I ntrnat. J. Hath. & Mh. Sci.
Vol. 2 No. 2 (1979) 163-186
163
AN INVITATION TO THE STUDY OF
UNIVALENT AND MULTIVALENT FUNCTIONS
A.W. GOODMAN
Department of Mathematics
University of South Florlda
Tampa, Florida 33620
U.S.A.
ABSTRACT.
We begin with the basic definition and soma very simple examples from
the theory of univalent functions.
After a brief look at the literature, we
survey the progress that has been made on certain problems in this field.
The
article ends with a few open questions.
AMS (MOS) SUBJECT CLASSIFICATION (1970) CODES.
Primy 30A6, secondary 30A2,
30A34.
i.
THE HEART OF THE SUBJECT.
We are concerned with power series
w
f(z)
n=0
in the complex variable
z
x
+
iy
b"n z n
b
0
+ bIz +
b2 z2
+
that are convergent in the unit disk
(i.i)
A.W. GOODMAN
164
zl
E
onto some domain
E
Such a power series provides a mapping of
< i.
(A) given the sequence of coefficients
Two questions present themselves:
what can we say about the geometric nature of
b0, bl, b2,
(B) given some geometric property of
D.
D
D:
and
what can we say about the sequence
b 0, b I, b 2,
An example of a nice geometric property is given in
DEFINITION i.
said to be univalent in
E, if it assumes no value more than once in
E.
a function is also called simple or schlicht in
in
E
is univalent in
E
for each complex
w
Stated algebraically, f(z)
E
in
f’ (z) # 0
E, then
in
f(z)
When
is
E.
Such
is univalent
is a simple (schlicht) domain.
f(E)
D
we say that the domain
has at most one solution in
E
f(z) that is regular (holomorphic) in
A function
if the equation
f(z)
If
0.
w
f(z)
0
is univalent
But one must be careful because the converse
E.
is false.
As trivial examples, we mention that
f2(z)
E
z
2
E.
is not univalent in
n.
for each positive integer
larger disk
PROBLEM.
<
z
f(z)
-=
+
z
The function
The function
is univalent in
z
sin z
zn/n
E
while
is univalent in
is univalent in the
/2.
Find a useful set of conditions on the sequence
both necessary and sufficient for
f(z)
n
that are
E.
to be univalent in
This open problem is extremely difficult.
{b
However, partial results have
been obtained, some of which will be stated here.
We observe that if
we may assume that
b
0
f(z)
0
in (!.i) without loss of generality.
this amounts to translating the domain
under the mapping
w
f(z).
f(z)
is univalent, then so is
D
so that
Next we note that
z
f’ (0)
b 0, and hence
Geometrically
goes into
0
b
w-- 0
so we can
I # 0,
UNIVALENT AND MULTIVALENT FUNCTIONS
divide by
bl,
f(z)
and then write
F(z)
z
.
165
in the form
+
n=2
anZ
n
a
Geometrically this amounts to shrinking or expanding the domain
rotating
When
D.
bn /b I"
n
(1.2)
D, and possibly
This does not disturb the unlvalence of the function.
F(z)
has the form (1.2) we say that the function has been normalized.
Other normalizations are possible, but the set of conditions
0, F’ (0)
F(0)
i,
is the most usual and the one we will use here.
We now give a very important example of a normalized univalent function.
We start with the function
l+z
I
z
g(z)
(1.3)
and from the properties of the general linear fractional transformation, it is
easy to see that
Re g > 0.
g(z)
is univalent in
If we square
univalent in
E
and maps
g(z)
E
i
and divide by
h’ (0)
and
g(E)
is the half-plane
we see that the function
2
g (z)
h(z)
is also
onto the entire complex plane minus the sllt
along the negative real axis from 0 to
h(0)
E
4.
-.
To normalize
E
we subtract
Thus we arrive at
K(z)
(1.4)
This important function is called the Koebe function and it maps
entire complex plane minus the slit along the negative real axis
-.
E
onto the
from-1/4
to
On intuitive grounds it is a "largest" univalent function because we cannot
adjoin to
K(E)
any open set without destroying univalence.
A short computation
A.W. GOODMAN
166
(starting with
In=0 zn)
i/(i- z)
K(z)
z
gives the power series for
+ 2z 2 + 3z 3 +
[
K(z),
nz
n--1
This power series for the "largest" univalent function suggests immediately
CONJECTURE i.
