Thus (23.1o) gives - By THE MAXIMUM MODULUS AND VALENCY OF FUNCTIONS MEROMORPHIC IN THE UNIT C

(23,II)

loglml<_logT _ + A - - k

I f m is the l a r g e s t odd i n t e g e r s a t i s f y i n g (23.Ii) we have p~(~)--< 2~n. T h u s we have

18 - 642128 Ae~a ~ / r 86

242 W . K . Hayman.

(23.I2)

log p~ (0) -< log I log 3, I - - ~

p , (e) < - -3(I - - e )

if k > A . T a k i n g # = M, M + I in (23.I2) we have f r o m this a n d (23.9)

M + I I

(23.13) Z, p~ (e) <

^- ^- ^"

/*=1 I - - ~

Also since (23.4) holds, it follows f r o m l e m m a 4, as we r e m a r k e d earlier, t h a t the circle I z l < p can contain no points which correspond to points ~ in the sheets R ji+2, R~x+3, . . . etc. Thus if p(0) denotes the t o t a l n u m b e r of roots of the equation (23.3)in ]z] < p we have f r o m (23.x3)

M + I I

p (o) = X p ~ ( e ) < - - ,

provided t h a t (23.8) holds with a sufficiently large c o n s t a n t A (c). Since we have a l r e a d y shown t h a t (i7.9) , (i7.8) hold in this case, the proof of Theorem I I I is complete.

Sets o f V a l u e s E H a v i n g the S a m e Effect as t h e W h o l e P l a n e . 24) W e now t u r n our a t t e n t i o n to the last problem of this chapter, problem (ii) of p a r a g r a p h I. I t has been shown in C h a p t e r I I , Theorem V I I , t h a t if f(z) takes none of a sequence of values w,~ which satisfy

(2 4. I) W o = O,

(24 .2) I w , + l l < ~ l w - I , ~ = I, 2 . . . ,

(24-3) ]w~[ --> c~, as n ~ c~,

more t h a n p(~) times in ]z] < ~ for o < ~ < I, t h e n we have (24.4)

where

I + 2 ~ Q.--

The Maximum Modulus and Valency of Functions Meromorphie in the Unit Circle. 243 No stronger result than this holds even if f ( z ) takes no value more than p(~) times, at least when

(24-5) P(e) = (I - - e ) -a, O ~ a < o o .

This was shown in Chapter II, paragraph 2 1 . Our problem is to what extent the conditions (24.I) to (24.3) can be relaxed, without weakening (24.4). W e show first that we cannot greatly weaken (24.2), (24.3) when P ( e ) ~ I, i.e., a -- o in (24.5). Clearly (24.:) represents a mere normalization. In this case (24.4) gives

= o(,oo

As a converse we have

Theorem IV. Suppose that r~ is a sequence of real positive numbers such that

r n + 1 r n

Then there exists f ( z ) regular nonzero in I zl < I and taking no value wn such that I Wnl----rn more than once and such that

log If(e) l + oo, as e log (:/(i -- e))

Although we cannot weaken (24.2), (24.3) much for all functions p ( ~ ) w e can do so if p(~) grows as rapidly as in (24.5) with a > o . In fact we showed in Theorem IX of Chapter I I that in this case we can replace (24.2) by the weaker condition

(24.6) I w n + : l < l w n l k, n-~ 1,2, k = c o n s . > I .

This condition is best possible in an even sharper sense than that of Theorem IV.

W e have in fact

Theorem V. L e t E be any set of complex values which does not contain a sequence of values w,, satisfying (24.3), (24.6). Then given a, o < a <--- I there exists f ( z ) regular nonzero in I zl < : taking no value in E more than ( I - - e ) -a times i n

I z [ ~-- ~ for o < ~ < I and such that

lira (I - - e) a l o g M [ e , / ] = oo.

q---~ 1

Thus the condition t h a t E shall contain a sequence satisfying (24.3), (24.6) is necessary and sufficient ill order that E shall have the effect of the whole plane when P(e) is given by (24.5) with o < a - - < I.

