ON DIRECT CRITICAL SINGULARITIES

AND REGULARITY OF GROWTH

By

W. K. HAYMAN London, England

Introduct ion. It follows from a classical theorem of Ahlfors [1], that if

f(z) is an integral function having k distinct asymptotic values then

(1) ct = lim inf T(r,f.____~) > O, rk/2

r -~ oo

where T(r , f ) is the Nevanlinna characteristic* of f ( z ) . Later Heins [4, 5]

investigated the case when e < + 0% and proved that in this case

T ( r , f ) �9 (2) rX/-'---T- -

if k = 1, and that f ( z ) has exact order �89 when k > 1. The latter result was

later improved by Kennedy, [8, 9] who showed that

(3) log T(r , f ) = ~-log r + o(log r) ~',

and that this result is best possible. The theorems of Heins and Kennedy were

actually proved for subharmonic functions which become unbounded in k

disjoint domains in the open plane.

Another type of generalisation was obtained by Ahlfors [2] somewhat

earlier. According to examples of Valiron [11] a meromorphic function of

* For the definitions and results of Nevanlinna Theory, see e.g. [10].

113

114 w. K. HAYMAN

finite order can have infinitely many asymptotic values so that a direct exten-

sion of (1) is not possible. However Ahlfors [2] proved that a meromorphic

function f ( z ) whose inverse function has k distinct direct critical singularities

must satisfy (1). According to Iversen [7] the inverse function z = ~h(w) has

a direct critical singularity at a, if a branch of ~(w) can be indefinitely

analytically continued without algebraic or other singularities in some neigh-

bourhood: 0 < [ w - a l < 5, if a is finite or R < [w[ < 0% if a is infinite,

but ~(w) cannot be continued to the point a.

The assumption that Ahlfors actually used, somewhat weaker than that

of direct critical singularity, was the following. There exist k non-overlapping

simply-connected domains A v in the open plane, such that each A v is mapped

by f ( z ) into a neighbourhood Nv of a point a,, while f ( z ) ~ a, in A~ and f ( z )

is bounded away from av on the boundary of A~. Since two distinct asymptotic

paths of an integral function f ( z ) are separated by a path on which f ( z ) ~ 0%

this condition is satisfied by an integral function with k distinct asymptotic

values, if we set a~ = ~ , v = 1 to k and take for N~ a disk f w I > R, where R

is sufficiently large. These assumptions were later weakened further by Gold-

berg [3] and Heins [6], without destroying the conclusion (1).

It is natural to ask whether under these more general hypotheses the as-

sumption that a is finite in (1) leads to restrictions on the growth or 'regularity

theorems' of the type (2) or (3) and this question was in fact recently raised

to the author in an oral communication by Professor M. H. Heins. We shall

see that the answer is negative. We have in fact the following

Theorem. It is possible to choose the increasing sequences of positive

integers Pn and positive real numbers r., such that if k is a positive integer and

(4) = c o s + + r . ) - * " + ( ,- . - ], n = l

then f (z k/2) has k direct critical singularities at infinity and satisfies (1) with

a finite value of a, while at the same time m(r,f), N(r , f ) and T(r , f ) all

exceed an arbitrary preassigned positive function ~(r), for a sequence of r

tending to infinity.

ON DIRECT CRITICAL SINGULARITIES 115

2. P r o o f o f t h e T h e o r e m .

We proceed to prove our theorem. We need the following

L e m m a 1. Suppose that for n = 1,2, ... p~ is a strictly increasing

sequence of positive integers and that the r, are positive numbers, such

that r 1 = 1,

(5)

Then i f

r.+x > r . + 4, n = 1,2 ...

g(z) = ~ (z + r.) -p", n = l

we have

(6) I g(z) l < 1,

i f I z + 1".1 > 2, for every n, and

(7) Ig(~)l > [z + r . [ -~" -

I g'(~) I < 1,

1, i I l=+r. l<2.

