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LIBRARY '

"fl,”ml'lfll"l"llfllifllfl'll‘l'l" ..
University
This is to certify that the
thesis entitled

A STUDY OF A CLASS OF FUNCTIONS
HOLOMORPHIC IN THE UNIT DISK
presented by

Philip James Pratt

has been accepted towards fulfillment
of the requirements for

_Ph...D..__degree in Mathematics

 

fig: (I, %
Major professor

Date—W—

0-7639

Q 75);? 45L

ABSTRACT

A STUDY OF A CLASS OF FUNCTIONS
HOLOMORPHIC IN THE UNIT DISK

By
Philip James Pratt

The non-constant holomorphic function f is in classB *
if and only if for each point of a dense subset of the unit
circle f is bounded on an are which ends at the point. The
classa * is investigated with respect to its closure pro-
perties under certain elementary operations.

An approximation technique of Bagemihl and Seidel is
used to show that any holomorphic function can be written as
the sum and product of two functions each having radial limit
zero on a dense subset of C. This result is used to show&*
is not closed under addition or multiplication.

Approximation techniques involving a repeated use of
Mergelyan's Approximation Theorem and modifications of a
technique of Earth and Schneider are used to show the
existence of a function f which is not in £2 * such that
ef is in B * and a function g which is in B * such that

foz g (t)dt is not in B *.

It is also shown that if f is in a *, ef is in B *.
If f is in E3 * and f omits the finite value a, then

l/(f(z)-a) is in B *. The fact that f is in B * does not

Philip James Pratt
imply that f' is in a *. There are no sufficient slow or
fast (infinite) growth conditions for a function to be in

B,*ornotin E *.
Finally, the possibility of extending theorems from nor-

mal functions to functions in B * is discussed.

A STUDY OF A CLASS OF FUNCTIONS
HOLOMORPHIC IN THE UNIT DISK
By
Philip James Pratt

A THESIS
Submitted to
Michigan State University
in partial fulfillment of the requirments
for the degree of
DOCTOR OF PHILOSOPHY

Department of Mathematics
1971

ACKNOWLEDGEMENTS

I wish to express my appreciation to Professor Peter
Lappan for his help and guidance throughout the preparation

of this thesis and to Mr. Edwin F. Smith for drawing the

figures that I used.

11

I.
II.
III.

TABLE OF CONTENTS
LIST OF FIGURES .................................. iv
INTRODUCTION ..................................... 1
OPERATIONS IN CLASS E1 * ........................ 8
FURTHER PROPERTIES OF FUNCTIONS IN CLASS B * 28
Rmuwmmy”HHHHHHHHHHUHHHHHH.37

111

LIST OF FIGURES
FIGUREl ............................................... Ill
FIGURE2 ............................................... 16
FIGURE3 ............................................... 16

iv

I. INTRODUCTION
Throughout this paper we shall let D denote the unit
disk, 52: lzl(1? , and let C denote the unit circle.

{2: ’2’ .. 1}.

By a boundary path in D is meant a simple continuous

 

curve 2 =- z(t) (O$t<l) where lim Iz(t)l == 1. If lim z(t)=
1:91 t-n
G, where C 6 C, we say that the path ends at ca . If there
is a path which ends at C; and along which f(z) tends to
the complex number c, we say that f has asymptotic value c
at G .
Q9

If jinx“ 1 is a sequence of continuous compact

curves in D and x is an arc of C. 52 écmlsarg zsfl}J :

 

we say that X“ converges to I and use the notation that
an X, if for each 6 )0 there exists a positive integer
N such that

(1) Kn _C_ {:1-e<|zl<1}

(2) Iinf argz - all ( e
Z‘
(3) I222: argz-fil<é

00
if n) N. We then call 5! I a sequence of Koebe arcs
n n-l

 

and X the end of the sequence.
If, for a complex number 0, there exists an are 2! con-
«:0
tained in C and a sequence of arcs SXngml contained in D

that converges to 3 such that

lim sup lr (z) - cl = 0 if c is finite and
11-)” zéxn

11m inf lr(z)l=°° 1fc=oo,
rfibupze

then we say that f has c as a Koebe value.

 

If f is a holomorphic function in D and S is the
family of all one-to-one conformal mappings Of D onto it-

self, then f is said to be ggrmgl if the family if(8(z)i3s£rs

is normal in the sense of Montel [37, p. 5%] . We de-
fine 7? to be the class of all non-constant normal holo-
morphic functions defined in D.

In Iggy , MacLane defined some subsets Of C as

follows:

A 2 5C 6 C: f has asymptotic value a at C i

B* = f C, 6 C: there is a boundary path which ends
at C) and on which f is bounded3

A a UAa where the union is taken over all complex

numbers a.

B=B*UA°,

He then used these to define two classes of functions,
a and B . We say the non-constant holomorphic function
f is in a ( B) if and only if A (B) is dense in C.
MacLane showed a =-= B and proved several properties about
class a .

These results have been generalized in various ways.

Barth [6] generalized many of them to meromorphic func-

3
tions. In [163 , Lappan defined a similar class Of

functions which he called 8 * by saying the non—constant

holomorphic function f is in B. * if and only if B* is
dense in C. He then proved it was possible to characterize
B * in a slightly different way.

