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```CHAPTER 2.
Special classes:
convex, starlike, real, typically
real, close-to-convex, bounded boundary rotation
We shall examine some special families for which we can determine the closed convex hulls and corresponding extreme points.
In
principle then, all linear extremal problems for these families are
elementary.
We shall restrict our attention to normalized analytic functions
N= (£EH(U)
Then
f(O)=O, e(O) =l}
.
nN
S = Hu(U)
(f E 5 :
£(U)
is convex}
S*= (f E S:
f(U)
is starlike with respect to the origin}
S = (f E S:
lR
T = {f EN :
lR
f (n) (0) E lR
K =
C
=
(f E N:
for all
(Jm z) (Jm f(z))
~
n
~
2}
O}
~e[ f' /eillq>'] > 0
for some
c:pEK
and
IlElR}
are the familiar normalized schlicht, convex, starlike, real,
typically real, and close-to-convex classes, respectively.
Functions in
Functions in
if
f E TlR
Tm
then
to the real axis.
K
and
S*
have obvious geometric properties.
have real Maclaurin coefficients.
f (z) = f (z)
Clearly
and
f (U)
is symmetric with respect
SlR c: T lR , but
nonunivalent functions (e.g.,
z
+ z3)
Consequently,
TlR
contains also some
On the other hand, close-
to-convex functions turn out to be univalent, and this is the motivation for the definition of
LEMMA2,l.
Proof.
Since
D
If
Suppose
is convex,
FEH(D)
C.
where
D
F (zl) = F (z2)
(l-t)zl + tZ 2 ED
is convex and
for some
for
ReF'>O,
then
zl' z2 ED, zl" z2 •
t E [0,1 ] • Therefore
7
1
S (~eF')dt
> 0
o
THEOREM 2.2.
Assume
.f.!:Q.Q1.
cp E K, and
If
Ct
E 1R
•
f E C, then
fEH(U)
f
and
is univalent.
F = fa f -1 E H (D)
Then
ReF'= ~ef'/q;/> 0, by Lemma 2.1
Since
t
e
where
D = f (U)
is
and
F.~=f
where
~ef'/~' >0
F
cp,
convex.
are univalent.
We shall need to convert the geometric definitions for
S*
io.
K
and
into analytic relations.
THEOREM 2.3.
If
f E s, then the following are equivalent:
(a)
fES*:
(b)
Dr = f(\z\ <r)
0 <r <1 :
origin for
(c)
Proof.
zf'/fEP.
o
(b)
=>
for some
(a)
Iw(z)
1
:s;
(a)
oDe arg f (re i9 ) = Jm oDe log f = Re zf' If
sufficiently close to
(b):
Izl
for
is a non-
i.e.,
zf' If \ z=O = 1 •
is obvious since each point of
For
O<t<l,
1f- l
0 < t < 1.
(tw 0)
So
1=
Dr
f (U)
= Cl(tf(Z»
Suppse woE Dr'
Iw
belongs to
1.
UJ(z)
by Schwarz's lemma.
1f- l (~) 1 < rand
tWoE Dr
e,
by the minimum principle, since
r
=>
:s;
arg f (re ie )
is starlike iff
(b)" (c):
decreasing function of
Re zf If> 0
is starlike with respect to the
(2 0 )
1
\$
1201 <
is starlike.
satisfies
Then
r.
IZol =
Therefore
Dr
THEOREM 2.4.
f E 5, then tbe following are equivalent:
(a)
f EK
(b)
Dr
(c)
1
(d)
zf' ES* •
Proof.
ae
f(lzl <r)
is convex for
~
(c):
Dr
is convex iff the tangent angle
R.e {l + zftl If'] > 0
(b) '" (a)
D
r
A
for
r
sufficiently close to
Suppose
hence, w,. = f(zA)
= g(Z2)
where
zl ;i z2
for
zA E U.
for some
g(z) = Af(zz l /z2) + (l-A)f(z).
wA
~~ '" ~e (1 + zftl If') •
[1 + zt" If' Jz=O '" 1 •
by the minimum principle since
= ).f(zl) + (1-A)f(z2) E feU)
g(U) Cf(U).
i.e.,
is obvious since any two points of
(a) '" (b) :
w
e,
is a nondecreasing function of
Jm oOe log
some
O<r<lr
+ zf tl If' E P r
(b)
Of (re i9 )
arg
If
By Theorem 1.1,
Then
feU)
.
1
I z l l1OI z 2 1 < r
0<1..<1
since
Then
is convex;
feU)
We must show that
gEH(U),
•
belong to
wI.. € Dr'
g(O) = D, and
g(lzl <r) CD ; in particular,
r
EDr •
(c)" (d):
f
satisfies
(c)
iff
g
zft
satisfies part (c)
of Theorem 2.3 •
COROLLARY 2.5.
and
C
(f E N:
ie[zf 'le iC1 gJ > 0
for some
9 E S*
a E lR} •
~.
