We begin with the basic definition and somavery simple examples from ... 165 divide by bl, and then write f(z) in the form . n bn. F(z) z +. anZ a. /b n=2 n. I". (1.2) ... cases. A complete account ofthese results is farbeyond the scope of this ... grateful to S. D. Bernardl, who has devoted much ofhis time to preparing an ... Page 7 ...
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.
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
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
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
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
