on nevanlinna’s second theorem and extensions

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.