Since p. is a strictly increasing sequence o f positive integers p . > n. Thus

if I z + r . I > 2, we have

I z + r . I -p < 2 -~, p , , I z + r . l -{p.+*' <p.2-{P-+*) < n2-{"+l ' ,

since n2 -" decreases with increasing n. Thus i f I z + r.I > 2 for every n,

we have

Ig(z)l < :~ 2 -"= 1, Ig'(z)l < ~ n2-'"+x)= 1. 1 1

The inequality [ z + r . I < 2 can hold for at most one n, in view of (5). I f it

does hold we have

116 W. K. HAYMAN

Ig(z)l [z+r.l-, - Z 2 -m, m # n m = l

which gives (7).

L e m m a 2. I f p. , r,, satisfy the hypotheses of Lemma 1, and f ( z ) is

defined by (4) then f ( z ~lz) is meromorphic in the plane and has k direct

critical singularities at ~ .

Suppose that z = x + iy, with I Y I > 2, then

(8) lf(z)[=lcosz + g(z)+ g(-z)]>=lcosz[-2 >= e I*l - e - l y l 2

- 2 > ; - 2 1 > 1 .

Thus f(z) has no zeros for l yl__> 2, and

q~(z) = logf(z)

is well defined in each of these halfplanes. Further

- - f ' (z) i e i ~ - i e - i ~ + 2 g ' ( z ) - 2 g ' ( - z ) f ( z ) e i~ + e-iz + 2g(z) + 2g( - z)

so that in view of (6) we have if y => 3

I [' q~'(z) + i'[ = 2ie'~ + 2g'(z) - 2 g ' ( - z) + 2ig(z) + 2ig( - z)

e--ZTz + < 2e-Y + 8

ey -- e - y - 4 < 1.

Thus ~b'(z) has negative imaginary part in the half-plane y > 3 and so ~b(z)

is univalent there. Again ~ = ~b(z) maps the line y = 3 onto a curve F in the

plane and if ~ = ~ + iq, then we have on F

l < e r < l c o s z 1 + 2 < e a + e -3 ~ - + 2 < e a,

so that 0 < ~ < 3. Thus qS(z) assumes equally often in y > 3 all values ( such

that ~ > 3, and since

l f (x + iy)l ~ oo as y ~ oo

ON DIRECT CRITICAL SINGULARITIES 117

uniformly in x, these values are taken exactly once. Hence ~b(z) maps a sub-

domain D + of the halfplane y > 3 (1,1) conformally onto the halfplane ~ > 3,

and so

w = f ( z ) = e ~(~

maps D + (1,1) conformally onto the infinite covering surface R, over the

annulus e3 < [ w [ < oo. Since f ( z ) is even it follows that f ( z ) also maps the

domain D - , which is obtained by rotating D § through an angle rc about the

origin, onto R. Thus f ( z ) has two direct critical singularities.

The function g ( Z ) = f ( Z ~) is clearly meromorphic in the plane, since f ( z )

is even. Also g(Z) maps D onto R, where D is the image of D + under the map

Z = z 2, which is univalent in D § Thus g(z) has one direct critical singu-

larity at oo. If k is a positive integer f ( z k/z) = g(z k) maps onto R each of

the k domains which correspond to D under the transformation Z = z k. Thus

f ( z k/2) has k direct critical singularities over infinity as required. This proves

Lemma 2.

It remains to prove that f ( z k/2) satisfies the growth conditions of our

theorem. To do this it is clearly sufficient to consider the case k = 2, since

m (r,f(z~12)} = m (r t'12 , f (z)},

N ( r , f ( ~ / 2 )} = N (r k/z, f(z)},

T (r,f(z kl2)} = T (r kiz, f(z)}.

Suppose then that [ z l = r, where rv + 2 < r <r~+, - 2. Then I z-T- r z l > 2,

for every ~ and so we have in view of (4) and (6)

1 e,: I le,, Is :)l < Icos:l + 2 =< -~-{1 + I) + 2,

(8) m ( r , f ) ~ 2re(r, e ") + 2 log 2 =--2r + 2 log 2.

Also if Pv = )_2 Pu, we have

118 w . K . HAYMAN

n(r,f) = 2P~,

f n(t,f)dt < 2P~log r. N(r,f) = t 1

Given P, , r v we chose rv+ 1 so large that

(9) 2Pv log (r~+ 1 - 2) < (r~+ 1 - 2) ~.