If
F(z)
is univalent in
E, and has the power series
(1.2), then
for
n
2, 3,
This conjecture has been the subject of research for more than 60 years and
still represents an open problem, although it has been settled in many special
cases.
A complete account of these results is far beyond the scope of this
paper but the interested reader can pursue the matter in the books by Spencer
and Schaeffer
[27],
Goluzln [8], Jenkins [14], Hayman [ii,
12], Pommerenke [25]
and Schober [28].
The best result known today is due to D. Horowltz [13] who proved that
anl
__<
(1.7)
1.0657 n,
using a very deep method due to Carl FitzGerald [7].
The question raised by Conjecture i, suggests a large number of related
problems, some of which are open while others have been solved completely.
The
Koebe function is the "heart of the subject" because it appears so often, that
a theorem in this subject becomes very interesting if it does
no___!t use the Koebe
167
UNIVALENT AND MULTIVALENT FUNCTIONS
We will mention some of these theorems and
function or some modification of it.
conjectures, but first we give
2.
A SURVEY OF THE LITERATURE.
The specialist in the theory of univalent and multlvalent functions must be
grateful to S. D. Bernardl, who has devoted much of his time to preparing an
exhaustive bibliography of the subject [3].
The first volume covers the period
from 1907 to 1965 and lists 1694 papers.
The second volume covers the period
from 1966 to 1975 and lists 1563 papers,
Each of these ,2 volumes has an exten-
slve index which lists subtopics in this field and those papers that touch on
Thus it is an easy matter for the specialist to determine the
each subtopic.
status of any problem up to the year
1975.
In addition, these two volumes give references to the many survey articles
on the subject that have appeared during this period.
Here we cite only three
such; one by D. Campbell [5], a second by Anderson, Barth and Brannan
pp. 138-144], and third, the article by the author [9],
I,
It is the purpose of
this paper to review this third survey and bring it up to date by reporting on
the progress made on the problems mentioned there.
3.
THE PRESENT STATUS OF SOME OPEN PROBLEMS,
Let
S (z)
n
be the n
th
partial sum of the series (1.2.
n
S (z)
n
QUESTION
an,
a2, a3,
valent in
i.
E,
z
+
k=2
akzk,
Thus
n > 2.
Find necessary and sufficient conditions on the sequence
such that
F(z)
and every partial sum
S (z)
n
is unl-
A.W. GOODMAN
168
It is an easy matter to prove that if
(3,2)
k;2
then
F(z)
F(Zl)
F(z2)
z
z
2
+
i
I
akPk(Z2,Z.l)
k=2
z2k-i + z-2Zl +
’k(Z2,Zl)
where
Indeed as Ao Hurwitz first observed,
Eo
is univalent in
+
zlk-l,
< k
for
Zl, z2
e E,
It
follows that if the condition (3.2) is satisfied then the right side of (3.3)
is never zero for
Zl,
z
e E.
2
Now suppose (3.2) is satisfied, then the same inequality is true for each
S (z)
and hence the condition (3.2) is a sufficient condition for
every
S
n
Let
which
n(z)
to be univalent in
PS(1)
F(z)
F(z)
and
E.
of the form (1.2) for
F(z)
denote the set of all functions
and all the partial sums are univalent in
F(z)
(3.2) is a sufficient condition for
to be in
E.
PS(1).
Thus the condition
However, the condl-
tion is not necessary as we will soon see.
Alexander [2] has proved that if
i
then
F(z)
=>
is univalent in
2a
=>
2
E.
n
3a
3
a
k
=>
__>
nan >=
f i>
For example
(- z)
in (1.2) and
0
.
n--i
z__
n
>= 0,
(3,4)
(.3.5)
UNIVALENT AND MULTIVALENT FUNCTIONS
In fact every partial sum also satisfies (3.4)
satisfies the condition (3.4).
F(z)
to be in
Thus (3.4) is also a sufficient condition for
z) e PS(1).
in (i
and hence
PS(1).
G(z)
169
Further Alexander proved that if
a3z3 + a5z5 +
z
+
3
=> 5a5 =>
+ a2k+iZ 2k+l +
(3.6)
and
I
then
G(z)
function
=>
3a
is univalent in
G(z)
to be in
=> (2k+l)a2k+l =>
PS(1).
is a difficult task.
(3,7)
0
Thus (3.7) is also a sufficient condition for a
E.
But neither of these conditions is included in
the other, nor in the condition (3.2).