Apply Theorem 2.4(d) to the definition of
COROLLARY 2.6.
~.
choosing
Let
a
K C S*
=D
and
C.
K C S* C C •
is geometrically obvious.
g= f
in Corollary 2.5 •
S* C C
follows by
9
It will be useful later to know that the analytic conditions of
Theorems 2.3(c) and 2.4(c) actually imply univalence.
We therefore
sketch the proof:
Let
THEOREM 2.7.
(a)
If
(b)
If
Proof.
fEN.
l+zf"lf'Ep, then
f ( I z I= r)
Yr
f
assumes
If
(a)
n(r/O)
so that
1
2TT
Yr
in
zf'/fEP
winds around
Iz
z = re
S2n
1
2TT
Yr
arg
Wo
and also the number
(1
i9
then
I
+ positive powers of z)d9 = 1
0
Yr
curve.
asa arg f(re is )
Since
and
1 + zf" I f ' E P
211
2TT
~~ld9 = S~e (1 + z{' )de = ~e f
o
z
= re i 9
I
then
(1 + positive powers of z)d8
= 2n
o
So the total variation of the tangent angle to
for e
dz
is a Jordan curve.
If
(b)
(z)
Wo ~ Vr
I < r.
and
S2n zf'
f de =
f'
2ni
f(z)-w o
Izl=r
winds once around the origin.
2TT
o
Wo
S
_1_
o
o
~ezf'lf > 0,
SI :e
S~
-w-w
2ni
gives the number of times
fEK
is a closed curve and for
1
n(r,wo )
of times
fES *
zf'/fEP, then
Yr
is
2TT
There-
can wind around any point at most once and is a Jordan
Since the total change in the tangent angle is also
2TT,
the winding is in the positive direction.
n(r/wo )
In both cases (a) and (b) the winding numbers
identically one for
in the exterior of
interior of
Yr
w
a
vr
of
Yr.
f
maps
Therefore
exactly once in
point of the exterior of
mapping,
in the interior of
I z\ <r
vr
in
Iz I
Iz I
vr
and zero for
are
Wo
assumes each point in the
f
<r
<r
and does not assume any
Since
f
is an open
in a one-to-one fashion onto the interior
This is true for all
r < 1.
Therefore
f
is univalent
•
10
in
U.
Since now
f E S, the respective assertions follow from
Theorems 2.3 and 2.4 •
THEOREM 2. B (Ruscheweyh and Sheil-Small [R4 1 ).
If
f
e K,
then
-L}
~e{......L.k.:L f(z)-f(t) _
z-C z-t f(C)-f(t)
z-C
Proof.
> 1
-
The function
z,
for all
2
C, t E u
f(z)-f(t)
f(C)-f(t)
~.k.:L
F(z,C,t) - z-C z-t
_
•
~
z-C
_ 1
is analytic in all three variables when extended by continuity for
z = c, z = t, and
~e
Since
F(e
ia
t, e
f(lzl ,;r)
arg
whenever
is
•
is convex for
if
0 <
f(el.Pt)-f(t)
if
0 <: a. <:
'Q
[-TT, OJ
(f(e~at)-f(t)}
Jm
and
I = I cI = I t I
z,C,tEU.
~e
the minimum principle implies
Actually
~eF>O
since
F(O,O,O)
{ Iz
and finally for
I "I CI
first for
Iz
It
< 1 .)
II
I (; \ I z \
I
=
COROLLARY 2.9 (Suffridge [S23J).
zf'(z)
Re { f(z)-f(C)
~.
Let
t
If we also let
~
z
-
-L}>l
z- C
'2
It I
If
I
~e
=1.
{ I z I = r} 3
I
for
(Toapply
is the
individually to
then for
f EK
for all
F "= 0
<: r} 3, or else one can
z,C,t
apply it successively in the variables
a
iteF(z,C,t) "=0
is a harmonic function
F(z, Ct)
distinguished boundary of the po1ydisk
Re F >
S <: 2TT
By continuity,
Since
< 21'1
have the same sign
f(el.St)-f(t)
the minimum principle one can observe that
obtain
~ <a
f(eiUt)_f(t) E {[ 0,1'1 ]
of all three variables I
all
then
r= Itl,
~eF(eiat, eiet,t) ~O.
Iz
a.1 S,
0 < a, S <: 21'1,
If
1
~ J [f(eiCtt)_f(t»)
t, t) _
- sin ~ (a-~) s~
mtf (ei~t) -fIt)
sin J., (a-[:l)
Therefore
so that
C= t
Z,
I
I CI , I z I ,; \ t
then
CE U
in Theorem 2.8 •
C ~ z, the above inequality implies that
\
I
11
1 + zf" If'
e P.