Koebe's Lemma [:13:] may be stated in the following
way: A non-constant bounded holomorphic function has no
Koebe values. This has been extended to normal functions

E5, Theorem 1, p. 10] and Lappan showed it could also
be extended to functions in B *. He actually showed more
than this; he Showed that it could not be extended to any
larger class Of functions. That is, B * can be
characterized as the class of all holomorphic functions
having no Koebe values.

It is the purpose of this paper to study class B *.
Before beginning, it should be noted that our definition is
not precisely the same as Lappan's, although it certainly is
equivalent to it. When he defined the set B* he required
the paths on which the function is bounded to begin at the
origin. Doing this enabled him to prove some theorems about
the set B*. For our purposes, however, it is simpler to
drOp this restriction.

Lappan gave another characterization for B * which
is more useful to us in this paper and that is the fact that
a non-constant holomorphic function is in B * if and only

if it does not have 00 as a Koebe value. In general, in the

4
results in this paper, when we know a non-constant holo-
morphic function is not in B * we use the fact that it
has 00 as a Koebe value.

Before getting to the results in this paper, it would
be useful to point out the relationship between B * and
two very well known classes, 72 and a . Since non-con-

stant normal holomorphic functions have no Koebe values,

72 C B * E 53. In [:14] , Lappan constructed a

non-normal holomorphic function that was a product of a nor-
mal holomorphic function and a bounded holomorphic function.
Since the product of a bounded holomorphic function and a
function in B * is obviously in B *, this function is
in B * and the inclusion is proper. Turning our attention
to class a , we note that if B* is dense in C, then so is
B (=B* VA, ), and this shows that B * Q B . Then
B * S 4, since 4 =3 . In [19, Example 3, p. 573,
MacLane presented a function in class a which has °° as
a Koebe value and thus is not in B *. This shows that
this inclusion is also proper.

The first question we answer in Chapter II is whether
or not the sum or product of two functions in a *, is
necessarily in B *. This question has been answered in
the negative for class 71 by Lappan E1143 and for a. by
Ryan and Barth E223 . The answer for B * is negative
as well as is the answer to the next logical question: must

the sum or product of two functions in B * be in class a 7

5
We answer both of these in Theorem 1, Corollary 1.1 and
Corollary 1.2 by showing that any holomorphic function can
be expressed as the sum or as the product of two functions
each having radial limit zero on a dense subset of C. (We

say that f has the radial limit a at C, e C if lim f
r-tl

( r C, ) = a ). In Theorem 2 we show a corresponding result

 

for meromorphic functions.

We show in Theorem 3 that if f is in B * and f omits
the finite value a, then 1 / (r (z) - a) is in B *. In
Theorem A, we show that if f is in a and f omits a finite
value, then r is in B *. This generalizes a result in [17]
which says that a holomorphic function which omits two fi-
nite values is in n .

Theorem 5 shows that if f is in B * then ef

is in B *.
This is not true for a as Barth and Schneider showed in
[8] . We cannot always define a logarithm but a related
question is: if f is in B * and we can define a single
valued holomorphic function log f, then is log f in B *?
We answer this question negatively in Theorem 6 by con-
structing a holomorphic function 1‘ which is not in B *

f.isinaih

but for which e
We define In (1') by I ( r (z) )=.- [Oz r (t) d t and

In (f) = I (In-1 (f))- We show in Theorem 7 that f e B. *

does not imply I (f) 6 a * and in Theorem 8 that f e B.*

does not imply f' E B. *. The same results were proved

for class a by Earth and Schneider E 9] . A related
question is: if f is in B * can we get out of a, * by

6
integrating or differentiating f enough times or, converse-
ly, if f is outside of a * can we get into a * by inte-
grating Or differentiating f enough times? we answer both

of these negatively. In Theorem 9, we construct a function

f in B * such that f(n) and In (f) are also in a * for

all positive integers n. In Theorem 10 we exhibit a func-

(n)

tion 8 811011 that g is not in B * for any positive inte-

ger n and in Theorem 11 we exhibit a function g where In(g)

is not in B * for any positive integer n.
In Chapter III we turn our attention first to growth
functions with the two growth functions
M (P) =’ sup | f (z) ‘
z|:= r

and

m(r)=(1/21T)joenlog + |f(re1’)'d9,

the maximum modulus and the Nevanlinna characteristic func-
tion. We show there is no sufficient condition involving
either slow or fast growth for a function to be in B * or
not to be in B *. By way Of contrast, there is a suffi-
cient slow growth condition for a function to be in £2
E 19, pp. 36-38] and a sufficient fast growth condition
for a function not to be in ” C17, section 18, pp.57-
58 :I .

There are uniqueness theorems for bounded functions and

functions in class ,2 that say if f - g has zero as a

Koebe value, then f 3 g. We show in Corollary 1.A that

these do not extend to g *.
Finally, we investigate the possibility of extending
certain types of results from the class % to the class
£1.*. We show in Theorem 1A the impossibility of extend-
ing theorems based on properties of normal functions along

sequences.

II. OPERATIONS IN CLASS B. *
Let FL denote the class of all holomorphic functions
defined in D that have readial limit zero on a dense subset
of C. Let Em be the corresponding class for meromorphic

functions.

THEOREM 1: Let h be a holomorphic function in D.