PS(1)
f i>
The determination of all functions in
Still more difficult is the generalization to
p-valent functions.
DEFINITION 2.
A function
F(z)
is said to be p-valent in a domain
times in
E
solutions in
and there is some
w
0
n
E
p
if it assumes no value more than
such that
F(z)
w
0
has exactly
p
E, when roots are counted in accordance %rlth their multiplicities.
Observe that the normalization
been dropped.
2-valent in
a z
a
I
i
(used for univalent functions) has
Indeed it is no longer available because
E.
f(z)
z
2
is
170
A.W. GOODMAN
PS(p)
Now let
S (z), with
the partial sums
F(z)
be the set of all functions
n > p
F(z)
such that
The deter-
p.
have valence not exceeding
and all
seems to be very difficult, and very little is known beyond
mlnation of
PS(p)
the trivial
PS(1) C PS(2) C PS(3)
....
.
proved that if
F(z)
z
k
In this direction Ozaki [24] has
+
n=k+l
anZ
n
(_3.9)
i _< k < p
and if
7. n(n+p-l) lan
>= n=k+l
k(p-k+l)
then
E.
F(z)
Thus
Sn,
and all partial sums
with
n _> k
+I
(3
are at most p-valent in
Fz) e PSi).
biunivalent in
F(z), given hy (1.2) when F(z) is
What can we say about
QUESTION 2.
E?
Of course we need
A function
DEFINITION 3.
F(z)
is said to be biunivalent in
if both
E
F(z) and its inverse are univalent in E.
On a first reading,
Let us examine this definition a little more closely.
it sounds as though
is 1-to-l, but l-to-i and regular is really univalent,
F(z)
so biunivalent must mean something different.
Suppose that
F(z)
is regular in
Schwarz’s Lemma that either
such that
F(eia)
< 1
and
F(z)
z
F(eiS)
E
and
f’ (0)
It follows from
i.
or there are two points
> i.
The image of
E
e
ia
and
under such a
el8
171
UNIVALENT AND MULTIVALENT FUNCTIONS
function
F
rw
Let
is shown in Figure i.
F(.)
be an arc of the boundary of
F(e i)
F(eiS)
w-plane
z-plane
Figure i
E
that lles inside
a suitable arc on
on
F
z
in the w-plane and suppose that
Izl
i.
If
F(z)
is
lwl
< i.
F(F
w
F
where
z
z
is
biunivalent, then it must be analytic
and such that it can be continued across
and regular throughout
F
Fz
so that
F
-i
(w)
is defined
This additonal requirement is very stringent,
and is very awkward to use in practice.
The concept of a biunivalent function was introduced by M. Lewin [19] who
proved that if
then
,,]a2[
for every
F(z), given by (1.2),
< 1.51.
is biunlvalent (belongs to the set
At one time, I believed that the bound
n, and that the extremal function should be
]an
< i
z/(l-z)
BI),
was true
=i
z
However, in a very difficult paper, E. Netanyahu [23] ruined this conjecture
by proving that in the set
BI,
max
a2]
4/3.
Jenson and Waadeland [15] proved that if
f(z) e BI, then
]a3]
<
2.51, 5ut
this result is not sharp (a lower bound may be the correct one).
Prof. Waadeland noted (in a letter to the author) that the interesting
example on page I0 of [15] is not in the set
BI
because it has a pole at
A.W. GOODMAN
172
w--
to Busklein [4] has the same defect
16/25, and that another example due
BI
although it was intended to show that in
nothing is known about
set of functions
BI.
QUESTION 3.
Let
nl,
max
a
max
,
+
8 > 0,
X.
X
First suppose that
E.
f(z)
DEFINITION 4.
1
(z)
We let
F(E)
f
intersection of
are arbitrary functions from
(z)?
<
,
<
D
g
8
in
CV
D
(z)
such that
S
are in the interval
has infinite
(0,1/(l+e)),
then
be the set of all normalized univalent functions
is a convex region.
A domain
(3.12)
l+e
and
or
e__:0 958
is unknown.
normalized univalent functions
to the origin.
etc, for the
S, the set of normalized functions that are uni-
,
However if
the maximum valence of
for which
f’(z)),
It is known [i0] that if
then it is possible to find
F(z)
At this time
(3.11)
g(z)
and
l+e
E.
n/4.