944 W . K . Hayman.

L a s t l y when a > I in (24.5) it follows f r o m T h e o r e m V of C h a p t e r I I t h a t even t h e set E given by w = o, x, co is sufficient to r e s u l t in (24.4). T h u s t h e proof of Theorems IV and V will dispose of problem (ii) of p a r a g r a p h I, com- pletely when a > o, a n d to a large e x t e n t w h e n a = o.

P r o o f o f Theorem IV.

25) W e prove first T h e o r e m I V which is m u c h simpler t h a n T h e o r e m V.

L e t r~ be t h e n u m b e r s of T h e o r e m I V supposedly a r r a n g e d in order of increasing m a g n i t u d e . W e m a y suppose w i t h o u t loss in g e n e r a l i t y t h a t

( 2 5 . I ) r n e = e

f o r some i n t e g e r n o . T h e n we choose

(25,2) ~ = log r n + ~ - l ,

Since the rn satisfy

we shall have (25.3)

7~ = I , 2 , . . .

- - ---> CK~

~ n + l - - ~ n ~ OO.

T h u s t h e n u m b e r s ~,,+1--~n have a positive m i n i m u m a n d so we can find ~ 0 > o such t h a t

~/o< I

2~7o<~n+x--~n, n = I , 2 , . . . W e t h e n choose

(25.4) ~n=*?0, n = 1 , 2 , . . .

a n d it follows t h a t the n u m b e r s ~n, ~n satisfy the conditions (5.4), (5.5),

W e shall define the curves Cn in accordance w i t h ( I I . I ) , so t h a t ~ = ~V(s), defined as in (I8.1) maps the strip i : I conformally onto a d o m a i n D, since the sheets Rn are non.overlapping. W e define f ( z ) by (18.1) to (18.4).

Suppose t h a t

(25.5) =

where I = I t f o n o w s t h a t either n < no, so t h a t

(25.6) = log Iw l <

The Maximum Modulus and Valency of Functions Meromorphic in the Unit Circle. 245 by (25.I) or n ~ n o so t h a t

(2S.7~ ~ = log l w . [ =

~,,-,~

Also if (25-5) holds,

log w , = ~ + i a r g w , + 2 m ~ i

m u s t lie in D a n d in case (25.6) holds we deduce f r o m (5.12) l a r g

w.

+ 2 mzc[ <

~,--~0

= 2

w h i c h can hold only f o r a t m o s t one v a l u e of m. Also if ( 2 5 . 7 ) h o l d s we deduce f r o m (25.4) a n d (5.I3) t h a t

] a r g wn + 2 m ~r[ < ~1o < I

which c a n a g a i n hold f o r a t m o s t one value of m. T h u s if ( 2 5 . 5 ) h o l d s we m u s t h a v e

lo~f(~) = log I ~ . I + i (a~g ~ + 2 = ~)

w h i c h c a n h o l d f o r a t m o s t one value of m in all cases. Since log f ( z ) gives a schlicht m a p p i n g o f I z l < I o n t o t h e d o m a i n D, we d e d u c e t h a t (25.5) has at m o s t one solution in I z l < I f o r each wn.

26) It remains to show that

log I f ( ~ t ~ + co, log I

as ~ - } I .

Since ~, ~, 0 are r e l a t e d as in (i8.1) to (18.4) this is e q u i v a l e n t to p r o v i n g t h a t

(26. I) -~-~ + co, as ~--> + co,

w h e r e ~ = ~ c o r r e s p o n d to s = o in t h e m a p p i n g of

(IS.I).

Suppose

(26.2) ~,, <_ ~ <_ t (~,, + ~,,+,)

a n d t h a t s = a c o r r e s p o n d s to ~ = ~. T h e n it follows f r o m (I 1.2), t h a t if as is defined as in (I8.5) we h a v e

~(a--a,)

< l o g +

-~--~"

+ A

(26.3) (a - - a~) < log + (~ - - ~,) + A + log + !