It follows then from Theorem 2.7 (b) that the condi-
tions of Theorem 2.8 and corollary 2.9 are not only necessary, but
also sufficient for
fEN
to belong to
are invariant when replacing
f
characterize the convexity of
K.
In fact, the expressions
Af + B.
by
f(U)
They therefore
independent of normalization.
COROLLARY 2.10 (Stroh"acker [S21J, Marx[Ml]).
~e zf'/f
.R!Q2f.
Set
(, = 0
> ~
fEK, then
If
Re f/z > ~ •
and
in Corollary 2.9, and let
t = 0,
C" 0
in
Theorem 2.8.
f
is called starlike of order
~
if
fore convex mappings are starlike of order
Re zf' If >
~.
There-
(l.
More importantly, the
second condition of Corollary 2.10 turns out to describe precisely
the closed convex hull of
are due to
K.
This and related results that follow
L. Brickman, T. H. MackGregor, and D. R. Wilken [B10J.
THEOREM 2.11 ([SlOJ).
coK
K
(fEN: Ref/z
and
co K
are compact,
>~)
( jz/(l-~z)d~; ~ is a probability measure on 1~1=1},
\'1")1=1
and
ECOK'"
Proof.
(z/(l-T'lz)
The mapping
homeomorphism of
H(U)
IT'll "'l} •
r,
defined by
r, (g)
onto the subspace
r,(P) = [f EN; Re f/z >J:i).
comes from Theorem
measure
~
f(z)
1.6~
namely,
P
A second representation for
f E '£(P)
~z f[l+ (l+T'lz)/(l-"z))d~
OJ.
By
onto a compact convex
.£(P)
iff there is a probability
such that
1,,1=1
is a linear
(h E H(U); h(O) =
Theorem A.2 it maps the compact convex set
set
"'!o:!z (1 + g)
Sz/(l-llz)d~
I~\"'l
12
It follows from Corollary 2.10 that
K
K
is a subset of
!(P)
is closed since the relation of Theorem 2.4(c) is preserved
K.
under locally uniform convergence of functions in
K
!(P)
is a compact subset of the compact convex set
co K C:.l: (P)
Observe that the mappings
C::co
.s:(P) =co E!(p)
Therefore
Therefore
•
By Theorems A.2 and 1.5,
and
Consequently,
! (p) =
co
E.s:(p) = £(Ep) = (z/(l-nz):
K.
belong to
z/(l-nz )
Hence
Inl =l} •
E£(p)C:K
by the Krein-Milman theorem (Appendix A).
K
K, and the proof is complete.
The extreme points
z/(l-nz)
whose boundaries have distance
@
map
U
onto the half-planes
from the origin.
~
/......z/(l-nz)li
~
/ / /
I ,/
i
+'1--1'l....+--f".;:-I
/
L(f) = f{n) (z)
Since the extreme values of the
must occur at an extreme point (Theorem A.3).
we have the following
immediate application.
TIiEOREM 2.12.
If
I'"
z +
cnz n E K, then
n=2
f(z)
If(z)1 "Izl/(l-Izl)
and
If(n)(z)1 "nl/(l_lzl)n+l
for all
and
In particular,
'..l
I c n I .. 1
for
zeu
n:;, 1
n = 2,3, • •• .
We turn to the class
s*.
THEOREM 2.13.
iff there exists a probability measure
f E S*
such that
f(z)
=z
exp[-2 Jlog(l-nz)diJ. ]
1T'l1=1
Moreover, the probability measure iJ.
is unique.
.
.
13
Proof.
If
fE5*, then by Theorems 2.3(cl and 1.6 there is a
unique probability measure
such that
~
[zf'/f-ll!z = S[(l+"zl/(l-"zl-lJ/z d\.l = S21l/(1-"Z)d\.l •
1,,1=1
Therefore
1111=1
log f/z = -2 Slog(l-11Z)d\.l
Conversely, if
f E S*
zf I If E P
and
by Theorem 2.7 (a) •
COROLLARY 2.14.
all
by integration.
Inl=l
f has the given form, then
f E 5*, then
If
lim arg f (re
r-+l
e.
Proof.
Represent
is )
arg f/z = -2 Sarg(l-11z)dIJ.
exists for
I" 1=1
limit exists by the Lebesgue bounded convergence theorem.
To determine the
co 5*
we shall exploit the connection with
convex mappings rather than Theorem 2.13.
THEOREM 2.15 ([BlOJ).
S*
co S* = ( !z/(1-11Z)2d\.l : \.l
Proof.
defined by
~
homeomorphism of the space
Theorems 2.4(d) and 2.7.
111 I=1} ,
I" 1=11
(z/ (l-11Z) 2 :
The mapping
are compact,
is a probability measure on
1,,1=1
and
co S*
and
= zg'
~(g)
is a linear
(hEH(U) :h(O) =O}, and
~(K) =5*
by
The results now follow from applying
Theorem A.2 to Theorem 2.11.
