(Zo, r, z,a) ~Ao r4- Iz-- zol log+ r r-- lZ--Zo[ t(a)'
where
(lO.4) ( , ) t ( a ) : m a x [Z - -Zo l ,~ r , l a - Z o l .
3 6 8 w . K . I.-i A u M ,A IN
C l e a r l y g ( z o , r, z , a ) can be positive only i f l a - z o l ~ r , so that we make
this assumpt ion in wha t follows. We have in this case,
(lO.5) - / - ( z - Zo) (a - Z o ) 1 a z I l o g + ~ . O. log 2 (z - - a) i - - 2 r (a - - z ) I
For the left hand side of (10.5) is a harmonic function of a in the circle
I a - - zo i ~ r so that by the maximum principle we may suppose l a - - zo I = r.
In this case (10.5) reduces to
log 1 -- a z ~ 0 , z - - a
which is true since the c i r c l e l a - - Z o l = = r lies by hypothes is i n l a I " ~ 1 .
It follows also that i l i a - z o I ~ r we may write log+ instead of log in
(10.5) so that (10.2) gives
(1o.6)
r ' - (z - Zo) (~ - Z o ) g (zo, r, z, a) ~- log+ - - , r ~ a - - z )
r 2 - ( z - z o ) (a - - Z o) l o g + 2 r (a - - z) --
We next put
(10.7) z -- zo a - - Zo : Z t , - - - 121.
r r
Then (10.6) becomes
(10.8) g(Zo, r , z , a ) ~ log+ 1 - - -z , a t _ log+ a t - - Z t I
1 - - g I Qt .
2 (0 t --. z I )
Suppose first that ?, ~ I at I ~ 1, where ~, is given by
~ - I z , I 1 1 .... ;~ l z , I - - 2"
ON NEVANLINNA'S SECOND THEOREM AND EXTENSIONS 36~
In this case we have clearly l at l ~ max (l z~ l, l ) ,
in (10.4). W e have also
so t h a t t ( a ) : l a - - Z o ]
t - a ~ z ~ 1 - 1 a ~ l l z ~ l 2
Thus in this case (10.8) and lemma 7 give
g (Zo, r, z, a) ~ log 1 - - i zi l l a~ [ < Ao l -+- ! Zx l log 1 , l a, I - I z, l 1 - 1 z ~ l l a~
which reduces to (10.3), using (10.7). Suppose next that I a~ I ~ ~.. Then we
have trivially from (10.8) and lemma 7
1 - ~ I z, j / < 1o 1 + t z~ l log 1 g (Zo, r, z, a) ~ log 2 ~- log ~. -~ i zi [ ] 1 - - [ z~ [ )-f'
Also by hypothes is I <_~ 1 _ __~ r and it follows from (10.9) that z la, f l a - - z o l
) ~ m a x , I z ~ l so that
r ) -~- ~ rain ' - I' - -
r I z zo l a - - Z o l t (a)"
Thus (10.3) holds in this case also and so holds generally.
11) W e are now able to prove the following fundamental
L e m m a 9. Suppose that O ~ l zo [ < 1 - - r < 1 and that l z --zo [ < r.
Then i f f (z) is meromorphic in l z l < 1 we have
log I f , (z) [ < r -~- I z - - z . t m (Zo, r, f ) -[- Ao n (zo, t, f ) r - - l Z - - Z o l o
where to = max ( 2 ) r, l z - z o l .
370 w. x. riAyraAr~
The poisson-Jensen formula applied to f(Zo -q- P e ~~ gives for 0 ~ p < r
( r ~ - - p~) d �9 1 ]"~ log I f (Zo -~ r ~'o) [,~ log I f (Zo -t-- [9 e~0)[ --- 2 r, g o - - - 2 r p cos (0 o) +
r2 (b; - - Zo) [9 e iO I r ' - (a~ Zo) p e'~ ~- ~ log+ - - - - E log+ --
r ( b y - - Z~ - - [9 e iO) I r (a~ -- Zo -- p e '~
where at, are the zeros and b, the poles of f (z). We write Zo ~ [9 e t~ z,
ignore the terms over the zeros, which are nonpositive, and replace the inte-
grand on the right hand side by the larger
r + [9 l o g + [ f (Zo -~- r eiq')] . r - - [ 9
In this way we obtain
r ps--L- log [ f (z) [ ~ ---------~ m (Zo, r, f) -f- ~ log~-
r - - p
r 2 - ( b ~ - - z o ) ( z - - z o ) .
Using (9.1), (10.2) we deduce
log I f , (z) I - log I f (z} I -- ~ Jog+ ~ ( : - t , 0
r§ r - - [ 9
m(Zo, r , f ) - [ - ~ - ' g ( : o , r , z , b ~ ) , v
and s ince [ z - - Zo ) [9, (10.3) g ives
I r] (11.1) I o g l f , ( z } l <r--[gr~-~ m(Zo, r,f)--[-Am ~ log+ t(b,~) "
Now by (10.4) we have, using the notation of lemma 9,
( , ) t ( b v ) : m a x t z - z o l , ~ r , [ b v - z o l - -max( to , lb , - -zo]) .
t. d t Also the contribution of each pole b, to the integral r n (Zo, t, f ) ~ -
ON NEVANLINNA'S SECOND THEOREM AND EXTENSIONS 371
r is just log+ t (b,)" Thus
/ '~ d t r n (z o, t, f ) - - - "-- ~ log +
,o t t(b,)
Now lemma 9 follows from (11.1) and the fact that I z - - Zo I -" P. We quote one final result
L e m m a l o . ( ~ ) Given ~,0 < ~ < 1, we can f ind t such that ~ ~ t 1
- - (1 -~- ~) and 2
l o g , f ( z ) [ < iog l f . (z) l + A n ( 3 - - + ~ ,f), l z l - - t,
where A is an absolute constant.