>
max (arg
What can we say about
0.042
valence in
n
f(z) + 8g(z)
i, and
some given normalized set
valent in
If’(z) l,
a
be the set of all functions of the form
A
(z)
where
max
F(z)
We let
for which
ST
F(E)
denote the set of all
is starlike with respect
is starlike with respect to the origin if the
with each ray (starting at the origin) is either the ray
itself or a line segment.
For example the functions z
z/(l
z)
maps
E
and z/(!
onto the half-plane
z)
are convex (in CV), since
Re w > -1/2.
The Koebe function
UNIVALENT AND MULTIVALENT FUNCTIONS
z/(l
z) 2
and
< w <
ST, sharp coefficient bounds are known:
for every
ST) because it maps
is not convex but it is starlike (in
the entire complex plane minus the slit
1
n
f(z) e ST, then
and
[an
+
!(1-)
< n
for every
n > i
-1/4.
E
onto
CV
For the classes
lanl
l
is an extreml function,
If
f(z) e CV then
If
2
173
and the Koebe function is an
extremal function.
It is clear from the definitions of
CV
and
ST
that
(3.13)
CV C ST C S.
We now return to question 3.
Styer and Wright [29] have proved that if
two functions
f
and
g
in
ST
8
e
(z)
for which
a conclusion may also hold for other values of
i/2, then there are
Such
is infinite-valent.
and
e
8, but at present the
range of such pairs in unknown.
In [9] Goodman conjectured that if
is at most 2-valent.
for which
"may very
(f + g)/2
f
and
g
are in
CV, then
(f
+ g)/2
Styer and Wright [29] produced a pair of functions in
CV
is 3-valent and they venture the opinion that this sum
well be infinite-valent for some
f
about such sums have been raised and answered.
and
For
g
in
furthr
CV."
Other questions
details see
[5, 9].
There are similar results for the geometric mean of normalized functions,
(3,14)
hut for brevity we omit the details [9, I0]0
We can replace the arithmetic or geometric mean in equations (3.11) and
(3.14) hy other forms of composition.
For this purpose we set
A.W. GOODMAN
174
(z)
z
+
a z
(3.15)
z
(3.16)
n
n-2
Z(z)
z
+
n
n-2
and we define the Cvo operators (convolution, Yaltung)
f(z)g(x)
H(z)
z
+
Z
n=2
(z)vvg(z)
J(z)
QUESTION 4.
If
f
what can we say about
g
and
H(z)
Z
+
[
f,g
and
f,g
by
(3.17)
a b z
n n
ab
n n
belong to the sets
n
Z
X
and
Y
respectively,
J(z)?
or
As an example consider
CONJECTURE 2.
then
f,g
If
is also in
f
and
g
E
S
(normalized univalent function in
E)
S.
If this conjecture were true, then it would be easy to prove (in several
different ways) that
anl
S.
for every function in
_< n
Unfortunately,
conjecture 2 is false, and indeed has been disproved on several different
occasions (see [6]).
One easy approach is to observe that
f(z)Z
Z
z
anzn
[ ’I
n:l
f (t)
t’
dt
(3.19)
UNIVALENT AND MULTIVALENT FUNCTIONS
f(z) e S, then the integral in
Now Biernackf attempted to prove that if
is also in
S.
175
Unfortunately his proof was erroneous, and a counterexample
was found by Krzyz and Lewandowskl [17].
This same counterexample disproves
the conjecture 2, but also disproves the much weaker conjecture:
and
g e CV
then
Suppose that
fg
f
is in
S.
g
are in
and
S.
Since
If
f e S
J(z), given by (3.18) is
not
necessarily univalent, the question naturally arises, what can be said about
the valence of
THEOREM I.
Campbell and Slngh [6] gave the very surprising answer
J(z)?
There are two functions
has infinite valence in
S
f
G e ST
and
such that
fg
E.
We outline the proof.
By a theorem due
to Libera
[20], if
(z) e ST
then
g(z)
is again in
ST.
I
2
(.20)
(t dt
We apply this theorem with
z/(l
(.z)
z)
2,
the Koebe
function, and find that
g(z)
-= 2z
is a starlike univalent function.
f(z)
-=
2n
ntndt
I
i- bi
n
+
1
z
n
(3.21)
Now take
[(i
z)
-l+bi
i],
0 < b _< i.
(3.22)
If we use the properties of the exponential function and the logarithm function,
A.W. GOODbL%N
176
(i- z)
it can be shown that
.