The starlike functions
and map
u
@
are called ~ functions
z/(l-11z ) 2
onto the complement of a ray from
I/U// i
!
..
'j /
I
z/(l-Tlz)
2
"----'"
"
.
,
to
i
...
i_\1T'L,i I.
I. '/' / ...-:-r
:
...
;'
0
/' ./
'
I
J
By examining just these extreme points we have the following application.
14
THEOREM 2.16.
If
I'" anz n E S*,
fez) = z +
then
n=2
If(n)(z)1
,.;; nl(n+lzl)/(1_lzl)n+2
In particular,
I an I ,.;; n
for
for all
SlR and
TJR.
TJR is precisely the closed convex hull of
n;'O.
We shall see that
SlR'
The following are equivalent:
(al
fET lR :
(b)
(1-z2)f!zEP
(cl
there exists a
[-l,lJ
and
n = 2,3, • •• •
We turn now to the classes
THEOREM 2.17.
zEU
and
f(n) (0) ElR
for
all
n"'2
(unique) probability measure
~
on
such that
fez)
Jz/(1-2xz + z2)dlJ
[-1,1]
Proof.
if
z
(a) => (b):
is real so that
If
f E T lR ,
fen) (0) E lR
then by continuity
for all
n.
f
is real
0 <r < 1
If
and
I z I = 1, then
Re[(1-z 2 )rf(rz)lz] = 2 Jm(rz) • Jmf(rz) "'0 •
By the minimum principle,
r
ot
Re(1-z 2 )f(z)/zl;,0.
l: then
minimum principle since
(b)
where
=>
Re[ (1-z 2 )rf(rz)/z];'0
(c):
pEP.
in
U.
Let
Zero is not possible by the
(1-z2)f/zl
= 1 •
z=O
Since all Maclaurin coefficients of
f
are real,
By Theorem 1.6 there is a probability measure
')
such that
fez) =
Sz/(1-2(~e TI)z+z2)d\l = Jz/(1-2xz+z2ldl-!
I TlI=l
where "I-l(x) = \I(TI)+\I(ii)," Le.,
AC[-l,l].
[-l,lJ
I-l(A) =
For the uniqueness we assume
'9
\I£e1.: cosBEA})
for
fez) = Sz/(1-2Xz+z 2 )dl-l k •
[-1,1]
15
i.e.,
"k(B)=J.,lJ.k([cos6: e
for
Bc::(lnl =l} •
i9
-i8
EB,os;es;n))
2
J(l+Tlz) / (l-llz)dvk= (l-z )f/z, and by the
=1
uniqueness of the Herglotz representa tion v 1 = v 2: hence U 1 = IJ. 2
(c) '" (a):
Then
EB,O",Bsn})+J.,lJ.k((cos9:e
\ nl
It is clear that
(1_\z\2)\1_2xz+z 2 \-2.9mz
that
fEN.
Since
.9m[z/(1-2xz + z2) ]
has the same sign as
Jm z, it follows
f E TlR •
THEOREM 2.18 ([BIO]).
co 8lR =
also convex,
8lR and
TlR
are compact,
TlR
is
T lR , and
(z/(1-2xz+z 2 ): xE[-I,I]).
lR
That SlR and TlR are closed, T]R is convex, and
ET
Proof.
SlR c:: T lR is clear from their definitions.
TlR
C
2
(zp/(l-z ):
Hence both
By Theorem 2.17 (b) ,
pEP}, which is compact since
SlR and
co
TlR are compact and
P
is compact.
SlR c;; TlR •
Since the representation in Theorem 2.l7(c) is unique,
= {z/ (1-2xz + z2): x E [-1,1]).
In fact, each function
]R
2
z/(1-2xz + z) belongs to SlR so that by the Krein-Milman theorem
ET
TlR =
co
ETlR C
co
S]R: therefore
z/(1-2xz + z2)
The functions
real slits from
(Note for
4
x =::1
co
(1 + x) -1
to
SlR = T]R •
®U!
/
appears and we have
O·
2
II,
1/
'
ET
These mappings are starlike, i.e.,
we have the estimates of Theorem 2.161
f(z) = z
'"
lR
/./
,
....-----)
/
If
~(I_x)-1
j
z/(1-2xz + z )
Koebe functions.)
THEOREM 2.19.
onto the complement of
and from
-'"
one
of the slits dis-
U
map
,
I
i
'
/ /' /
,I
,'
1 _'1
; -~ ( 1 + xl
I
C
/1' ,/
0/
I
;' (l-x)
I / ! I .' ! !
!
/
.",
S*, so as an application
+ ~ anz n ET lR , then
n=2
1/ /' 1/,:
/ / ,i i ,J
I'.