12) We can n o w prove Theorem VI. We s h o w first that if (9.5) ho lds
T (z,,, r, f ) cannot be uniformly bounded in [ z o I < 1 - - r for any r. Suppose contrary to this that we have for some fixed r, where 0 < r < 1
(12.1) T ( z ~ , r , f ) < K, I Zo I < 1 -- r.
Then if z
This gives
1 is real, 1 - - r < z < I choose zo so that z, Jr- r : - - (1-[- z}.
2
1 (l - z ) . r-(Z-Zo)--r- Iz-zol:-~
Also lemma 9 and (1.1), (1.2) s h o w that
loglf,(z) l<Ao r-]-IZ--Zol T(zo, r , / )< 2 A ... . . . . . K = . . . . r--tz---zol 1(1 --z)
2
4 A o K l w z
Similarly by considering f (z e i~ instead of f (z) we have generally
i~ e ~
4 A ~ l - - r , log I f . (z) l < 1 - - I z l
log 34 (~, f , Cz)) < - - 4 A ~ - - r < ~ < 1. l - - p
372 w. x. uA~rra.~r
This contradicts (9.5) and so (12.1) must be false.
Suppose next that (9.6) holds but (9.7) is false. Then we cannot have
(1 - - p) n (P,3') -~ 0% a s p -)- 1,
since this would imply (9.7). We can thus find ~ arbitrarily near 1 such that
3 @ p , f~ < K 4 K / 4 1 3 --I- p 1 - - p '
4
1 ( l § where K i s a constant. Hence by lemma 10 we can find t in p ~ t <~ ~-
such that
4 A K log I f , (=) ! --> log I f (z) i , t z ] = t,
l - - p
and so since t > p we deduce for some t arbitrarily near 1
log M (t, f . (z)) log M (t,f) 4 A K 1 - - t
Now (9.6) implies (9.5). Thus with the hypotheses of Theorem VI either (9.7)
holds, which gives (9.4) with z~- - -0 provided (9.3) holds; or T (zo, r, f ) is un-
bounded for every fixed r, in which case given any sequence r~ satisfying (9.3)
and 0 <: r~ < 1, we can find z~ such that I z, ! < 1 -- r~ and
1 )2, T ( z ~ , r , , f } > log 1 -- r,~
and this again implies (9.4). Thus we can always tind a sequence of circles
[z - - z~ ] < r~ exhausting ] z I < 1 properly w. r. t. f (z) (even with assigned rfl)
and so Theorem V! is proved.
P r o o f of T h e o r e m VII. 13) To prove Theorem VII, we again need a subsidiary result
L e m m a I[. Suppose that f (z) is rneromorphic of finite order in [ z 1 < 1.
Then given a positive integer p, there exists a constant K depending only on f (z)
ON NFVANLINNK'$ SECOND THEOREM &ND EXTENSIONS 373
and p such that i f O ~ ! Zo I < l ~ r < 1 we have
m Zo, r, f ~ - / < K i O g r ( 1 - - r ) '
We shall denote by K any constant independent of zo and r, not necessarily the same each time. Since f (z) has finite order in I z I < I we have
(13.1) T (r, f ) < K (1 - - r)-% 0 < r < 1.
l Taking r - - - - (1 q- I z I) this gives, using lemma 9
2
4 (13.2) log I .t, (z) l < A o - - - - 1-1~!
T ( r , f ) < K ( I - - l z l ) K, I z l < 1.