 

Then there exist functions fo , f1 , g0 , and gl 2 class

R SUChthatha fo+80andh= floglo

.0 00
Proof: Let {an 3,131 and £3n3n=l be
two disjoint dense subsets of C. Let Hn be the radius to

n
o‘n intersectedwith 52:1-(1/2 )5. ’2! <1}

and Kn be the radius to fin intersected with the same set
c0

60
(n=-=l,2,"’). LetH- U H andK=UKn.
n31 n {la-=1
We define a continuous function 3 on H U K by

s (z)
s (z)

h(z) irze-H

N

O ifzéK.

“

Since 3 is continuous on H U K and H U K is a closed

subset of D, we can extend s to be a continuous function in

all of D by Tietze's extension theorem [12, pp. 1149-1513 .
The collection of radii to the dn's and fln's form

9
a tress E 2 3 so that by [2 , Theorem 1, p. 181-] there

exists a holomorphic function f such that

(1) lim 'f(r a! )- s(r at )]= O
r-tl n n
(2) r1331 |f(r fin ) - s(r fin)! == 0

Remembering what 8 is equal to, these become

(1') $221 lr(r an)-h(r can)! = .0

(2') i131 lf<r flu)! = 0

Thus, if we let fO ==f and go =: h - f, h == fo + go and
to e R by (2' ). so e R. by (1').
The result for the product has been proved by Brannan

and Hornblower in [:le .

COROLLARY 1.1: There exist functions f0, f1, gO , and

 

 

g1 }_r_1_ glagg B * such that fo + go and fl ' gl are not in
B *.
Rather than prove this corollary, we prove this more

general result.

COROLLARY 1.2: There exist functions fO , f1 , g0 and

 

 

81 in class a * such that f0 + g0 and f1 ° g1 are not in
Proof: Let h be a holomorphic function that is not
if! (I. (For the existence of such a function see [:20,

Example 16, p. 80:]. By Theorem 1, there exist functions

lO
f‘o ’ f1! go : and 81 in class R such that h = f0 + Bo
and h = f1' 81 . Since R is certainly contained in B *,

the result follows.

COROLLARY 1.3: There exist functions f and g in F1

 

 

such that f - g has a zero Koebe value, but f :% g.

 

Proof: Let h be any non-constant holomorphic function
that has a zero Koebe value. Using Theorem 1, we get func-

tions f0 and g0 in class ‘2 such that h =: fO + gO .

Letting f =‘fo and gt: - gO , the conclusion follows.
Since R g; B *, we also have the following trivial

extension.

COROLLARY 1.1;: There exist functions f and g i_r_1_ B *

 

such that f - g has a zero Koebe value, but f:# g.

 

 

The result of Theorem 1 is also valid for meromorphic

functions, with the use of some functions from F3 m'

THEOREM 2: Iet h be’a meromorphic function £3 D.

 

Then there exist functions f0 and f1 in R and functions

 

 

goandglin Rmsuchthath=fo+goandh=fl°gl.

 

Proof: We observe that in the proof of Theorem 1 the

only properties of h that we used in constructing f0 were

that it was continuous and finite valued on the set H. If
oc>
we pick gain 3n a 1 so that none Of the poles of h lie

on H, this will still be true. We now get fo Just as before

and let 80:“ h - fO . Now gO is meromorphic rather than

ll
holomorphic, but it still has radial limit zero on a dense
subset of 0. Thus, h=f0+go , f0 5 R , go 5 Rm
and the theorem is proved for the sum.

For the product, we pick $091an a 1 in the same
way but we define our continuous function 3 a little dif-
ferently. We still define s (z) to be zero if 2 5K. For
each positive integer n, we now define s (2) on Hn to be
a real-valued continuous function that tends monotonically
to 00 as ‘2' tends to l and which satisfies

's(r°(n)‘ > (sup( V293? tn‘lr)
for 1 - (1 / 2r1 ) £ r 4 1. We again have that s is con-
tinuous on H \l'K. By the same argument as in Theorem 1,
we Obtain a holomorphic function f such that

(3) lim ,f(rdn)-s(roln)lz o
r-n

(Ll) lim1 If(rfin)-s(rfln)'=0

By the definition of 3, these become
(3') 11m lh(rdn)/r(rotn)/ = o
r-9l

(M $15,“. If (r Ah.) I

If we let f1= f and g1 =:h / f, then hi: fl . g1 , where

1‘1 e R by (3')ands,é Rm by( (u').

THEOREM 3: _I__r_ 1? EH * and if. r omits the finite

 

12

waflg(z)=l/(f(z) -a), theng é a *.

£29.93: If g é a*; then g has 00 as a Koebe value
so that there is an are I contained in C, a sequence of
arcs $3.13 I: 1 in D and a sequence of positive real

so

numbers EMn 3 n = 1 such that 3n "" X ’fln -9 OO ,
and ’8 (Z)! Z [an on In , n=1, 2, But then
lf(z) ‘3 l 5 1//"(non In,n=l, 2, "',and
1 /M n --) O. This says that a is a Koebe value for f so

that f f a *. This contradiction proves the theorem.