. '/,i"
-1
i
I,'
I
•
to
16
~ nl(n+ Izl)/(1_lzl)n+2
If(n)(z)1
In particular,
I an I
S
n
for
for all
zEu
and
n;;'O.
n = 2,3, • •• •
The following theorem is a fundamental tool in the further study
of special families.
THEOREM 2.20 (Brannan, Clunie, and Kirwan [BB]).
F = [fEH(U) : f-< (l+cz)/(l-z)}
Fet = [fet:fEF}
let
Proof.
F
Since
GC1. = (
and
probabili ty measure on
for fixed
I Tl I=l}
ITlI=l
Then
.
P.
do not vanish so that
po.
Since
Go.
on
is weakly compact.
\-l
Go.
are compact,
(l+z )/ (l-z) +J.,(l-c),
F
is compact and
I cis: 1, the functions in
The
is compact since the set of probability measures
Suppose now
measures
is a
\-l
is well defined and compact.
convex set
I Til = 1
~(l+c)
Therefore
EF = «l+cTlz)/(l-Tlz) : ITlI =l}
F
F, Fa. , and
a ., 1
For
Icl sl •
SC (1 + cTlz) I (l-Tlz) Jet d\-l
(l+cz)/(l-z) =
is an affine image of
c,
Let
on
g E EGa..
I Tli = 1
Let
be the set of probabili ty
Pg
such that
S[ (1 + cTlz) /(l-Tlz) ]ctdl-l
= g(z)
ITl I =l
"
9
I- ¢
and is compact in the weak topology.
the Krein-Milman theorem.
Let
some probability measures
VI
C1.
1 dv 2 •
Since
vEEr.' If
9
and v 2 ' then
Therefore
E" '" ¢ by
'1
v = A. VI + (I-A.) v 2 for
g(z) = ASC
JC1. d v l +
g E EGet, this is impossible unless
v l ,v 2 E "g ; but the latter is impossible since
v
must be a point mass and
v E Ef>. Therefore
Ct g
EGa. c [[ (1 + CTlz) I (l-Tlz) 1 : I ril =l} •
The sets are actually equal since i f
Agl(z) + (1-A)g2(z), then e v ery
(1-A)g2(nnoz)
If
and
f E F-E F '
[(1 + cTloz) I (l-Tloz)
C(l+cTlz) / (l-Tlz)]
et
it
=
= A91(T11loz) +
EGa = ¢ .
then distinct
f l , f2
eP
and
I. E (0,1)
exist
17
such that
Af1 + (1-A) f 2 •
~-lfk' k
where
maps
=
f
U
= 1.2,
Now
fa.
=
are distinct.
onto a convex set and
0.:;"
Afa.-1f1 + (1-)..)
Since
~-lf2
loge (1 + cz) / (l-z) ]
1 ,
10gf + (l/a)log fk < log[(l+cz)/(l-z)]
(1-1/0.)
so that
Therefore
By the Krein-Milman
The next theorem was verified in many cases in [ BB] and in
general first by D. Aharonov and S. Friedland [A3].
The following
elementary proof is due to D. A. Brannan [B7].
THEOREM2.21.
Then for
0.:;"
1
Suppose
the coefficients of
sponding coefficients of
Proof.
g
f«l+cz)/(l-z)
By
g
'it< h
fo.
[(1 + z)/(l-z)
for some
~
Fix
Jo. •
we mean that the coefficients of
0.:;,,1.
1>< [(l+z)/(l-z),a
If
Icl "1
is obvious.
and employ induction.
If
and
Iclslo
are dominated by the corre-
are dominated by the corresponding coefficients of
O.:k':n.
c,
zk
for
for
h
f< (l+cz)/(l-z), then
fa « [ (1 + z) 1 (l_z)]o.
We assume
n
Icl sl, then
[(l+Cz)/(l_z)]l-lIa[(l+z)/(l_z)]l/a < (l+yz)/(l-z)
for some
y,
I yl S I .
Therefore
[(1+cz)/(1-z)]a-1[(1+z)/(1-Z)] «[(l+z)/(l_z)]o.
n
by the induction hypothesis.
Since
[(l+z)/(l-z)]o.
and
1/(l_z 2)
have all nonegative coefficients, it follows that
(1 + cz)o.-l = (1 + CZ)a-1(~) __
1_ «
(l_z)(l + 1
'l-z
1-z
(1_z2) n
Now
azo [(l+cz)/(l-z)]
0.
<~
(1 + z)o.
1
1-z
(l_zZ)
= o.(l+c)(l+cz) a -1 l(l-z) Cl + 1
20. (1 + z)
(1-1
/(l-z)
0.+1
.
(l+z)o.-l
(l_z)o. + 1
18
By integra tion,
[(1 + CZ)/(l-z)Ja <<.
n+.l
[(1 + z)/(l-z)Ja
If now
f< (1 + cz)/(l-z), then by Theorems A.3 and 2.20 we may dominate the
coefficients of
fa
by dominating the coefficients of the extreme
1
ri~l [(l+z)/(l-z)]a, we have
fa ~h [(l+Z)/(l-z)](1, and the
induction is complete.