Again (0.1) gives
(13.3) log I f (z) I = log I f . (z) 4 - ~ l o g ' ; 1 - b ~ z 2-iz---~,-;~"
where b, are the poles of f (z) satisfying b v -- z 1
1 - -b~ z ': <- 2 ' and this latter implies
�9 _ i ~ l - - i z l l - - l z l . Ib~ ~ 1 + 2 1 z l 1 , b , ; > 2-q--i z I ' 2 + ] e l 3
Thus by (13.1) the number of terms in the sum in (13.3) is at most K (1 - I z [)-K,
and if ~ (z) denotes the distance of z from the nearest pole b~, then (13.2), 13.3) give
(13.4) l i ) ( n - I z I) -~c. IoglfCz) l < K 1 -t- log+ ~ (z
We now use Theorem 1 applied to f ( z o - t - z ) i n s t e ad off (z ) with r instead of R, and we take for D the domain I Zo + z I < 1. Then
dr (O) = l - - I Zo -~- r e iO I,
and since f (z) has finite order in J z I < 1, nr (0), the total number of roots of
the equations f (z) = 0, 1, oo in I z I < 1 - - 1 dr (0), satisfies 2
,,r (0) < K [dr (0)l -~,
374 w. K. .AvM,r~
so that
(13.0) 1 = I (r) = ~ log+ n, (0) -t- log+ dr (0)
< K (1-J- ffr~ log I dO) 1 - - I Zo - } - r e iO [
Also applying (13.4) to lff(z), which also has finite order in z] < 1, as well as f (z), we deduce that
( ! log If(Zo + r e'~ ! I _< K 1 + ~og+ ~
where ~,. (0) denotes, as in lemma 2, the radius of the largest circle centre z, -~- r e i~ in which [ (z) is regular and unequal to 0 or 1. Thus we have
(13.7) log+ t log[ f (Zo + r e'~ [ : d 0
< K 1 log 1 - - I Zo q- r e iO [ ~- I~ I~ ~r- d 0 . o
Again it fo l lows from lemma 4 that
f z~ 1 f ~ 1 log+ log+ d 0 < log< o ~ (0) ~ (0)
- - d O < K ( l ~ - I ~ ' r
where i is defined as in (13.0). Combining this with (13.0), (13.7) and Theorem l we deduce finally that
(13.8) m Zo, r, ~ : ~ ] < K l ~ l o g - - + r o log l _ ] z o _ J _ r e , 8 I . d O .
We proceed to estimate the integral in (13.8). We have to show that
f ~ 1 1 (13.9) log .-- d 0 < A log 1 - - Zo-~-re'~ r(1 -- r)'
where A is an absolute constant. Lemma 11 will follow at once from this and
ON NEVANLINNA'S SECOND THF~OREM AND EXTENSIONS 375
(13.8). To prove (13.9) it is clearly sufficient to take z o real and positive,
0 < Z o < 1 - - r . Then
�9 __ 2 r 2 (13.10) 1 - - I Z o - ~ - r e '~ 1 - - z o - - - - 2 r z o cosO
-- 1 - - (Zo-}-r) 2 - ~ - 2 z o r ( l - - c o s 0 ) ;
1 (1 - r). Then Suppose first Zo < 9~
�9 1 1 1 - - 1 Z o ~ - r e ' ~ : (1 - - r), 0 < ~ <: 2 rq 2
I (1 --- r), (13.10) gives so that ( 1 3 . 9 ) i s trivial. Next if z o > -~-
�9 1 (l_[Zoq_re,O[,) l - - [ Z o + r e ' ~ > ~ -
1 > Zor ( l - - c o s 0 ) > ~- r(1 - - - r ) ( l - - cos0),
since zo-1-r < 1, so that we have in t h i s case
fo ;~ 1 2 f ] ~ 1 log . ~- log d 0 1 [Zo-~-re'~ dO<zrcl~ - - r(1 - - r ) l - - c o s 0
< A log ~-1 < A l o g r (1 - - r ) r (1 - - r)"
Thus again (13.9) follows. This proves (13.9) and lemma 11 follows from this and (13.8).
14) The proof of Theorem VII now follows from standard arguments.
We have from Theorem B as n ~-co
q ( 1 (q--2)T(z~,r~,f)< Z N z~,r~,r_av
376 w. x . .AY M^r~
w h e r e
q
u /" ) + o(z).
f - - - 12v
It thus fo l lows from lemma 11 that
l 1_~ S (z,,r., f ) - - O l og 1 -- r . t - - - o { T(z , , r , , f ) l, a sn -~oo ,
since the circles I z - z. I < r.. exhaus t [ z i < 1 p roper ly . This p r o v e s (i).
Nex t if a~ (z), a~ (z), a.~ (z) are unequal func t ions of b o u n d e d character is t ic
in l z l < 1 put
(z)-- f ( z ) - a t(z) a 2 ( z ) - a 3 ( z ) f (~) - a= (z)' a, (~) -- a, (z)"
Then �9 (z) z 0, 1, ~ on ly if f (z) == a, (z), a2 (z), as (z) or a I (z) = az (z), a~ - - a ,
a.~ :-= at. T h u s
(14,1)
( ) ( , ) 1 -i- N z . , r . , - ~ - N r . ,
' ( '~) ( ' ) < v~ N z,, r,, f _ a , (z Jr- ~ N z., r., a~ (z) - av (z ) ----- l ~p .<v<z I
' ( , ) < ~ ~., r., + o(l). ,=, f - at (z)
For s ince the a~ (z) - a , (3) have b o u n d e d character is t ic in I z l < 1, w e have
a s ~ -),- ~:)
( '2 ( ' ) z . , r . , - - ~ 0, 1, - < co. a~ a F a v
For a s im i l a r reason we have as n ~ cx~
(14.2) T (z,, i",, [ ) : T (z,, r,,, �9 (z)) q-- 0 (1).
ON NEV/INLINNA~S SECOND THEOREM AND EXTENSIONS 377
We can n o w apply (9.8) to (I) (z) with q --:- 3, a I ~--- 0, as • 1, a8 = oo and the
results for f(z)follow f rom (14.1) and (14.2). T h u s (if) is proved.