THEOREM ll: Iet f G a . If f omits a finite value,

then f E 3*.
Proof: Let c be the omitted finite value so that

 

g (z) = 1 / (f (z) - c) is holomorphic in D. If f has an
asymptotic value at a point, then so does g. (If f -b 00 ,
g -:b 0; if r—oc, g—soa; if r+bandb7£c, oo ,
g-)l / (b-C)). Thus g £- a.

If f 6 B *, f has 00 as a Koebe value. A similar
argument to the previous theorem shows that g has zero as a
Koebe value. By [19, Theorem A, p. 18: , this shows that
g fl 4 . This contradiction proves the theorem.

Putting Theorems 3 and A together gives the following

corollary.

COROLLARY 4.1: _I_§ f e a and .11 r omits the finite

 

value c, _t_1_u_ep e (z) —-= 1 / (r (z)-c) e E *.

l3

THEOREM 5: _I_f_ f e 3* 333g (2) =ef(z), mg
6 B *.

2522:: Let u (2) be the real part of f (z) and suppose
C, e B* for the function f. Then there is an are P
that ends at z) and a constant M such that 'f(z) ‘ < M
on [-7 Thusl8(z)l = eu(z) so that lg(z)l < eM on P
Thus C: 63 B* for g and B* for f is contained in B* for g.

Since B* for f is dense in C, B* for g must also be dense in

Candg e B*.

THEOREM 6: There exists é. function h not 3211 B * such

 

 

that if g (z) =- eh(z)’ then g E- H *.

Proof: We pick two subsets of C as follows:

H-513, 1.3-5-13

11:3 {1, e21T1/3, euTri/33 , H: _ {8 "1/3, -1, 85 171/3]:

H; T 5821411'1/93 :0 {(2191) 11' 1/928

K = O
1 2K'ITi 3”":l 3n-31 2K 1 ni/(3n- -1 3’“l
unafe /< 2‘30; :21; +> ph-l

O

on
L81: A1 = nUI Hn , 1 a: l, 2, SO that Al and A2 are

countable dense disjoint subsets of C. (See Figure 1).

For each non-negative integer n, let 8n =- 1 - 1/(n+1)

i a“
and let 3,11 be the line segment from the point 3 en /

l'l
'lTi/u

to the point snfl e and let In be the line

114

Figure 1.

15

segment joining the conjugates of these points. Let r1 1 =
on
X: , i 2—. l, 2, and let T be the domain bounded by [1,

F2’ andfily. let 1:1 > 132 > 153 > L 0 be a sequence

picked so that

tn < 1/2 min San - and, Sn+1 - an} and
Fn-fz=x+iy:(YI stn,(sn-1+3n)/2<x

s(sn + sn+1) /23 9 '1‘.

let bn==upper boundary Fn n 52; ‘2‘ as“;
Cnalower boundary an £2: IZI ”Sn;
An :- line segment from bn to °n
an=An ߤz=1mz==03
dn=$z=x+iy:x=an,tn/2£
fln=fz==x+iy: Xaan,-t Syé-t /23
gnafz:Iz-an]=tn/2,7T/2éarg(z-a)é3‘IT/2?

(See Figure 2)
LetTn===T/)fz:|2l 23n_1,n=1:2:"'. If

10
p = e E- A2 , there exists N such that p 6 H3 but p f

2
Hn-1 . Define Tp to be Tn rotated through the angle '6 .
P
For n ZN, let Fn : «3,13%: 0’: , afi be the rotations of
1In ’ “n: An , O'n , a.n respectively through angle 6’ .

Iet (tn: 52: IZI=3n3 n C( U2 Fi)

p eHn

16

 

Figure 2
P P
U (“a U Ba
/ PEH:
2
H2
.1.
T:

Figure 3

17

andtg=ltn U ( U2 (dp U I93) ).

n
péHn
where C: represents complementation with respect to the

disk. (See Figure 3).

If p 6 A1, there exists M such that p 61-11}1 but p 4‘

1

Hm-l . Define Rp to be the intersection of the radius to p

and {2:3méz<13 andletR=UR.

Then R U( U I"; ) is a network that does not disconnect
n:. 1

the plane ( i. e. its complement is connected ) , so that

we can repeatedly apply Mergelyan's Approximation Theorem

[22, Theorem 20.5, p. 3863 to the function i / (1 - lzl )

to get a holomorphic function f that satisfies

(1) lf(z)-inl <1 ifzé’t';
(2) f (z) --i 00 as '2! --9 1 along HP in such a
way that He f (Z) < l for all p E A]. , (For examples of

such a use of Mergelyan's Theorem see Theorem 14 in this

paper, [2, Theorem 1, p. 187:], [7], and [8]).

We now construct another holomorphic function g as

00
follows. Enumerate A2 as fpm 3 "E: 1 . If m 13 a positive

integer, there exists N such that pm 6 HS but pm ¢ HE 1 ,

The constructions have been virtually identical to those
of Barth and Schneider [ 7:) so that by the same technique

that they used we get a holomorphic function gm such that

(1) lsm(z)l < 1/2m+lon C( U rfm)
JZ:N

18
I - p pm
(2) m(8m(z))7 l/eonjg/N(ot3mU/BJ )

(3) l8m(z)l > firtpm (|f(z)()+3on 0‘me

J
if j 2 N.
00
let 8 (Z) =3 2 gm (2). If K is any compact subset
m =- l

of D, K intersects only finitely many Fmp's, so that for all
but finitely many m's we have (gm (2), < 1 / 2m + 1 for

all z in K. The series thus converges uniformly on each

compact subset and g is holomorphic in D.

