An Area Theorem for Starlike Functions

Full text

AN AREA THEOREM FOR STARLIKE
FUNCTIONS
By R. R. LONDON and D. K. THOMAS
[Received 6 May 1969]
1.
Introduction
A function/ that is regular in the open unit disc A, and such that/(O) = 0,
/'(0) = 1, is said to be
starlilce,
if, and only if, for all points z e A,
Denote by S the class of starlike functions. It is well known that all
functions in S are univalent in A, and map A onto a domain which is
star-shaped with respect to the origin.
Every function / e 8 has the representation (4)
f(z) = *exp(i
j\gT-}—2dti(t)},
(1.2)
where z e A and where /x is a positive non-decreasing function on [0,
2TT],
satisfying dp[t) =
2TT.
Hence fx is continuous on [0,2n] except for at
most a countable number of jumps. Let
noc
denote the maximum jump
of
JU
on [0,
2TT].
In (4), Pommerenke showed that
log M(r)
«=lim , ,, ,, (1-3)
log(lr)
where M(r) = max|/(z)| (0 < r < 1). Using (1.3), Pommerenke (5) then
Isl-r
showed that
where M' denotes the left-hand derivative of M. Clearly (1.4) implies (1.3).
Let
TTA(r)
denote the area of /(Ar), where Ar = {z: \z\ ^
r}.
It was
shown in (3) that a is connected with A(r) by the relationship
which is analogous to (1.3). It is therefore natural, in view of (1.4), to
Proc. London Math. Soc. (3) 20 (1970) 734-48

AN AREA THEOREM
FOR
STARLIKE FUNCTIONS
735
ask whether
one has
2«-lim(l-r)2g.
(1.8)
This problem
was
left open
in (3), and its
proof is
the
object
of the
present
paper.
The
proof,
although
in
essence simple,
is
rather long
and
will require
a
number
of
lemmas. These constitute
§2.
2.
Preliminary results
Let/
G
S;
then from
(1.1) we may
write
zf'(z)=f(z)F(z),
(2.1)
where
F is
regular,
F(0) = 1 and
ReF(z)
> 0 for all z in A. It
follows
from this
and (1.2)
that,
for
z e
A,
We shall assume, without loss
of
generality, that
fx
is
continuous
at 2n.
Our first three lemmas concern
the
function
F.
LEMMA
2.1. Let
F
be
defined
by
(2.1). Then:
B 1
—r2
(i)
Re
Fire*0)
> ^
\l_reno-9)\i»
*/ P
JumPs
nP at
?>'
(ii) M(r,F)
2
l-r'
(iii) M(r,F)
=
———,
where
TTOC
is
the
maximum jump
of
p;
p2
(iv)Jo
(Here,
and
elsewhere
in the
paper unless otherwise stated,
0 < r < 1,
and
the
o,
0,
and
~
notation refers
to
behaviour
as r
->•
1.)
Proof,
(i), (ii), and (v)
follow immediately from (2.2);
(iii) and (iv)
are
to
be
found
in
(1).
LEMMA
2.2. Let
pjump
nfl > 0 at
cp,
and let e(r) > 0, e(r)
=
o(l).
Then

736
R. R.
LONDON AND
D. K.
THOMAS
uniformly when
\6
—
<p\
< e(r), and
Proof.
Without loss of generality, we may assume that
<p
= 0.
Choose
r)(r)
> 0, such that 8(r)
=
e(r)
+ r)(r)
= o(l), and
l-r = o(\l-reiii{r)\2).
For any
r
such that 8(r) <
IT,
put
P(r)
=
[0,£(r)]u[27r-£(r),27r],
Q(r) = (8(r),27r-8(r)),
Then, for 0
e
P(r),
t e
Q(r),
\9-t\ > \6-1r\-\t-1r\ > (^-'
Let
2n->q(r).
7r£ (0
< t^
2TT),
0 («
=
0),
so that v is continuous
at 0
and 27r. NOW
/ )8
1
+
rei0\
C
1+
reii0~l)
I
»
'91
lypiOt
I , i
roi(Q—l)
^ '
and so, for 6
e
P(r), we have
27
F(re")-$
pi+reie
2
\-reie
T
f
L
— rj
0(1)
1-r'
(2.3)
from the choice of
r)(r)
and the continuity of
v
at 0 and
2-n.
This proves (i).
To prove (ii), observe that, from (2.3),
r
r
l—r2
C C 1—r2
=
I r-s
av(^)aa+
:
J
P(r) J Q(r)
I
1 ~?"fi
I
J
P(r)
J
R(r)\*-~1'£
<o(l)+f dv(«)
=
JjR(r)