-l+bi
e
C-l+bl) in (l-z)
other items in (3.22) merely normalize the function,
J(z)
f(z),,
2n
n--1
f(z)
We apply this with
n
z
i
n
n-- anZ
n=l
2
+
[i
bi---
e-2n/b
J(z n)
i, 2,
n
The
Now
2
n
(3.23)
f (t)dt.
z
(i- z) bi- i
(3,24)
]"
iz
E, because at the points
This function has infinite valence in
1
E,
given by (3.22) and find that
J(z)
Zn
2
[
is univalent in
always assumes the same value.
This theorem supplies one more counterexample for Conjecture
2, and as
far as the author can see, it is the simplest of the many counterexamples now
known [6, 9],
Returning to question
Y
4, we observe that different selections for X and
and different properties for
H(z)
and
J(z), yield
a tremendous number
However the most interesting ones have been settled in a
of open problems.
very important paper by Ruscheweyh and She i l- Smal l [26].
Here we merely state
some of their results.
THEOREM 2.
given by
(3.17),
THEOREM 3.
J(z)
If
f(z)
and
gz)
is also convex in
If
f(z)
and
E.
g(z)
given by (3,18) is also starlike
THEOREM 4.
If
normalized convex in
f(z)
E, then H(z),
are normalized convex in
are normalized starlike in
in
E, then
E.
is normalized starlike in
E, then H(z), given by (.3.17)
E
and
g(z)
is starlike in
is
E.
UNIVALENT AND MULTIVALENT FUNCTIONS
DEFINITION 5.
Kaplan [16].
to be close-to-convex in
E
of the form 63015) is said
f(z)
if there is a univalent starlike function
(z)
E
such that in
Re
We let
A function
177
zf’ (Z) > 0
(3.25)
-(z)
denote the set of all functions that are normalized and close-to-
CC
E.
convex in
Geometrically, the condition (3.25) implies that the image of each circle
zl
r < i
is a curve with the property that as
tangent vector does not decrease by more than
-=
8
increases the angle of the
in any interval
[81,82 ],
Thus the curve can not make a "hairpin bend" backward to intersect itself.
This means that each function in
CC contains CV
are not in
CC.
THEOREM 5.
and
CC
is univalent in
E.
Furthermore the class
On the other hand there are univalent functions that
ST.
Ruscheweyh and Shell-Small [26] extended Theorem 4 by proving:
If
f(z)
rrmalized convex in
E:
THEOREM 6.
f(z)
If
normalized starlike in
is normalized close-to-convex in
then
E
and
g(z)
is
E.
H(2), given by (3.17) is close-to-convex in
is normalized close-to-convex in
E, then J(z)
E
and
g(z)
is
given by (3.18 is close-to-convex in
E.
It is worthwhile to compare Theorem 1 and 6 and to observe that when "close-
to-convex"
1
is replaced by
"univalent", the
maximum valence of
f**g
Jumps from
.
to
We now look at the bounds for the coefficients if
In his thesis, the author initiated
f(z)
is p-valent in
E.
A.W. GOODMAN
178
CONJECTURE 3.
[n-I
f(z)
If
is p-valent In
a z
n
E, then
[ant-< k=l pk)’. (p-k).’ (n-p-l).’ (n2-k2) lakl
for every
<3.261
n > p.
The reader will find in [9] some historical notes and an account of the
progress on this conjecture up to the year 1968.
anl <_ n[all
yields
When
p
i, inequality (3.26)
Certainly, we
and this is equivalent to Conjecture i.
can hardly expect to prove (3.26) in full generality, as long as the inequality
lanl <= nlal{
for univalent functions is still unsettled.
However,
there are
some recent advances that are worth recording.
We generalized Definition 5 of close-to-convex univalent functions so that
CC)
the new class
detail, a function
f(z)
is regular in
Jr0,1)
(z)
E and if there is some r
zl
the image of the circle
D) and the curve makes
Now a function
convex of order
f(z)
ST)
belongs to the class
starlike with respect to the origin
in
Without going into too much
includes p-valent functions.
p
0
under
r
(arg f(re i8)
< i
f(z)
(starlike p-valent) if
such that for each
r
is a curve that is
is monotonic increasing
complete turns around the origin.
is said to belong to the class
p) if there is a function
(z)
in
ST(p)
CC(p)
(close-to-
and an
r
I
< i,
such that
zf’ (z}
>
Re
0
(z)
for
r
I
<
[z
< i
The main point
(we omit some minor refinements here) [21].