Set 1'1 (Z) x f (z) + g (z) andvna Tit U U a’np ,
“ pen“2

00
so that 511,, in a 1 is a sequence of closed curves tend-
ing to C.

cl?
On (rim , '8,“ (z)' 2 If (2), + n and 2 Ign(z)/

Ila-:1, 11%!“

co
5-. <.£-:1/2n =1sothat
11:31 Go

lh(2)(=’f(z)+8m(2) + Z 8n(z)l

n31, n¥m
M

2 lem(z)/ - 3H2)!- Z [8,, (z)/

n‘1:n#m

2 lf(z)l +n -If(z)/-l

zn-3
chaff“ upnpm, lf(z)-in I <1,Imgm(z)>-l/2
sothatwehave

19

Ma

h() = f
z ((2)+sm(2)+n=1’n#msn(z)(
.09
2 If (2) + 8111(2), i Ian (2)]
n=-1,n7€m
2(n-l)-1/2 -1
2 r1"3 ,0
0n Tn, |f(2) -1nl -< l, (a (2)1 2’; £1 1/2n
nae

= 1, so that
,h (2)! 2 f (2) -s (2)

Thus, on )1“ , [h (z)! 2 n - 3 so that h has 00 as
a Koebe value and is not in 8 *.

pr 6A1 andz éRp,Re (f(z))<land lg (2)!

g 2 1/2n also thatRe(h(z))<23nd thus [611(2)]
n==l

R h
e e (z) < e2 Since Al is dense in 0, this shows that

NZ) 5 B * and the result is proved.

For notational purposes, let Io (f) a. f, I (f) (z) =

3
‘0 f (t) dt , and In (f)==I (In- 1(1‘)).

THEOREM 7: There exists a function f in B * such

that I (f) is not in B

, 1
Proof. Let A1 , A2 , Hn

p p
(rn, an , “En, 73;, and yn be defined exactly as in the

:Hr'lsTsTpstadp :flp:

proof of Theorem 6. ( n a: 1, 2, ...)

Let 77: z = z (t) ( 0 it <1 ) be an unrectifiable

20

curve lying in T and Joining the points 0 and 1 such that
t1 < ‘52 =>IZ(t1)I < 12(5),, lim lz(t)l—
131.,
and let 91 be sufficiently smooth so that if :0: is para-
meterized with respect to arc length 8, then 93% is contin-

d z
uous and arg ~55- (O) as o, Lef, 9 (s) = arg 9.5%. (s) and
- 1 9(3)

define a continuous function r by r (z (s) ) a: e

If L (z) is the arc length along ’7 from o to 2, then

2 ()
’0 I‘(W)dW =I: zr(z(s)) d—5§(s)ds

IMZ) e -1 0(3) 13 (s) d 5

(Z)
3‘0” 1%, d8

= [01“) lazl = W)

so that I (r) (z) is a real valued function that tends mono-
tonically to 00 alonng . Further, 'r(z)’ =- 1 for all
z in q . We also require that the length of the inter-
sectiononand 52:3n S ’2' 5s gbeat

least one for each positive integer n.

1¢

n+1

If p = e 6 Al , let 42 p be the rotation of 7

through angle ? . Define r on 07 p in exactly the same

manner that we defined r on 07 . (r is well defined since
in both cases arg gag (o)= O and thus r (O) a: e'1 ' O_ l

and O is the only point of intersection of ”p and 02 )

Let S = U 7? . We observe that any component of
p 6 A1 p

21
\J/ Tfn* intersects S at exactly one point. We define r
11:1
to be constant on each component (namely equal to the value

of r at the point of intersection). We now have r con-

tinuous on S U( &1 tn" ) which is a network that does

not disconnect tgezplane so that we can again use the approxi-
mation techniques we used in the proof of Theorem 6, this

time being careful to keep the integral of our holomorphic
function f "reasonably close" to the integral of r on this

network. We thus obtain a holomorphic function f such that

(1) For each p in A1 . r is bounded on 7] p ,
(2) II (“MI 7 n and Im(1(r)(z)) < 1

on Tn“ n a 1’2, 0..

We again construct another holomorphic function g as follows.
The constructions have been virtually the same as those Barth
and Schneider used in the proof of [:9, Example 1, pp. 15-22;?
so that, Just as they did in that proof, for each positive

integer m we get a holomorphic function gm such that

m p
(1)’sm(2)l < 1/2+1 ona U FJm)

a g N
p
(2) 11 (8mm)! < 1 / 2’"+1 on m u FJ m)
3:2”
(3) II (sm)(2)l > :13me (I (0(2)) +3
pm
on (T J '2. N.

J 3

22

p p
(4) Re (I (sm)(2)) > - 1 / 2 on JyNUIJm U 53 m)
2 2
where N is the integer such that pm 6 HN but pm é HN 1 .
0° -
Let 8(2) = : m(z) so that, as before, g is holomor-
m = 1

phic in D. Also, if p 6 A1, [3 (z), _<_ mZ'1(1/2n) = 1

so that, if h =- f+g, h is bounded on W p (both1 f and g are)
and hence h 6 B *.
We now show I (h) has 00 as a Koebe value along Vn's

so that I (h) $3 *.