AN AREA THEOREM
FOR
STARLIKE FUNCTIONS
737
LEMMA
2.3.
Let
/xjump
TTJ8
>
0
at
cp.
Then
(i)
if
e{r)
> 0 and
e{r)
=
o(l),
ie)\2
\F{re
ie
)\
j
uniformly when
\0 —
<p\
<
e(r),
and
(ii)
if K is
a
positive constant,
| F{rei0) |2
-
^°(1)
Re
F{rei0)
uniformly when
\ 9 —
<p\
< K{\
—
r).
Proof.
We may
again suppose, without loss
of
generality, that
cp
=
0.
(i)
For any
r
such that
e(r)
<
TT,
put
U(r)
=
[0,e(r)]u[2Tr-e{r),27T]
and
X(r)=
inf
—^L-
so that
Now
let
^(r)
=
|7r(yA(r))~*,
and
for
the
remainder
of
this
proof,
consider
only values
of r
such that
e(r) + r){r)
<
TT.
Put
V(r)
=
(e(r)+i7(r),27r-e(r)-77(r)). Then,
for
6
e
U(r),
t e
F(r),
we
have
\B-t\
>
\B-v\-\t-n\
>
(rr-e^-in-eir)-^)) =
v{r)
and
|^-«|
^ |0-7r| + |«-7r| ^ TT+{7T-e{r)-7)(r)) < 27T-7){r),
so that
Thus,
for d
G
E7(r),
Next,
put
W{r)
=
[0, 2ir]\V{r)
so
that,
for
9
e
U{r),
5388.3.20
CC

738 R. R. LONDON AND D. K. THOMAS
Now, from (2.2),
and so, for 6 e U(r), we have by (2.5)
by (2.4) and (2.6). Thus, for 6 e U(r), the Cauchy-Schwarz inequality
gives
since /^ jumps
7TJS
at 0 and is continuous at
2-n.
Hence
\F{reie)\2
\-r
and this proves (i).
(ii) Put P(r) = [0,K(l-r)]u[2w-K(l-r),2TT]. Then, for 0 e
P(r),
we
havel+rei0 _ 1-r2 ^ 1+r
1
_ fgiO ~ I j _ ^et512
Thus,
by Lemma 2.2 (i),
and
uniformly for 6 e P(r). Hence
\F{reid)\2(l-r) B\l+rei0\2 n
T>
jpf iO\— ~ o —1 "*" ^ aS T "*"
uniformly for 0 6 P(r). This proves (ii).
We now prove three lemmas concerning the function /.
LEMMA
2.4.
'2/7
rin pr • p2
i) 2^(r) = 2 \ \f'(peP)\*pdpdO = \
Jo Jo Jo

AN AREA THEOREM
FOR
STARLIKE FUNCTIONS
739.
and
(ii) rA'(r)
= f
n\f(rei0)F{rei())\2dd.
•JQ
A proof
of
(i)
can be
found
in (2); (ii)
follows from
(i) on
using (2.1).
LEMMA
2.5.
//
the
function
g is
defined
in
A
by
where
v is a
positive
non-decreasing
function
on
[0, 2v], with maximum
f27T
jump
TT(3
and
such that
dv(t) ^ 2v,
then
Jo
for
all e > 0 and all
sufficiently large
r.
p2n
Proof.
If
dv(t)
=
2TT,
then
g e S, and the
result follows from
(4)
Jo r-2n
Theorem
1. If
dv(t)
<
2TT5
let v* be a
continuous non-decreasing
Jor-2rr
function
on
[0,
2TT]
such that
d(v{t) +
v*{t))
=
2TT.
Then
0,
defined
Jo
on A
by
/I
C2n 1
O(z)
=
zexp
-
log——&d{v{t)
+
v*(
\7T
j
o
J. ze
belongs
to 8, and so
M(r,g)
=
0(l)M(r,G)
< 1/(1
-ry+e
for all e > 0
and
all
sufficiently large
r.
LEMMA
2.6. Let fx
jump irfi
> 0 at
<p,
and let S(r) =
(<p —
e{r),(p + e{r))
and
m(r) = sup
\f(reie)\,
where
e(r) > 0, e(r) = o(l).
Then
OeS(r)
(i)
for any c e (0,1)
there exists
A
=
A(c) such that
\f
(rei0)\
> cm(r) and 6 e S(r)
imply \6-<p\
<
A(l—
r),
and
(ii)
for any
A
> 0
there exists
c = c(A) > 0
such that
\6
—
<p\
<
A(l
—
r) implies
\f{reie)\
>
cm(r).
Proof,
(i)
Again
we may
assume that
cp
= 0.
Assume that there exists
i(j(r)
E
S(r), and a
sequence
S?,
such that
\f{re^
r(r)
)\
>
cm(r),
and
A(r)
=
| ift(r) |/(1 — r)
->
oo
as r ->
1
through
the
sequence
£f.
We show that
\f(re^{r))
|
=
o(l)( |/(r) |)asr^lony. Throughout
the
remainder
of the
proof
let re^.