.here
is that
ST(p) C CC(p)
(any starlike function is
in
UNIVALENT AND MULTIVALENTFUNCTIONS
179
close-to-convex) and hence any theorem about close-to-convex functions includes
the same result about starlike functions as a special case.
Two results by
A. E. Livingston deserve mention.
THEOREM 7.
[21].
Let
f(z)
be close-to-convex of order
Then the inequality (3.26) is true for
n
p
+ i,
p
in
and this inequality is
In other words the bound for the coefficients of a p-valent function is
f(z) e CC(p)
true in the special case that
and we look only at the
(p
+ l)st
coefficient.
THEOREM 8.
in
E, and that
for every
n > p
[22].
a
a
I
f(z)
Suppose that
2
and every pair
ap_ 2
ap_ 1
O.
and
is close-to-convex of order
p
Then the inequality (3.26) is true
ap
not both zero.
The inequality
is sharp in both variables.
For the latest results about close-to-convex functions of order
p the
reader should consult the paper by R. Leach [18].
4.
MORE OPEN PROBLEMS.
Let
A denote the set of all functions
f(z)
z
+
n= 2
E, and let
that are regular in
are also univalent in
S
a z
n
denote the subset of those functions that
E.
It is not too difficult to ask questions about the set
about the relation of
S
to
A.
S, or questions
Here some caution is necessary.
When one
considers the 3257 papers listed in Bernardi’s bibliography together with those
published since 1975, and those few that Bernardi may have missed, it is clear
A.W. GOODMAN
180
that a question selected at random may be answered already either explicitly or
With this note of warning to the
implicitly in some work previously published.
reader we pose a few problems, that w believe are still open.
f(z)
Suppose that
1.
L*
For flxed
L.
able curve of length
f(E)
and that
S
is in
>
is bounded by a rectifi-
S(L*)
2, let
denote the subset of
those functions for which
L
L*. We
can ask many questions about
example, in this set find
sup
If(z) I,
find
y
omits
2
in
Let
E, find min
lI,
max
a
nl
for each
n.
S(L*).
If
For
f(z
etc.
f(E)
A denote the area of
S
note the subset of those functions in
A*
For each fixed
for which
A < A*.
same type questions for this set that we asked about the set
let
S(A*)
de-
We can ask the
S(L*).
Here, we
have the help of the well-known formula for the area of the image of the disk
E
Izl
<r
2
[ nla
(r +
A
12r2n)
(4.2)
r< I,
nffi2
but there is no way to find an analogous formula for
(4.3)
L
f(Er).
the length of the boundary of
3.
Zl.ffi rle
in
S?
Let
s
and
and
8
z
r2e
2
be fixed and suppose that
What can we say about
(s,t)
More generally if
the minimum or maximum value of
it is known that if
8
s
+
,
f(z)
omits the two values
mln (r
r
I + 2)
for
f(z)
is any fixed function what can we say about
(rl,r 2)
then
for
f(z)
in
S?
As a simple example,
UNIVALENT AND MULTIVALENT FUNCTIONS
Suppose that
in
E.
course
If
f(z)
max
anl
4.
z
z
and
I
is in
S, find
f(z)
in
M(r,f)
If
r
(0,i)
in
f(z)
lanl
max
is in
max
A,
E
5.
f(z)
If
M(r,f)
f(z)
and
z
2
Here, of
n > i.
2.
set
(4.5)
r
(4.6)
2
i- r)
Can we say anything about the valence
is sufficient to insure that
f(z)
is
(or in some fixed smaller disk)?
is in
A
and
Re(!
then
I
z
sup M(r,f).
O<r<l
S, it is well known that
What condition on
univalent in
z
and
I
If(re i8)I
Suppose, conversely, we are given M(rf).
f(z)?
for each fixed
z
M--
and let
M(r, f) <
of
omits both
is a function of the omitted values
As is customary, for
for each
f(z)
are fixed and
2
’81
is univalent and
+
f(E)
z
f")
>
f-,)
0,
z
E,
(4.7)
z
E,
(4.8)
is a convex region.
If
Re
then
fz)
is univalent and
(z)
zf’
(z}--)
f(E)
>
O,
is starlike with respect to the origin.
A.W. GOODMAN
182
Now in (4.7) and (4.8) replace
some fixed positive constant.
sarily univalent).