246 W . K . H a y m a n .

The Maximum Modulus and Valency of Functions Meromorphic in the Unit Circle. 247

248 W . K . Hayman.

The Maximum Modulus and Valency of Functions Meromorphic in the Unit Circle. 249 for v > n fixed a n d W~+I fixed a n d small, a n d since t h e n a increases with ~ a n d b e c o m e s very large if ~ = ~+1--~7n+1, we can find 6' so t h a t

~'. <_ ~' _< ~.+~

a n d ~' corresponds to ~' where

W e t h e n change '7~+1 to the value given by

(28.6) ~/,+1 = ~,+x - - ~'

so t h a t a' becomes o,+1. I t follows from l e m m a 6 t h a t , however t h e ~7~, v > n + I are chosen this c a n n n o t alter o' by more t h a n I so t h a t

(28.7) ] ~ . + , - o'] < ~.

U s i n g (28.5), (28.6) (28.7) we s h a h have

f r o m which (27.5) follows f o r n + I provided t h a t A~ > 2 which we m a y assume.

Also we have clearly

(28.8) ~ . + ~ < ~.+~ - ~ = 89 (~.+1 - ~.)

so t h a t (5.5) is satisfied. F u r t h e r (27.I), (27.2) imply

since ~o-- - - I by (5.I), so t h a t it follows f r o m ( 2 8 . 8 ) t h a t ,7,+1 also satisfies (5.4).

F u r t h e r if

(28.9) A1 > 3 e ~ + I

we shall have f r o m (27.3)

so t h a t (28.8) yields

Thus w h e n n is even (I3.I) is also satisfied provided t h a t (28.9) holds so t h a t in this case we can define t h e curves Cn in accordance with t h e conditions (iii).

T h u s t h e conditions (i) to (iii) of l e m m a 13 can all be satisfied provided t h a t we can satisfy (28.4) for all n > o.

250 W . K . Hayman.

29) N o w w h e n n = o, we have f r o m (28.3) , (5.I) a n d (27.I)

= 8 9 +

so t h a t always a; = o a n d (28.4) becomes

( ; - - I ) log ~ + K > 2 - - A ~ ,

w h i c h is always t r u e , since ~ , K are positive a n d we have a s s u m e d A s > 2.

T h u s we can define ~7~ to satisfy t h e c o n d i t i o n s of l e m m a 13 with A,. = 3- S u p p o s e n o w t h a t V2m-1 has Mready been defined to satisfy (27.5) with a c o n s t a n t A~ = 3. T h e n it follows f r o m l e m m a 9, (I 1.2) t h a t we h a v e w i t h t h e n o t a t i o n of (28.3)

( a ~ - , - p a2~-~) < l o g ~2,~-x - - ~2.-~ + A

~ ] 2 m - - 1

= lOg .~2m- ~2m--1 2~- A,

~]2m- 1 i.e.

log (~2m ~m,--l) ^-^- ~ O'~ra--1 + 7/: a ~ m - 1 - - log ~2m-1 + A > o.

Using (27.5) , which holds by h y p o t h e s i s f o r n = 2 m - - I , A s = 3 a n d (28.3) this becomes

- - ' - - - 0 " 2 m - 1 - [ - - - I log g2~ + K > - - A ,

which yields (28.4) f o r n = 2 m - - I , with As = A on n o t i n g t h a t g2m < g.~+1.

T h u s if (27.5) can be satisfied f o r n = 2 m - - I w i t h a c o n s t a n t A s = 3, t h e n (28.4) holds f o r n = 2 m - - I , w i t h A s = A a n d h e n c e (27.5) can be satisfied f o r n = 2m, w i t h A s = A, where A is a n a b s o l u t e c o n s t a n t .