It remains to prove (i i i). We note fo l lowing MiUoux(s) that
r. ~,r. , 7 _<,,, ~,,,r., 7 + . , z.,r,,,
( , (14.3) ~ lV (z~, r,,, +) -[-- N z,,, &, ~ - - _ - {
where
S(z . , r . ) - -m z.,&, -~-2m z~,r~, ~-m z~,&,~__ 1 q- 0 ( 1 ) .
This follows on app ly ing Theorem B to + (z). N o w
l
* (z) ----- ~ a , (z) f(') (:),
and since the a v (z) have by hypo thes i s b o u n d e d characteristic and so finite
order in I z I < 1, and f(z) and all its derivatives have finite order in I z I < 1, qJ (z) also has finite order in l z I < 1. Also
m z,,r,, : m z , , r , , ~ a,(z) (z) ,=o f (z)
( ' ) - ~ a s / / . - ~ c ~ := O log 1 ..... r,,
as iemma 11 shows , On apply ing lemma 11 deduce therefore that
14.4 ( ' / S(z , , , 1 , , )= O log I - - r ,0 '
also to ~ ( Z ) - - 1
a s / / . ~ o o .
and ~ (z) we
378 w. K. HAYMAN
Again the poles of + (z) occur when f (z) or one of the av (z) has a pole so that
1
(z., r., ~ (z)) ~ N (z., r., f) -" ~ N (z., r., a, (z))
<_ N ( z . , , . . , f )+O( l ) .
Using (14.3), (14.4) we obtain
and adding N (: . ,
( , ) ( N ( z , , , r . , f ) + N z, ,r , , ,~--~-~ 4 - 0 log
') r., [ : to both sides we obtain
~_N(z,,,r.,f)~-N(z~,r.,, -~-N z,.,r,,,~-(z)_
+o(,og, _1,)
, ) )
1 &
,)
) / [ (1 - - !".) 2 - - [ z . !~] A (z . , r . , f ) -~ oo .
Then the Riemann surface of f (z) is regularly exhaustible in Ahlfors' sense, in
particular this is always the case i f there exists a sequence of circles properly
exhausting[ z [ < 1 in the sense of (9.4).
Corollary: I f (9.5) or (9.6) holds the Riemann surface of f (z) is regularly exhaustible.
Our proof is based on
(15.1)
Theorem VIIL Suppose that f (z) is meromorphic in [ z I < 1 and that there
exist a sequence of circles ] z - - z, I < r, exhausting I z I < 1 so that
Now Theorem VII (i i i) follows on making n ~ c~. This completes the proof of Theorem VII.
THE AHLFORS EXHAUSTION THEORY.
15) We recall the definitions and notation of the introduction, paragraph 3.
We proceed to prove the following
ON N E V A N L I N N A ' S S E C O N D T H E O R E M AND E X T E N S I O N S 370
1 L e m m a m2. S u p p o s e t h a t 0 ~ [ z o [ < 1 - - r, - - ~ r < 1 a n d t h a t
2
(15.2) A (Zo, r, f) > 4 ~ h ~
(1 - r ) ' - I ~o I ~
T h e n t h e r e e x i s t s a c i r c l e [ z - - z~ I < ri c o n t a i n e d in I z I < 1 a n d c o n t a i n i n g
[ z - - Zo ] < r s u c h t h a t
(12.3) A (zt, r~, f ) > h L (z~, r~, D .
This result is a slight generalisation of a corresponding one of Ahlfors, (~a)
w h o showed that (12.3) holds for some rl, r < r i < 1 and zl ---- o if
(15.4) A (0, r, f ) > - - 2 z c h ~
- - r o
W e proceed to derive our result from that of Ahlfors. W e may suppose wi thout
loss in generality that Zo is real. Then Zo -~- r, Zo - - r are ends of a diameter of the cir-
cle I z - - z [ , < r . W e map I z [ < 1 onto [ w [ < 1 byab i l inea r t r ans fo rmat ion
which keeps the real axis invariant and maps Zo ~ r on to two points ~ to.
Then the circle [ z ~ Zo [ < r b e c o m e s ] w I < ro and
(15.2) A (Zo, r, f (z)) = A (0, ro, f [z (w)])
L (zo, r, f (z)) = L (0, ro, f [z (w)]).
Also the crossratio of four points remains invariant under the transformation so that
l - - 1, Z o - r, zo-~-r , 1 1 - - - / - - l , - - r o , ro, l f
(1 - - r -+- Zo) (1 - - r - - Z o ) _ (1 - - ro) ~
2 �9 2 r 2 �9 2ro
i~ e~
r _ _ r o
(15.6) (1 - - r) ~ - Zo ~ - (1 - - r o ) ~
1 It is clear from (15.6) that ro ~ r. Suppose n o w that r ~ - - and (15.2)
2 holds.
380 W . K . ItAYMAN
Then we deduce from (15.5), (15.6) that
A (0, ~o, f [z (w)]) > 4 ~ h~ t 1/(1 r - - r ) ~ - I Zo i ~
Y i r~ > 2 ~ h ~ 2 - - 4 ~ h ~ 1 - - r o ) 2 1 - - ro'
and this is (15.4). Thus for some p such that ro < p < 1 we have by Ahlfors ' result
A [0, p, f [z (w)]] > h L (0, p, f [z (w)]),
and so (15.3) follows, where I z -- : i I < ri is the inverse image of the circle
[ w I < P in the w plane. This proves lemma 12.