S
D
The Koebe domain
that is contained in
M
f(z)
C
is
must be uni-
as a function of
M be some collection of functions each regular in
Let
6.
if
f(z)
where
C?
the constant
D
We can no longer assert that
Can we give some upper bound on the valence of
valent.
C
on the right side by
0
f(E)
for the set
M
is the largest domain
for every function in the set
then the Koebe domain is the disk
lwl
<
(not neces-
E
M.
For example,
1/4.
M,
Koebe domains have been determined for a large number of different sets
for which the Koebe domain is unknown.
but we mention here two sets
M
first recall that a function
f(z)
all
z
in
E
in
We
is said to be typically-real if for
A
we have
(4.9)
(Im z) (Ira f(z)) > 0.
This merely states that the upper half of the unit disk must go into the
upper half-plane under
the lower half-plane.
f(z), and the lower half of the unit disk must go into
We let
The interval (-i,i) goes into the real axis.
TR
denote the set of all typically-real functions with the normalization (4.1).
The Koebe domain for the set
is the domain bounded by the curve in
TR
polar coordinates
sin
4e(
e
0 <
e)
e
<
(4.10)
together wlth lts reflection in the real axis.
One open problem (suggested by E. Merkes) is to find the Koebe domain for
the set of odd typically-real functions, with
f’ (0)
I.
UNIVALENT AND MULTIVALENT FUNCTIONS
TR
in
Let
7.
+
i
1
D1
/.
<
valent in
D
f(z)I
for which
< M
for
z
E.
in
be the subset of functions
Find the Koebe domain for the
TR(M).
set
z
TR(M)
M > i be a fixed constant and let
Let
183
D
1
D
and that
8.
p
in
/
<
is the largest domain with this property.
1
p
D
find
and
D
such that every
Thus
TR.
TR.
the domain of p-valence for the set
P
words find the largest domain
at most
i
Goluzin first proved that every typlcally-real function is uni-
is called the domain of univalence for the set
For fixed
z
be the domain common to the two disks
f(z)
in
TR
In other
has valence
D.
Finally we examine families of rational functions of the form
f(z)-k=l
where the
and
a
k
z
a
k
are subject to certain restrictions.
As P. Todorov pointed out to me in a letter, Thale [30] was the first to
prove that if
> 0, and
-i < a < i, then
k
f(z)
zl
is univalent in
and this is a maximal domain of unlvalence for the family of functions
> i
t.hat
satisfy these conditions.
Later, Cakalov rediscovered the same theorem.
that if
> 0
and
akl
<
I, then f(z)
Further
is univalent in
Cakalov proved
zl
>
/,
and this
is the maximal domain of univalence.
Distler gave a beautiful generalization of these two theorems by finding
the domain of univalence of the family (4.12) when the
lie in some fixed bounded set, and
arg
I
<
00/2
a
k
are restricted to
for some fixed
00
<
.
A.W. GOODMAN
184
If
denotes the domain of univalence in any of the above families
D
Todorov [31, 32] proved that one can add all of the boundary points of
f(z).
without destroying the univalence of
D
(with one exception) to
D
Suppose that in (4.11), we assume that
< a
and that all
a + aj+
each
j
1
2
Todorov [31] proved that,
< a
Let
are real.
Z
I
-<-=
f(z)
< a
2
3
<
an
(4.12)
<
K. be the closed disk
3
aj+1
aj
2
j
1, 2,
given by 4.11 is univalent in
n-1.
Kj
{ajl
(4.13)
for
n-l.
i, 2,
As interesting as the above results may be, the really difficult problem is
the domain of p-valence, for each p > i for the
P
family of functions of the form (4.12) for some reasonable set of conditions
still untouched.
and a
on the constants
conjecture for
D
Find
D
k.
For this problem, I can not even suggest a
P
Finally we observe that Todorov [32] has investigated the applications of
sums of the form (4.11) to magnetic fields.
In this work, some of the residues
may be negative.
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2.
Alexander, J. W.
Functions which Map the Interior of the Unit Circle upon
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Simple Regions,
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185
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Bernardi, S. D. A Bib!$ography of Schlicht Functions, New York University,
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5.
Campbell, D. A Survey of Properties of the Convex Combination of Univalent
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6.
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FitzGerald, Carl. Quadratic Inequalities and Coefficient Estimates for
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Goluzin, G. M.
9.
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I0.
The Valence of Certain
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Hayman, W. K.
1958.
Means, Jour
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13.
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A.W. GOODMAN
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