On 0pm, ’I(gm)(z)’ 2 'I(f)(z)l +nand

n

go
‘2'. #'I(Sn)(z)'££1/2n=lsothat
T1131: n 11“].
00
{100(2) l= lurxz) + I(sm)(2) + Z I(gn)(Z)/
n =-= Loon 74m
2 new): - mm 2)! - Z {1(gn)(2)l
=1: n #m
2 11 mm! + n - II(r)<:)I - 1
Z n - 3

«pm U flim II (f)(2)l > n . ‘Im (I(f)(z)) I < 1,
and Re (I (8m)(z)) ) - 1 / 2 and f I(gn)(z) <

n31,n#m
2: l/2n=lsothat

n==1
w

II<h><z> l = lurxz) + I(sm)(2) + Z I(gn)(z)l

n=1,n7‘m

23
an?

[I me) + I (emxz)! - n “Zn ’I‘Ilglgnflz”

IV

2((n-1)*%)-1
Zn-3
00
On Tn , )1 (r)(z)l > n, II (s)(2)' 5- £1(1/2n) = 1
. 1'1"

so that
!I mm! = I I mm + I (gxz)!

z I I (f)(z)"' II (s)(2)’
2 n - 1

Z n-3

Thus on vn , II (h)(z)' Z n - 3 and the theorem is
proved.

Lappan, in [153, constructed a function that had
0° as a Koebe value, but whose integral was uniformly nor-
mal, hence normal. Since normal functions are in a *, we

note the following result.

THEOREM 8: There exists a function 1' is B * such

 

that f' _i__s_not .12. B *.

 

 

THEOREM 9: There exists a function f in B * such that

n n
I (f) and f( ) are in B * for all positive integers n.

 

Proof: If lf(z)| 5.- M in D, then IIU'HZ)! "

z
”o f(t)dtléM'lzl éMinD. Thus,forthe
integral any bounded function will suffice. We need a lit-
tle more care for the case of the derivative, but this is

easily rectified by taking a function holomorphic in 52 :

24
‘2 l < 2 3 . Then each of its derivatives is also holo-

morphic in {z : [2 I < 2 3 so that they are all bounded

in B which is a compact subset of this domain.

THEOREM 10: There exists é. function g not i_n_ B *

 

J
such that 8 is not in_ E2 * for any non-negative inte-
ger J.

Proof: For each positive integer n we define three

subsets of D as follows.
10 ‘
Tn“ ire : r== l - 1 /(n+l), - ”/14 5 65777113

sn= fx:(1-1/n)+1/(4'n-(n+i)) < x <
(1-1/n)+3/(4°n°(n+1))3
Gm” Sz: [2‘ <(1-1/n)+1/(8'n‘ (n+1) ) 3

We now construct a sequence of functions inductively.

Let V1 be the continuous function defined on GIL/Tl

which is equal to zero on G1 and 1% on T1 , we approxi-

mate V1 to within § by Mergelyan's theorem [:22, Theorem
20.5, p. 386;] to get a holomorphic function g1 (z) with
the property that

[31(2)] < 1/2 irzeq

[81(2)] 71 ifz éT1°

If 81 : 82 9 °" : gn-l have been defined let

25

n - l (
Mn== n + l + 1:: max ( sup ((33
le. OSlsn zflb

1)
(2H).
Let Vn (2) be a continuous function which is zero in a; ,
sufficiently small and real on Tn’ sufficiently large and
real on Sn, and linear on the two parts of the real axis

from 5; to Tn that are not in Sn and approximate Vn(z) to

within 1 / 2n by Mergelyan's theorem to get a holomorphic
function fn(z) with the property that

(In(fn)(2)l < 1/2nifz e 6"”

n
J
02§1n(:2§n(h (fn)(z)l)) > Mh

Then, setting gn = In (fn) we see that

)8n(z)l < 1/2n if z 66;

(J)
02?er (:fin (Ign (2)“) 2 Mn

«09
Let 8 (z) = E gn (2). Since ‘gn (z)((1/2n in
n‘= l

Gn this series converges uniformly on each compact subset

so that g is holomorphic.

Let J be a non-negative integer and m > J and look at

g(J) on Tm.
(J) °° (J)
'g (Z)'=(n§l 8n (2”
m - 1 (J) (J) 00‘
=ln§1 g” (2) + gm (Z) + n =Zm+lgfld)(2)‘

26

- ca
2 (3:33)”), ' E lséj)(2)l - Z 1 /2n
n “‘1 n ==m+l

m-l
m (J) “‘1
?- ‘ *1 " Ellgn W" ‘2: 1855” (2)! -1

n:= 1
:2 m
Thus, for all non-negative integers J, 3(3) has 09 as

a Koebe value and hence is not in E *. This proves the

theorem.

THEOREM 11: There exists a function g not in B *

such that IJ (g) is not in; Ei * for any non-nggative inte-

 

ger J.
Proof: Let Tn , Sn , and On be as before. Again, we

shall define a sequence of functions inductively.