16) We can now complete the proof of Theorem VIII. Suppose that there
exists a sequence of circles [ z - - z, I < r, satisfying (15.1) and exhaust ing
I z l < 1. It then follows from lemma 12, taking
h,2 :: I/[-~ - - r-) 2 - - I z,~ 1 ~] A (z,, r.,, f (z)) 4 ~
that for a sequence of circles I z - - z," I < r,,', which contain ] z - - z,, ! < r ,
and so also exhaus t J z ] < 1 we have
L (4, ,:,, i (z)~ A (z~,, r'~, f (z))
I �9 < - - - ~ 0 .
//.
Thus the Riemann surface of f (z) is regularly exhaustible in this case.
It remains to s h o w that if a Riemann surface is properly exhaustible in the
sense (9.4), then there exists a sequence of circles exhaust ing I z J < 1 so that
(15.1) holds. Suppose contrary to this that there is no such sequence. Then we
can find constants ro, O < ro < 1 and K > 0 such that if r o < r < l a n d
I z, [ < 1 - - r we have
(10.1) A (Zo, r, D < K
r - r ) ~ - I Zo I '
O N N E V A N L I N N A ' S S E C O N D THEORE1VI AND E X T E N S I O N S 381
N o w we have by the Ahlfors-Shimizu identity(24)
T (Zo, r, f ) = A (zo, t, f) T -~- O (1) = -[- 0 (1). o o
1 Hence if I zo I < 1 - r, ~ (1 + r,,) < r < 1, we have
~ dt (16.2) T(zo, r , f )= A (Zo, t,f) T + 0(1).
r
1 In fact if [ zo I < 1 - - r, r > ~- (l -+- ro), t ~ r o then
1 1 (1 + to), I z o l + t < r o + l - r ~ r o + - ~ ( 1 - - r o ) : : : ~-
. 1 ( l + r o ) < l and so so that the c i r c l e l z - - z o[ < t is conta ined i n , z l < _ 2
A (Zo, t, f ) is uni formly bounded . We n o w apply (16.1) with t > ro instead of r t
in (16.2) and deduce
f r Kd t T(zo, r , D ~ "~ 1/[( 1 -- 0 2 - 1 z o l 2 ]
+ o 0 )
1 = K log + 0 (1)
1 - - r + l/I(1 - - r) ~ - - I z o I~A
1 < K log - - -~- 0 (1).
1 - - r
1 This result ho lds for ~ (1 --{-ro)< r < 1 and [ Zo I < 1 - - r with a u n i f o r m O(1)
2 and this contradicts the existence of a sequence of circles which e : 'haust I z ] <: 1
properly. This comple tes the proof of Theo rem VIII. The corollary n o w fol lows
from Theorem VI. 25 - R e n d . C i r c . M a f c m . P a l e r m o , - - s e r i e I f - t o m o I I - a n n o ~953
3 8 2 w . K. HAyblAN
P r o o f of T h e o r e m IX. We now prove Theorem IX. We shall denote by K a constant independent of zo and r, not necessarily the same each time it occurs. We need two more lemmas.
L e m m a 13, With the hypotheses of Theorem IX, given ~ > 0 we can
find ro < l, such that if ro < r < 1,1zo l < 1 - - r we have
q
N Zo, r, < (b-l- log ~=, [ a,, ') - - 1 - - I Z o l - - r "
.It follows from our assumption (3.8) that if we write
nlzo, r):=Y~,n zo, r,[__ov,
then for r > re (~), I Zo I < 1 - - r, we have
n(Zo, r ) ~ n ( O , I zo I + r ) < b + ~
l - - l z o l - - r
Hence i f l ( 1 + q ) ~ r < l then 2
q( " f+I/ E N zo, r, < n(zo, t)--~ = V ~=1 �9 ~" I"
2 2
< n(O, I Zo I + r , ) log 2 + - - - - - b--]-~ f ~ d t
1 i z o l - - t " r i .~
1 (1 + rt) < 1, so Also I zo l < 1 - - r ~ l _ _ ( l _ r e ) a n d h e n c e l Z o l + r e ~ _ 2
that the first term on the right hand side is uniformly bounded�9 Further we
may suppose that 1/r e < 1 -t- ~. We then obtain for re < r < 1, I Zo I < 1 - - r
q
~=~ av < (1 --[- e)(b -l- e) log + K. - - 1 - - I Z o l - r
ON NEVANLINNA'S SECOND THEOREM AND EXTENSIONS 383
Next we can find rz such that r, < rz '< 1 and
1 K <( ~ log
1 - - r e
Then (17.1) gives if r e < r . < l , [ z o[ < l - - r
q ( l ) N Zo, r, F=--a~ < [(1 + ~)(b --I- ~) -t- ~] log
1
1 - - I z o l - - r
Since ~ is arbitrary lemma 13 follows.