S u p p o s e n o w t h a t (27.5) holds f o r n = 2 m with A , = A. T h e n ( I 4 . 4 ) o f l e m m a IO gives with t h e n o t a t i o n of (28.3)

(e9.1) 7 ( a ~ - - a , ~ ) < log ~ " - - - E 2 ~ log+log+ (g;~-~-~2~) + A.

N o w we h a v e

so t h a t f r o m (28.3) , (27.3) we deduce

The M a x i m u m Modulus and Valency of Functions Meromorphic in the Unit Circle. 951

2 5 2 W . K . H a y m a n .

The Maximum Modulus and Valency of Functions Meromorphic in the Unit Circle. 253 Thus i t follows f r o m l e m m a 4, t h a t in t h e m a p p i n g of ( I 8 . I ) t h e p o i n t + i , c a n n o t correspond to a p o i n t ~ l y i n g in t h e sheet •n+l. T h u s t h e value of l o g f ( z ) lies inside or on t h e f r o n t i e r of one of t h e sheets R0, R I , / ~ , . . . , / ~ .

I n p a r t i c u l a r if [z] --< Or, l o g f ( z ) ' l i e s in R 0 so t h a t we have f r o m (5.12), (27.I) ] a r g f ( z ) ] < ~t - - ~o = 2.

T h u s f ( z ) is sehlicht in }z}--< et a n d so takes n o value more t h a n once a n d a fortiori n o t more t h a n (I --~)-~ times in Izl < 0 if ~ < Ql.

Suppose n e x t t h a t

(30.4) ~ ~ ~ ~ e ~ i + l , n ~ I .

L e t w, given by (3o.I), be a value of E , consider t h e roots of (30.3) which lie in ]z [ < Q and let p (5) be t h e i r t o t a l n u m b e r . W e divide these roots i n t o n + 1 groups according as t h e corresponding value of

= log w = log f ( z )

lies in t h e sheet R~, p - - o to n. l As we r e m a r k e d above, ~ c a n n o t lie in R~

with p > n, if (3o.4) holds. W e denote t h e corresponding t o t a l n u m b e r of roots of ( 3 o . 3 ) i n Izl < ~ by Pt,(0).

I f ~ lies in R~ we have

~-- l o g w + 2 m ~ i

for some i n t e g e r m. I t follows f r o m (5.x2) t h a t we have in B t, I,~ r < ~,~+1 - ~ .

H e n c e t h e r e can be a t m o s t

~ ( ~ . + 1 - ~ ) +

different values of m. E a c h of t h e s e gives rise to exactly one r o o t of t h e equation (30.3) , so t h a t we have

p~ (5) -< ~ (~+1 - ~ ) + I < ~+1 - ~ ,

m a k i n g use of (27.1) to (27.4). T h u s we have

n--2 n--2

(30.5) X p ~ ( ~ ) < X ( ~ . § = ~n-i + i, . -> 2.

/.*=0 ~ 0

1 A point on the frontier segment of Rn-1,_l{n we consider as lying in Rn.

fi54 W . K . H a y m a n .

The Maximum Modulus and Valency of Functions Meromorphie in the Unit Circle. 255 if (30.6) holds, using (30.4). W e define the left h a n d side of (30.8) to be zero if n < 2 .

31) (3I.I)

Consider now p,(q) for /, = n, n - - 1. I f (30.3) holds for w in E and if

~ = log w = ~ + i~ 7 + 2 m : , r i

(3t.4) l o g [ F + 2 m ~ [ < - ~ a + l o g 2 F---] + log Ve - - 2 ae + A. , - - I i Making use of (27.5), (31.4) gives for any /~ ~ I

(31,5) log 1~7 + 2mz~l < - z a + log 2 ~---] I --I r + I -- ( ~ ) log ~ e + ' - Also we have from

(5.12)

if (31.1) holds. T h u s (31.5) gives

K + A .

(3 1.6)

/7I log ] ~/+ 2 m ~[ < ~-2 a + log I--I~1 + A - - K .