Let g1 be as in the proof of Theorem 10 and suppose g1,

"': gn_1 have been defined. This time we will let
”-
Mn=n+l + Ina an (1)
J=- 1 0315.5 2:,“ < '1 (she)! ))

and, in exactly the same manner, construct a holomorphic

function gn such that

(85qu < 1/2n irzea';

.31;an ( inf (II;I (gnsz) > Mn

zeTn
so
We set 8 (Z) =‘ 15::. gn (z) and again the series con-
n==1

‘Verges uniformly on each compact subset so that g is holo-

rnorphic.

27

Let J be a non-negative integer and m j) J and look at

IJ (g) on Tm.

00
lfl<s><z>l=l i IJ (an) (2)!

n an‘1
_ [m - 1 J J (x:
_' n2§;1 I (8“)(2) + I gm(z) +n Egg+1lj(gn)(z) I

- l 09
2113(gn)(z)l - :gl h‘j(gn)(z)l - S l / 2n

n ==m+l
m - 1 J m - l J
.2Knu1.+h:E;i )I (gn)(z)l) -n:E;; [I (gn)(z)l -1

2m

Thus IJ (g) has 00 as a Koebe value and is not in B *

for any non-negative integer J.

III. FURTHER PROPERTIES OF FUNCTIONS IN CLASS B. *
We now turn to growth conditions. As usual, we use

the following two measures of growth

M<r) =-= sup (Ir (2)! )
'z :. r
and

Tr
m(r') =(1/21T) S: 108+ (fheieflda

THEOREM 12: There exist functions $3 3 * 9;; arbi-

 

trarily slow (infinite) growth and functions in B * g

 

arbitrarily fast (infinite) growth.
Proof: we first observe that

m(I‘) = (1 / 211’) {:Trlog + If ( reio)’ (1'9-
217 +
.4. (1/2Tr)(O log Im(r)lo9

= log + M (r)
so that for slow growth it is sufficient to look at M (r)
and for fast growth it is sufficient to look at m (r).

‘ In [19, Example 6, p. 60] , MacLane constructed a
function f for which M (r) grows arbitrarily slowly and for
which A0 is dense in C. This function is thus in B * and
the result for slow growth is proved.

If there were an upper bound on the growth rate for func-

tions in a *, then there would also be an upper bound on

28

29
the growth rate for functions in R . Since any holomor-
phic function h can be expressed as the sum of two functions
f and g, in class R and m (r, h) 5 m (r, f) .+ m (r, g) +
bounded terms, this would say there is an upper bound on

the growth rate for all holomorphic functions. This con-

tradiction proves the result.

THEOREM 13: There exist functions not in a *

 

93
arbitrarily slow (infinite) growth and functions not in

 

 

g; arbitrarily fast (infinite) growth.

 

Proof: As in the proof of Theorem 12, for slow growth
it is sufficient to look at M (r) and for fast growth it
is sufficient to look at m (r).

InEl, Theorems 3 and 53 , Bagemihl, Erdos, and
Seidel constructed a function f for which M (r) grows
arbitrarily slowly and which has ‘2’ as a Koebe value and
is thus not in B *. This proves the slow growth result.

For the fast growth, let/‘4c (r) be a positive,
strictly increasing function defined on [O, 1) that tends
to co as rtends to 1. Let rn - 1 .. 1/2n andsn a
irneio: - ‘lT/2 5 6' 5 11723, n31, 2, .
Through a slight variation in the technique of repeatedly
using Mergelyan's Theorem we obtain a holomorphic function
f such that

)r(z) - ee,«(rn + 1) - l I< l for all z 6 sn

so that f has co as a Koebe value and is thus not in E *.

30

Furthe r

W
m(rn)=-(1/2TI') [02 log+ If(rn 51’), d9

"72

3(1 /21I') I‘ll/2 log + I f (rn e1.” d0:
11/2

Z (1 / 2") 4772 log + le2~'(rn+1), d6

Z (1 /2) ° 2 ‘f‘(rn+l)
=- Mrnn) n-1, 2,

But 1“. and m are increasing functions so that, if rnér
£rn+1 , m(r) _>_ m(rn) 2 A4. (rn+1) _>_ /“~(r). Since
any r that is bigger than r1 is in some interval E rn ,
rn+1) , this shows that m (r) 2. M (r) and the result is
proved.

00
THEOREM 114: Let fzngn a 1 E a sequence 93 points

EDsuch thatO < lzll < [22] < ’2 l< :--, lim Iznl“1
on 3 n-Oue
and let ("n3 n i: 1 be a sequence of complex numbers. Then

There exists a function f in 8 * such that lim 'f(zn) - w I
‘— ""'"— "" — ""'"" n-vao '1

=0.

Proof: Let fr 3 00 be a sequence of real numbers
__._._' n n - 0

such that r0 < )zl\<r1<lzel 4 r2 (Iz3|<°'°. Let

31
00
S N 3 be a dense subset of C such that the col-
n n a. 1
lection of radii to the Cln '3 do not contain any 21 's.
We denote $2 = ’ZI‘rIby Cr and 52: (z! < rgby Dr-

Rn will denote the radius to a!“ , n =2 l, 2, . We set

‘; U(Rl I) Crl) Ule3

U

—__.. n —.
Fn— Drn U Ejyl RJ)/) (Dr.n+1 - Drnfl

U (Rn+l n Cr'n+l ) V {Zn 3

Then each Fn is a compact set which does not disconnect the

plane.