Our next result is
L e m m a 14. I f ro is defined as in l emma 12 and ), as in Theorem IX, then we have /'or ro < r < I, I Zo l < I - - r
1 1 (17.2) m (zo, r , / ' ) < (~. -4- r log . . . . . . . . . . ; K log - -;
1 - - [ Z o [ - - r 1 - - r
(17.3) N (0, r, ]') .< K log - - 1 - - r
We have by Theorem B and lemma 11
(17.4) q ( ':/ ' (q - z ) r (zo, r, f) < ~:~ X Zo, r, I' :--rJ + / r l o g r . . . . . . . . (1 - - r)"
Taking Zo - - 0 and us ing lemma 13 for r > ro this gives
N(0, r , f ) < ( b + r 1 1
- - --l- K log . . . . . . . , 1 - r 1 - - r
which proves (17.3). Also (17.4) gives, us ing lemma 13,
(q - 2) m (zo, r, f) < (b -t- ~) log 1 1
- - r - - II'Zo' - + - x l ~ . . . . . l - - r " 1
This proves (17.2) if one of the a, is 0% so that ~ , - - - - b . If a~ to aq a r e all
q - - 2
384 w. K. HAYM&N
finite we put aq+, : oo and apply (17.4) with ai to aq+l instead
a, to a~, This gives
' ( ,) (q--1) T(zo, r, f) < ~../=l N ZO IF, /" =-a: + N(z*I r, f) -+- K log . . . . . 1 - - r
Subtracting ( q - 1) N (Zo, r, f ) from both sides we obtain
( q - - l ) m ( z . , r , f ) < N Zor, f f - - - - ( q - - 2 ) N(zo, r , f ) + K l o g ........ 1 - - r
V=I
of
1 1 < (b ~- e) log - - -~- K log - - ,
I - - [ z o l 1 - - r
using iemma 13. Now again (17.2) follows, since X - - - -
proves lemma 14. q - - 1
in this case. This
18) We can now complete the proof of (3.12). We take a fixed k ~ 4 and
given p, 0 < p < 1, we define p, by
1 - - p (18.1) I - - p, - -
k
Then for any fixed k and = > 0 we can find p as near 1 as we please such that
( 18 .2 ) n (o , p~ f ) < K k K
1 - - P k 1 - - p
For i f (18,2) is false for all p, > p' then for p' < p < I we have
~1_ d t N (0, p, f ) --- n (0, t, f ) --
t 2
n ( O , t , f ) d t > K d t t l - - t
- - K l o g . . . . . .
1 - - p t
I - - p '
ON NEVANLINNA'S SECOND THEOREM AND EXTENSIONS 385
and if K is large enough this contradicts (17.3).
Suppose then that p is so chosen that (18.2) holds. We next choose z : z (p)
so that I z (p) I - - P and
I f , (z) I - - M (p, f , (z)),
W e also take a fixed r so that ro < r < 1, where ro is defined as in lemma 13.
W e suppose p > r and choose Zo so that
(18.3) arg zo --- arg z (p),
(18.4) 1 - - I z , , I - - r = l I p ~ .
It fo l lows that
(18 .5) r-- l z ( p ) - - z o l - - r - - p ~ l z o l - - p ~ - - p - - (l - - 1 ) ( l - - P ) �9
W e next apply lemma 9 with z - - z (p). W e deduce, using (17.2)
log M (p, f , (z)) < r + I z (p) - - z o I m (Zo, r, [ ) -~- A,, n (Zo, t, f ) t r 1 I z (~) - zo I
2k [ (18.0) < (k . . . . l ) (1 p) (~" -~- r log --] - - iZo I - - r
+ A. -~- - t o to - - - n (0, I zo I -t- r, [ (Z))].
/ 1 N o w to - - max ~-2
so that by (18.5)
A lso
r, I z (p) - - Zo I ) - - I z (p) - - Zo I, if p is suff ic ient ly near 1,
. . . . ( 'i Ao r - - t o Ao 1 - - - - (1 - p). to to k
k K n (o, I Zo I + r,/') - n (o, p~, f ) < 1-U~p"
386 w . 1:. ,~A*MA.~
T h u s
Ao . . . . r - - t o
lo
Ao ( k - - 1) K n ( O , l z , , l + r , [ ) < . . . . . . . . . . < K k ,
to
1 since r is fixed and to ~-_- .... r. Again by (18.4), (18.1)
2
1 1 (~, + s) log = (~. q-- ~) log . . . .
1 - - I Zo [ - - r 1 - - , o k
1 . . . . . . . . . ~ (;~ + ~) log k - 4 . , ) t o g :
1 P
Thus (18.6) yields finally
(18.7) log/14 (~, f , (z)) < 2 k [ 1 q _ O ( 1 ) ] (k - - 1) (1 - - `O) ()' -~- 6 ) l o g 1 ~
provided `o is sufficiently near 1 and (18.2) holds. If f ( z ) has only a finite num-
ber of poles, and we put I z [ - - - ` o , then we have (18.2) and f (z) = f , (z) for
all p sufficiently near 1. Hence also (18.7) holds for all O sufficiently near to 1.
Since ~ can be made as small as we please and k as large as we please we deduce
(3.12) with lira instead lira, as required in Theorem IX.