Recall that
f
is close-to-convex if there exist
~e[ f' /e iClcp '1
cP E K and
> 0, or equivalently,
ex. E lR
such that
<~.
We first extend this definition in a way that will be useful
also for functions of bounded boundary variation.
DEFINITION.
of order
~
(tl > 0)
C(a) = {fEN:
Note that
f
e C (5)
P EH(U)
iff
with
The class of normalized close-to-convex functions
is
larg[f l /e i (1cp'JI
C(l)
fEN
Denote
>
for some
cpEK
and
is the old close-to-convex class
and
~ep
<~8TT
f'
=
e iCl p 8cp'
for some
ClElR}.
C.
Also
a E lR , til E K, and
o.
f
= t[ (1-~Z)/(1-T\z)]e+l_11l( (6+.1 ) (T'I-s)}
if~=Tl
\ z/ (l-f'\z)
where
lsi = iT'll = 1 .
Observe that
Cl E lR
k(z;~'Tl'S)
such that
Then
eC(e), for choose
cp(z) = z/(l-T'lz) EK
~e[e -io/S (l-~z)/(l-Tlz) J > 0
in
U.
and
19
THEOREM2.22«(B8 ] l.
Then
C(S)
COC(~)
and
Let
B~l
>l
EcoC(~) c (k(z:s,n,l3) : \ e;\ = In\ = 1 ,
Re p > 0 •
Since
[\ ~\ =1} x (lnl = 1}
is a probability measure on
T
and
T
.
are compact,
= (.fk(z:~,'I'),~)du :
coC(~)
and
l=e ia p(O) S
feN,
for an appropriate choice of
c
with
is by Theorem 2.20 a probability measure
eia.p(z)~
e; 1T1} •
e ia/l3 p -< (1 + cz) /(1-z)
and
I cl
T),
= 1.
'1
on
Since
S
1 e;1 = 1
~
1, there
such that
= S[1+ce;z)/(1-e;z)]6C\11
\e;1=1
By Theorem 2.11 there is a probability measure
that
!p'
2
(z) = Jl/(l-n z ) C\l2 •
\n1=1
£'(z) =
\J2
on
1T11 = 1
such
Therefore
J(1+c~z)/(1-e;z)J6/(1-T1Z)2 dt.-t 1
x \J2'
T
Since
10g(1-z)
[~/(B
maps
+ 2)1
U
onto a convex set,
10g(1-~z)
+ [2/(13 + 2>1 10g(1-nz) -< log (l-z)
~I'ld 1/[ 1-e;z) 13/(8+ 2 ) (l-nz) 2/(13+2) J -< l/(l-z)
c=O
and
By Theorem 2.20 (with
0.=6+2) there is a probability measure
\J3
on
le i =1
such that
Therefore
[(1+c~zl/(1-~Z)J6/(1-T1Z)2
belongs to
:z
=
r(1+c~z)S/(1-ez)I3+2<\.t3
I e1 =1
e;,T1dl) C\I: jC4J. = l}.
B I = {J
k (z:
Since B' is convex
T
T
and compact (the set of probability measures on T is cnmpact in the
weak topology), f "'EB'.
By integration, fEB
= (jK(z: ;,n.e) :
T
fdtJ = 1}
T
20
Clearly
B
CIs)
fn E C (S),
is also convex and compact.
is a closed subset of
I arg[
roC (S ) c: B
B: hence compact.
f~/ei(Irtp~1 1 < ~eT1',
t+l n E K, and
suitable subsequences
S
Hence
t(l
~
.. t(l
(Indeed, if
f n " f, then for
I arg[ f Ie io.t+l ' ] I
E K, and
I
If the latter is a strict inequality, then
~s'IT.
not, then by the rnaximmn or minimum principle
fEC(S): if
f '" «:p E C (S) . )
Just as in the proof of Theorem 2.20, extreme points of
must be generated by point masses.
1; 1 = 1,, 1 =
Hence
l } c: CI s ) •
Therefore
EBC: (k(z : C",B) :
By the Krein-Milman theorem,
BCcoC(S)
EcaC(S ) c: (k(z:t;,Tl,S):
the function
lsi =
k(z:",,,,a) = z/(l-Tlz)
ITlI
=
I} •
However, for ~=Tl
cannot be an extreme point of
since it is not even an extreme point of the subset
*
Kc: SeC
C C (8)
(Observe the chain
=
•
coCCa) = B
Now
1;1
B
PROBLEM.
Determine
PROBLEM.
Show that each
1,,1
= I
,
6>1.
roC (6)
for
and
8;' I
.)
EeoC (6)
~=l
0 < e <1 •
for
k(z:~,Tl,e) E EcoC (S )
For
- *
coS
for
~;i" ,
this fact is contained in the
following theorem on the close-to-convex class .