~ o w if

I-'1

= ~' and z is a p o i n t such t h a t f ( z ) = w, w h e r e log w lies in R e and z corresponds to a + i z in the m a p p i n g of (18.2), then we have

I I + ~ ' ~ff I I

(31.7) 2 log > + - log A

' - d 4 2 1-1 1

m a k i n g use of l e m m a 7. Also Pc(O) does n o t exceed the total n u m b e r of values of m (positive, n e g a t i v e or zero) for which (31.1) holds with ff corresponding by (,8.,), (,8.2) to a point z in [el < e. Thus (31.6), (31.7) give

lies in Rt,, then we m u s t have either /~ = o, or /~ odd, or /~ even and # : > o and

(3,.2)

^_<

making use of (27.4). I f /~ = o (31.1) can hold for at m o s t a single value of m as we have already seen. Suppose next be odd. Then C e is defined by ( 1 1 . i ) a n d hence if ~ = log w corresponds to a + iF in the s plane we have from l e m m a 9, (I 1.3) (31.3) 7 ( a _ _ ae) + log ~ - 1 ₁ 1 [ > I~ 1~7 + 2mz~ I -- A.

Similarly if /z is even and (31.2) holds, so t h a t C e is defined by (,3.2) to (I3.9) we have (31.3) from l e m m a IO, (I4.5). I n either case we deduce

256 W . K . Hayman.

_I log [p~,(~)-- I] < log ! + Q +

A--K.

a I - - O

We can take A (a) in (3o.6) so large t h a t this gives [ I - k O l o g i c ] l o g [ p t , ( Q ) - - I ] < a log x - - O

a log - - I log 5, I - - ~

(3,.8)

p~(~) < .~(I --~)-a +

^I.

Now ~ given by (3I.I) can only be interior to R 0 if ~ < ~i and in this case cannot lie in R, since C1 is given by (I I.I). Hence (31.8) applied with/~ = I, gives

P0 (~) -b P l (~) < I -[- 1. (I ~)--a.

Since

Po(O)+

.~, (0) is an integer, we deduce from this that

if ( I - - ~ ) - ~ 5 and

otherwise. Hence in any case we have

po( ) + p , -< (, -

Since ~ = l o g f ( z ) cannot lie in R~ with p - - 2 if I z [ ~ this proves t h a t p(~), the total number of roots of

f(z)=-w

in I z[ < Q, is at most ( I - - ~ ) - a , for ~ < ~2 and any w. Suppose next t h a t (3o.4) holds with n >--2. Then we have

from (30.7). Hence we have from this and (3o.8), (31.8) 0--2

Z < Z +

p,,-,(Q)

~=o ~=0

< l.(i __~)-a + 2 Ix + I(I __ ~))-a] ~ (.} + ~)(, __ ~)-a.

Thus again the equation

f(z)=w

has at most (I--O) -a roots in [z[ < ~ when w lies in E. Hence this is true in all cases and the proof of Theorem V is complete.

The Maximum Modulus and Valency of Functions Meromorphic in the Unit .Circle. 257 I n d e x o f L i t e r a t u r e .

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M. L. CARTWRIGHT, I) Math. Ann. Ixx, (i935) , 9 8 - - i i 8 .

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~ , 2) Prec. Cam. Phil. Soc. 44, (I948), x 5 9 - - I 7 8 . - - - - , 3} Prec. Lend. Math. Soc. (5x), 4 5 0 - - 4 7 3 . - - - - , 4) Quart. Jour. of Math. (i9), (x948), 3 3 - - 5 3 .

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K. L6WNER, x) Math. Ann. 89 , (i923) , x o 3 ~ I 2 x .

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R. NEVANLInNA, X) Le thgor~me de Pieard.Borel et les fonctions rae~romorlohes (Paris, i9z9) - - - - , 2) Eindeutige Analytische Functionen (Berlin, i936}.

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19- 642128 A ~ m~g]~m~a. 86

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