Suppose next that f (z) has infinitely many poles in I z I < 1. We choose .~
1 so that (18.2) holds with 2 k instead of k. Then if O -~_ r __~ - - (1 --[- `O), we
2 have rk ~ ~k, so that
2 k K k K n (0, r~, f ) <_ - - - - - - .
1 - - rk 1 - - r ~
Thus it follows that we can find `o as near 1 as we please such thai
2 k [ 1 + 0 ( 1 ) 1 (18.8) log M (r , / ' , (z)) < ik .... 1 ~-(i---" r) O, + ~) log 1 ~
1 for all r such that p ~ r ~ - - ( l -Jr P). It now follows from lemma 10 that
2
ON NEVANLINNA'S SECOND THEOEEM AND EXTENSIONS 387
there is an r in this range for which
3 ~ ~,f). log 34 (r, f) < log M (r, f, (z)) + A n O, 4
We have assumed k ~ 4 so that ~ ~ - - 3 + ~ and so 4
/'/ 0, 3--[-~ ) K k K k 4 ' f ~ - - < = - - " 1 ~ 1 ~ r
Since also (18.8) holds for some r arbitrarily near 1 we deduce
iim (1 - - r) log M (r, f)
r..~l log 1 - - r
2 k O. + ~) k - - I
and since k is large and ~ small as we please, (3.12) follows.
19) It remains to show that equality is possible in (3.12) under the hypothesis
(3.11) for any finite set of values a t to aq and b < c~. We may suppose without
loss in generality that ai :--O, since otherwise we can consider f ( z ) - an instead of ]'(z). Consider then
(19.1)
We put
and quote the following
1-{--z z
f (z) -" M e x ~ log
J + z Z = X - ~ i Y = l - - z
L e m m a x$. (~5) The /unction
- - ~ + i ~ = :p (z ) = z log (1 + z )
is schlicht in X > 0 and further if ~ ~ 0 in this region, then
X ~ + Y~ 2 X
388 w. K. HAYMAN
Using this lemma it follows that if I w l < M, the equation
f (z) - - M e xz'~ - - w
has exactly one root in z for each root of the equation
W X Z l o g (1 -]- Z) -~- log~.l~ + 2 n ~ i
for some positive integer n, i. e. one root for each integer n at most. Again if
the principal value of the logarithm is taken the integers n satisfy the inequality
X ~ + Y~ (19.2) i 2 n ,~ I <-- ~ + x
2 X
by lemma 15. Now if z, Z are related as in (19.1), we have
4 X 4 X 1 - - ] z [ ~ _ _ - - <
(X-~- 1) ~ + Y~ X ~ -[- Y~
Thus if z,~ is the root of the equation f ( z ) : w corresponding to the integer
n, then (19.2) gives
2~;~
and hence if this root lies in ] z ] < ~ we must have
(10.3) In I < z ~ - + 1 - ~"
The number of integers positive, negative or zero satisfying (10.3) is at most
2 - ] - 2 ~ . / (1 - ~e). We now suppose M so large that those a v which are finite
satisfy Iav I < M. Also the equations f (z) - - 0, oo have no roots in [ z ] .< 1.
Hence if p (~) denotes the total number of roots of the equations [ (z) __-- 0, a~,..., a~,
we have
p (p) < (q - - 2) 2 + ~ _ or p (p) < (q - - l) 2 + ~ _ 1
ON NEVANI'.tNNAJS SECOND THEOIIE.'~I AND EXTENSIONS ~
according as one of the a~ is infinite or not. Thus
iim (1 - - ~ ) p ( p ) ~ ) , ( q - - 2) o r ) , ( q - - 1),
according as one of the c v is infinite or not. This is equivalent to (3.11) with
the relation be tween k, b, given in Theorem IX. Also we have clearly from (19.1)
log M (~, I) ~:: ~, I -1 -~ log __2 __ _~_ log M, 1 - - O 1 --,~
so that equality holds in (3.12) in this case. The inequality (3.12) is therefore
sharp. This completes the proof of Theorem IX.
P r o o f of T h e o r e m X. 20) In proving Theorem X, we shall denote b y K 1
constants independent of Zo for [ zo I < - - . W e also put 2
(20.1) g (z) = f (z)/,~ (z).
Then g ( z ) : l : 0 , 1 in I z l < 1 a n d g ( z ) has poles only at the zeros of ~ ( z ) .
Clearly g (z) has finite order in [ z [ < 1, since �9 (z) does b y hypothesis . W e
therefore have b y (1.6), (1.7), taking q - - 3, at = 0, a 2 - - 1, a 3 - - oo
C, Zo, 2' 7' m - - g ( z ) < m zo, + r n z,,, -FK,
1 where ~ (z) : g (z) (g (z) - - 1). Using lemma 11 with r - - - - this gives
2
( ' ) (20.2) m Zo,~-,g(z) < K, z., I <-2- .
1 W e now suppose - - :~ r < p < 1 and choose 0 = 0 (r) so that
2
(20.3) . f (r e i~ = NI (r,/').
1 Clearly Theorem X is trivial if r < - - .
2
300 w. ~ . nAYr,,A~
Further let Z o - Zo (r) be such that
(20.4) 1 �9 1
I zo 1 - + - ~ = p, l r r 1 7 6 - Z o I - r - I z o I = r -k- ~ - - ~.