THEOREM 2.23 ([BIOl) •
coC
where
2
U[z-~(g +,,)z J /(I-Tlz)
T
T ={\~\=11
C
2
t\l
and
coC
are compact,
: 11 is a probability measure on T}
x (In\=l}, and
I,
;;iTll .
21
Proof.
Since
C = C (1)
e=1
everything is a special case of Theorem 2.22 for
\ ~o \
For
= \ T]o \ = 1,
except the
1;0
1"0 '
we
must show that
fe\! = 1,
T
is possible only if
(l-~oz)/(l-"oz)
atives we have
and let
is a point mass at
\.l
r ... 1 .
3
f(l-~z)/(l-T]z)
T
3
Then
where
Set
To
O!l:as1
~oTio -
and
Set
~
1 - ~ o~ 0
By the Lebesgue bounded convergence theorem we have
S(l-~no)~
Taking deriv-
(~o'''o).
(l-a) = ~o
S ~~ .
Then
a = Sdl-l •
To
On the one hand,
50'
a, and on the other hand,
! Sono-(l-a) \ ~l-(l-a)
both inequalities must be equality.
is possible only if
only if
a
= 1.
is a unit point mass at
\.l
~on 0 =
(~
o
2 2 t h
and their n
-
!o:i(n-1)~
-
~T] zJ/(l-T]z)
n+2
Therefore
nor
~C\J
is possible
To
,T] )
0
By expressing the extreme points of
C
in the form
derivatives as
-!o:i(~-'T'1)z /(l-T\z)
a
=
I!:o;:j 011, the latter
Since
Moreover,
\nJ~<\l\
To
To
n!"
n-2
z/(l-T]z)
[~(n+ 1)T]
and using elementary estimates, we have
as an application the following extension of Theorem 2.16.
co
THEOREM 2.24.
If
(n)
If
f(z) = z
(z)\ s:n!(n+\z\l/(l-\z\)
In particular,
\ a \ "n
n
for
n
+n~2 anz n
n
+2
= 2, 3,
E C,
then
foraH
zeU
__ . _
and
n:<:O.
22
PROBLEMS.
There are many intriguing extremal problems that
are not linear.
One example is the conjecture of M. S . Robertson
[R3] that
J. A.
holds for the coefficients of close-to-convex functions.
Jenkins [J2] has shown that the conjecture fails in the full class
S.
C
However, Robertson (R2J has verified the conjecture in
case
n-m
is an even integer and in some other cases .
sufficient to find a single functional
la n
holds in the starlike class
functional
'X : K'" C
-X
an- 11
x : 5 * .. C
in
It is
such that
cl
5*, or equivalently, to find some
such that
Ina n - (n-nxa n- 1\ s 1
K
holds in the convex class
for all
n.
Perhaps the inequality
of Theorem 2.8 can be used profitably to attack the last problem.
~lli.
is
A natural candidate for the functional
since it works for
~a2
EcoS*'
X : S* .. C
However, verify that the in-
equalities
la n
are violated for each
if
e
n:;, 4
-~a2a n- 1\ s 1
by the functions
is positive, but sufficiently small.
z(l+z)
-e
(l-z)
e-2
~
and
v
a finite signed measure if
a
=~-v
where
are finite (nonnegative regular Borel) measures and
denote its corresponding variation measure by
*
This observation is due
to R. Barnard.
We will call a
ES
lal .
23
The class of functions of bounded boundary rota-
DEFINITION.
tion is
z
V = (f exp[ Jlog(l-~z)dcrJdz : a is a finite signed measure on
o
ITll=l
and
is the subclass of
Vk
V
for which
By Theorems 2.13 and 2.4(d),
For
f E Vk
the constant
k
{dial
ITl =1
V2
S
(k ::e 2)
k
1 ~1=1
Jda = -2) •
ITl1=1
•
is just the convex class
K.
restricts the rotation of the
boundary in the following sense.
y
r
= f<\zl=r)
tangent along
at
f(re iS )
Yr
is then
is
The tangent to the level curve
~ f(re ie )
The variation of the
09
S21'Td larg.2....f(re i9 ) 1= S21'T l l
e arg..Q....a f(re ie ) Ide
o
e
Since
09
0
f E v,
0
0
the harmonic function
u
~ {Re[ (l+Tlz)/(l-Tlz)]da, and it is known that
lTi =1
creases to the limit
ZfH
= Re (1 + ft) =
S2TT lu (re i9 ) IdB
o
in-
TT Jdlo I as r -+ 1. If f E Vk ,then kTT is
ITll=l
a bound in a limiting sense for the variation of the tangent angle
at the boundary.
The characterization of domains of bounded boundary rotation
has been studied by V. Paatero [Pl).