By Mergelyan's Theorem, there exists a polynomial po(z)
such that

“10(2), <1/2 ifz 6-5:; U(Rlncrl)
lpo(2})-wl|<i/2

To construct pl (2) we first define a continuous func-

tion h1(z) on R1n( Dr.2 - Drl ) by

hl(r0(1)=po(r1 d1)(r2-r)/(r2-rl)

Then, if we define a function g1 (2) on F1 by

81(2):“ 0 ifz e DI.1

a. h1(Z)-PO(Z) ifz 6 R1 n(fi;-pr)
l

32

== w2 ~po (2) if 2-22

== - 90(2) if z 6R2 n Cr2

81 (z) is continuous on F1 and holomorphic at each interior
point of F1 so, by Mergelyan's Theorem, there exists a poly-

nomial pl (2) such that

‘p1(z)' < 1/4 ifz 5 131.1
lwz-(po(22)+pl(22))' < 1/’4

|h1(z)-(Do(z)+p1(z))l < i/uir

z 6R1/l('fi;; -Dr.l)

Ipo(z)+p1(z)'<1/u ifz 6R2nCr2

For the inductive step, suppose that polynomials p0,

p1, '°' pn have been defined such that

’

(pn(z)I< 1/2n+1 if z e 5;;

1
lwn+1 - (p°(zn+1) + p1(zn+1) + "° + Pn(zn+1))(<1/2n+

lhn(z) " (130(2) + p1(z) + .o. + 1311(2)) ! < l / 2n+1

n __
ifz £8le RJ)/](I>I,n+1 - Dr)

n

'Po(z) + 131(2) + + pun)! < l / 2""1

if z 6 Rn+1 fl OI.n+1

33
where hn (z) is defined by

hn ( r “J ) “(130(1'11‘3) + P1(rn 0%) + H. + Pn_1(rn°fj))

‘ (rn+1-r) /(rn+1 - rn) J -= 1,2,-°°, n

n + 1
We define a continuous function h n+1 on ( \./ RJ )
__ , J"1
D - D b
/r) ( n+2 rn+1 ) y

hn+1 (r on) a (po (rn+1°‘J) + p1(rn+1 0%) +
+ D“ (rn+1ch) ) (rn+2 ' r) / (rn+2'rn+l)

J =- 132’ ...’n+1

and then a function gn+1 on Fn+l by

 

8n+1(z) =- 0 if z E Dru“

=- hn+1(2) - (190(2) + + pn(2) )

n+1 -——-
if z “ng R3 m< Drn+2 - DI. )

n+1

= "n+2 - (po(z) + ---+ pn(z)) if z .- zn+2

- - (po(2) + + pn(2))1f z 6 lime/10%“

Again, 3n+1 is continuous and holomorphic at each interior

point so that we can approximate it to get a polynomial

pn+1 (2) such that

34

lpn+1(z)' < 1/22n+2 ifz 5: BE”

'Wn+2 ' (po(zn+2) + ... + pn+1(zh+2) )' < 1/2n+2

”Wm ' “’0‘” + °°°+ pn+l(z) )l < 1 / 2M2

n+1 __
if 26(JEJ1RJ)/\( Drn+2 - Dr. )

n+1

... n+2
(90(2) + + pn+l (z), < 1/2 if zeRn+2nCrn+2

”This completes the inductive step. We now define f(z)

=J:£O DJ (2). IfzéDrn andm 2, n, thenIpm (z). <

1 / 2m+l so the series converges on each compact subset and
f is holomorphic.
If z 6 R:j and I2 I > rJ, there exists n 2 J such

that I'm £ '2. <rn+1 so that hn is defined at 2. Further

We! = 1:». W + + can

. (rn+l ‘ r) / (rn+1 ’ rn )
5 1 / 211+].

We have

00
Inna) - f(z) l=~ Inn <2) - JZO 193(2),

n 00
=lhn(z) - 32;) 133(2) -J 331 1pJ (2)]

00'
5 bank) - 55:0 p52)! + z: loJ (z)!

J==n+l

35
.4. 1/2n+1 + Z 1/23“1

J=- n+1

= 1/2n+l + l/2I’H—l

= 1/2n
Putting these together gives
[f(z)] 5 1/2n + 1/2n+1

3 / 2n+l

fl

4 1 / 2“‘1

A8 2 ‘9 1 the I‘n involved tends to l and n -—> 09 . Thus

11ml 1‘ (r «J ) .. O. This shows f has radial limit zero
I"?

at each point of a dense subset of C and hence is in B *.
(It is actually in F‘ ).
Next

00
If (23) - wfl - Ingo pn (2,) - le

< J- 1 oo
_. {5:0 I», (2,) - m + 5131me

5 1/2J+1 + E: 1/2’”1

= 11/2“1 + 1/2J

1 / 23"1

Thus, 11m (f (23) - W3 ‘ := O and the result is
.1400

proved.

36
Remark: This is an example of the technique of re-
peatedly using Mergelyan's Theorem. It is essentially the
same as that used by Bagemihl and Seidel in [: 2. Theorem 1.:]

and is generally attributed to them.

BIBLIOGRAPHY

10.

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