With such a choice o f the quantities p~, r,, we have for r = r '+x = r~+l - 2,

r t ~ (10) N(r,+ l , f ) < r, + 1,

and in view o f this and (8) we have (1) with k = 2 and e < 2/re.

On the other hand suppose that r = r;' = r , + �89 Then

( 1 1 ) " = N(r~,f) >

r'tv

f n(t,f)dt t

r v

- - > p~ l o g - ~ = p, log 1 + .

~p iO Also if z = rye , with t 01 < 1/(4r~) then

1 r,[ 0 3 [ z - r~[<[ ( r ; ' - rOe i~176 [ < ~---.

Thus in view o f (7) we have

I g ( - z)l > - 1, [g(z)l < 1.

Hence

Hence

( 4 r ~ ) - 1

m[r~, g(z) + g( - z)] =>

- ( 4 r ~ ) - 1

[ 4 ] > (4~zr,) -1 p ~ l o g ~ - - - 21og2 .

(12) m(r",f)

ON DIRECT CRITICAL SINGULARITIES

> m[r",g(z) + g ( - z)] - m(r",cosz) - log2

>= --lP~A _ A 2 r ~ _ A3, rv

119

where A 1, a2, A 3 are absolute constants.

Suppose now that r is an arbitrary positive function of r and that

rl,pl,r2,P2,...pv_l,r~ have been assigned to satisfy (5). We then chose Pv

so large that

Alp~

P~l~ ( l+2--~f) > r (r~ + 1 ) ,

- - - A2rv- A3 > r (r~ + 1 )

and pv > P~-l. Thus in view of (11) and (12) we have with r~= r~ + �89

/ ! tv m(r~,f) > r N(r , , f ) > r ,

and so

t ! t t r(rv,f) > 2 r

Thus m(r,f), N(r,f) and T(r,f) grow faster than the preassigned function

r on the sequence r = r'v'.

On the other hand having chosen Pv, we now choose r,+l to satisfy (9)

so that in view of (8) and (10) we have (1) with k -- 2 and a __< 2/re. This proves

our theorem.

BIBLIOGRAPHY.

1. L. V. Ahlfors, Untersuchungen zur Theorie der konformen Abbildungen und der ganzen Funktionen, Acta Soc. ScL Fenn. nova Set. 1 (1930), No. 9.

2. L. V. Ahlfors, Ober die asymptotischen Werte der meromorphen Funktionen endli- cher Ordnung, Acta Acad. Aboensis Math. et phys. 6 (1932), No. 9.

3. A. A. Gol'dberg, On the influence of algebraic branch points of a Riemann surface on the order of growth of a meromorphic mapping function, Dokl. Akad. Nauk. S S R (N.S.) 98 (1954), 709-711, correction 101 (1955) 4. (Russian).

120 W.K. I-IAYMAN

4. M. H. Heins, Entire functions with bounded minimum modulus; subharmonic function analogues, Ann. of Math. 49 (1948), 200-213.

5. M. H. Heins, On the Denjoy-Carleman-Ahlfors theorem, ibid. 533-537. 6. M. H. Heins, Asymptotic spots of entire and meromorphic functions, Ann. of Math.

66 (1957), 430--439. 7. F. Iversen, Recherches sur les fonctions inverses des fonctions m6romorphes,

(Th~se, Helsingfors 1914). 8. P. B. Kennedy, On a conjecture of Heins, Proc. London Math. Soc. (3) V (1955),

22-47. 9. P. B. Kennedy, A class of integral functions bounded on certain curves, Proc.

London Math. Soc. (3) VI (1956), 518-547. I0. R. Nevanlinna, Eindeutige analytische Funktionen, Berlin (1936). 11. T. Valiron, Sur les valeurs asymptotiques de quelques fonctions m6romorphes,

Rendiconti del Circ. Math. di Palermo 49 (1925), 415-421.

IMPERIAL COLLEGE, UNIVERSITY OF LONDON, LONDON, ENGLAND

(Received February 28, 1966)