This is a l w a y s poss ib l e on c h o o s i n g a r g z o - - O . Let m,, v - - 1 to N be the
zeros of r (z) in l z - - Zo [ < 1__ wi th correct multiplici ty. Let 2
Then as
on ly at z = av while
Wri te
2 (z - - av) L (z, av) = I - - 4 ( Z - - Z o ) ( a v - - Z o ) "
a funct ion of z, L ( z , a ~ ) is regular in I z - - Zo I ~ 1 and van i shes 2
1 I L(z, av)] - - l, [ z - - z o l - - - ~ .
N
(:zo.5) go (z) = p (z) g (z).
1 Then go (z) is regular in I z - - Zo [ <~ and l go (z) ] --- [ g (z) [ for [ z - - Zo [ - - -~ - ,
so that
( ' ) m Zo,~,go(z) = m T h u s l emma 0 gives
1
(20.0) l o g l go (r egO)] < ~ - @ I r e i~ - - Zo ! - - m
• jr e,0- ol 2
1 (z)). Zo, ~ - , g
1 us ing (20.2), (20.4). Fur ther on [ z - - Zo [ = - - w e have
2
I r (z)) - - log A4 (~, r (z;), logl ~ ( z ) l ~ l o g M Izol-Jr-~- ,
1 and also I P (z) I - - l. Thus w e have for I z - - Zo I = - -
2
ON N E V A N L I N N A ,~ S E C O N D T H E O I ~ E M A N D E X T E N S I O N S 391
(20.7) log cI~ (z--) i - / l o g M (p, ,I) (z), p (z)
! and since (P (z) /p (z) is regular in [ z - - zo I ~ "~-,(20.7) ho ld s a lso in
I z - - Zo I < 1 and in part icular for z : r e 10. Also us ing (20.1), (20.5) we have 2
log I f (z) I - - log I g (z) a) (z) I : log I go (z) �9 (z)/p (z) I
: log leo (z) I + log -(I)(z) " p (z)
N o w Theorem X fol lows, pu t t ing z : r e i0 and us ing (20.3), (20.6) and (20.7).
Exeter, May 1952.
1 ) . - -
2).--
3 ) . - -
4) . - -
5 ) . - -
0) . - -
7 ) - - $) . - -
B I B L I O O R A P H V
L. V. AaLFORS, Zur Theorie der ~)berlagerungsfldchen. Aeta Malh. 65 (1935) 157-I94.
L. V. AHLPOR8, Uber die Anwendung DifferentialKeometrischer Methoden zur Untersuchuns
yon Crberlagerungsfldchen. Acta Soc. Sci. Fenn. Nova Series A, Tom. !!, nr. 6.
A. DmOHAS, Fine Bemerkung zur Ahlforschen Theorie der ~)berlagerungsfldchen. Math. Zeitschr. 44 (1938), 568-572.
W. K. FIAYMAN, Maximum Modulus and Valency of Functions Meromorphic in the Unit Circle. Acta Mat. 86 (1951), 89-257.
J. E. LITTL~WOOD, Lectures on the Theory of Functions, Oxford (1944).
H. MmLoux, Les Fonctions M#omorphes el leur Dgrivds, Paris (1940).
R. NmVAt~r.m~A, Le Th6or~me de Picard.Borel et les Fonctions Mdromorphes, Paris (1929).
R. N~vAsr~tI~sA, Eindeutige Analytische Funkctionen, Berlin (1936).
N O T E S
(') Nevanlinna 8), p. 157. We shall denote this book by E. A. F. in the sequel. (2) E. A. F., p. 158. (a) E . A . F . , p. 232. (4) E. A. F., p. 268. ~5) See Nevanlinna 7) p. 138. (6) Littlewood 5) p. 90. F) See Nevanlinna 7).
302 w. x . ~AYMAN
(s) See Milloux 6). (9) Ahlfors 1). See also E. A. F. Chapter 13, from which the following results are quoted. (i0) E. A. F., p. 332. We write A instead of Ahlfors' S. (li) E. A. F., p. 334. (~2) E. A. F., p. 335. (13) E. A. F., p. 336. (i4) E. A. F., p. 343. (i5) Ahlfors 2). See also Dinghas (3). (i6) If the a~ reduce to 0,I oo, (3.12) with A ~. instead of X follows from Theorem V
p. 174 of Hayman 4). This paper will be denoted by M. M. in the sequel. (i~) See e. g. Littlewood 5), Theorem !!0, p. 114.
(i8) If f (Re i0) ---- 0, I or oo we put ~R (0) : 0. Thus ~R (0) > 0 except possibly for a
finite number of values of 0. (~9) A proof is given in M. M. p. 108. The result follows from the arithmetic-geometric
mean Theorem.
(s0) M.M. lemma 7 p. 109. For n (0), d (0), d o (0) in M. M. read n R (0), d R (0), ~R (0) here.
(:~) See for instance Nevanlinna 7) p. 138. (23) M. M. Theorem VI, p. 131. The difference in the two definitions o f f . (z) does not
affect the proof. (23) E. A. F. p. 343, formula (30). (~4) See e. g. E. A. F. p. 167. 1~5) M. M. p. 195, lemma 1.