In order to- show that functions of bounded boundary rotation are
close-to-convex of some order, we shall use the following theorem.
THEOREM 2.25.
If
\-1 1
and
are probability measures on
ITiI =1, then there exists a constant -c,
Icl =1, such that
exp( {lOgn-TlZ)d(\.1rIJ2)} -< (1 + cz)/(l-z) .
ITl =1
24
Proof.
Assume first that the probability measure
generated by a continuous strictly increasing function
[0,2""J
onto
[O,lJ,
Since
loge (l-e
width
,.,.,
ia
j = 1,2.
z)/(l-e
is
~j
of
gj
Then
is z)J
maps
U
onto a horizontal strip of
1
sup Jm I(z) -inf Jm I(z) ~S [sup arg( ) -inf arg( )Jdxs,.,. •
zEU
zEU
Therefore
containing
0
U
U
is contained in a horizontal strip of width
I(U)
1(0) = O.
The types of measures considered are dense
I 'nl
in the weak topology on the set of probability measures on
Therefore by a normal family argument
I(U)
izontal strip of width
I (0) = 0
bility measures
TI
,.,.
~1'~2
containing
= 1 •
is contained in a horfor general proba-
So there exists a constant
c,
\ c\ = 1,
such that
.r10g(1-'nZ)d(~1j.l2) -< loge (1 +cz)/(l- z)] , and the re1111 =1
suIt follows by exponentiation.
THEOREM 2.26([B81).
For
k>2,
VkC::C(~k-1)1
functions with boundary rotation at most
of order
are close-to-convex
•
~k-l
Proof.
k
that is,
Let
f E Vk .
Since
V k c: Vk
I
whenever
k < k', we
may assume that
fl (z)
Then
~l
exp( {lOg(l-11Z)dO}
111 =1
(101 +0)/(k-2)
and
where
~2
rdo = -2
1111=1
and
(Iol-o)/(k+ 2)
Jd\ol = k
l'nl=l
are both
25
\ rd = 1
and
rexp(-2 f1og(l-~Z)~2JdZ
E V2
probability measures on
~(z)
=
z
o
1,,1=1
- 2t-L2 ' we have
~
I (z)
= exp( (~k-l)Jl09(l-"z)d(lJl-\J2l) •
\,,1=1
~
(fl/~/)l/(~k-l)
By Theorem 2.25,
is subordinate to some half-plane
mapping, and the result follows from the definition of
COROLLARY
2.27.
C(~k-l)
Functions of bounded boundary rotation
•
k s 4
are close-to-convex, hence univalent.
We now determine the closed convex hull of
terms of
~.
at
k
~
for
k« 4
in
ODC{B), which is known from Theorem 2.22.
THEOREM 2.28«(B8]).
For
Vk
Let
"="0
(]
For
4,
ODC (~k-l)
COVk
and mass
the Krein-Milman theorem
PROBLEM.
~
be the signed measure on
\,,1
= 1
with mass
Then
4, we therefore have
CoVkCODC(}zk-l)
k
EeoC (~k-l)
ODe (~k-1)
C
C
Vk
ODVk
I~
{do = -2
=1
by Theorem 2.22.
By
On the other hand,
by Theorem 2.26.
Determine
and
respects this is the more interesting case.
resul ts to the case
k
~
4 •
for
2<k<4.
In many
One would expect similar
26
We now solve the coefficient problem in
method is due to Brannan, Clunie, and Kirwan
THEOREM 2.29.
C(S)
and
Vk • The
[B8,B7].
The coefficients of functions in
are dominated by the corresponding coefficients of
C(6),
e >0,
k(zr-l,l,e)
E
C(e) •
E.!QQi.
~e
p > O.
If
>
c ,
Icl =1, depending on
1, the coefficients of
by the coefficients of
[ (l+z)/(l-z)
,,2j.reio.p(z)6(1 +"z)/(l-T1Z)du.
h\l =1
fl
h
were
0.,6, and
eillp(z) 6 (l+nz)/(l-"z)
is a probability measure and
U
e illp 8 cp 1
f'
cp E K
and
By Theorem 2.11
some choice of
6+1
fEC(S), then
I"
Js + 1
J, = 1,
2"
p.
Since
are dominated
by Theorem 2.21.
Since
the same is true of
Consequently, the coefficients of
are dominated by the coefficients of
I
z2 j [(l+Z)/(1_z)JS+l
(1+Z)S/(1-Z)6+ 2 =!.k(Zr-l,1,1l),
J=O
and the result follows.
COROLLARY 2.30.
k ;;, 2
2.29.
2.28.
Vk
for
are dominated by the corresponding coefficients of
Proof.
for
The coefficients of functions in
k = 2.
Since
For
V2
k >2
= K, the result Is contained in Theorem
2.12
the result follows from Theorems 2.26 and
That k(zr-l,l,~-l) E V
k
was observed in the proof of Theorem
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