740 R. R. LONDON AND D. K. THOMAS
Let/(z) =
zh{z)/{\
-z)P for z e A. Then, for sufficiently large r,
so that
|/(re^(r))|
= O(l)([A(r)(l-r)]-/?r|%e^(r))|). (2.7)
Now
where
nfi (0 < M 277-),
0
(« = 0).
Since
i/ is
continuous
at 0 and
2TT,
we can
choose
8
such that
0 < 8 <
TX
and
Uv(0+ dv(t)^-£. (2.8)
With this choice of 8, we have, since
ifj(r)
e S(r),
[2n-8
I
J $ 11
-
re
I
For
6 e
[0,
2TT]
and
sufficiently large
r, we
have
1
d 1
^7
log:;
s<ft
at 1
—
re"
2
—
•'o
11
10 r(2/5)A(r)
< 31ogA(r). (2.10)
Writing
D =
[0,8]
u
[2TT
-
5,
2TT],
we
deduce from
(2.8) and
(2.10) that
f logA(r). (2.11)
From (2.7), (2.9),
and
(2.11),
on
estimating
h, we
obtain
|/(r
which proves
(i).

AN AREA THEOREM
FOR
STARLIKE FUNCTIONS
741
(ii)
Let
\f{reir(r))\
>
\m{r),
then
by (i)
we
have |r(r)-p| <A1(l-r)
for
some constant
Xv
Thus,
by
(ii)
of
Lemma 2.1,
log \f(reie)
|
-
log
|
f{reir{r))
|
= - f
Im F(rei0) dd
Jrir)
2|0-r(r)|
1-r
1-r
if
10 — 9? | < A( 1 —
r). Thus,
for any
such
6, we
have
as required.
The last two lemmas we require concern both
the
functions
/
and
F.
LEMMA 2.7.
Let
<pv <p2,
...,
<pN
denote
the
values for which the jump
of
/x
is
equal
to
the
maximum jump
TTCX
>
0, and
choose
<p
so
that
<pk
e
{<p,<p +
2n)
(k
=
1,
2,
...,N).
Then there exists
e(r),
positive
and
o(l),
such that
Air)
I T I
ft*/)
fcl/
l
H 'I n*p
t\J
l \£t ri M
— f\
I
I 1
T(r)
l~r
and
(i)
f
\f(reie)\2~ReF{reie)d9
=
f
JT
T(r)
where
T(r)
=
[f»,?»
+
27r]\U (<pk-e(r),
k=l
Proof. We
have
)Tl\l-reii9-^\-<*,
(2.12)
where
g is
a
function defined
as in
Lemma 2.5, with
j8
< a.
Thus
M(r,g)
< (j^-^
(2.13)
for some
8 > 0 and all
sufficiently large
r.
Now
let
e(r)
be
such that
e(r)
>
0,
e(r)
=
o(l)
and
(2/£(r))4aA^
<
1/(1
-r),
and
define
T{r) as
above.
Then, from Lemma 2.1,
f
\f(rei<>)F(rei°)\*dd
<
0(l)(max |/(r^)l)Y,-^— log-^
JT(r)
\0eT{r)
/ \l-f 1-

742
R. R.
LONDON
AND
D. K.
THOMAS
and, since R,eF(reie)dd
= 2n,
f \f{rei0)\2ReF{reie)dd
<
27r(max \f(rei0)\)\
JT(r) XdeTir)
)
From (2.12)
and
(2.13),
we see
that,
for
sufficiently large
r,
I
l \a-S N
\f{reie)\
< - max n
11
-rei{9-*k)|"a
V~rJ 6eT(r)k=l
I l \oc-S
<
max
\l-reie\-*N
1—
rJ
max
6eT(r)
-r)2
r)2
+
4rsin2\e{r
a-f«
by
the
choice
of
e(r).
Thus, using (1.5),
we
obtain
I
5
|
\f(rei°)F(re«>)\*dd
=
0(1)
and
|/(f6«)|«Re J(rc«)£W
=
0(1)
This proves Lemma
2.7.
Our next lemma improves part
(ii) of
the last lemma.
LEMMA
2.8. Let a,
9?,
and
<pv
<p2,
...,
<pN
be as in
Lemma
2.7. If e > 0,
then there exists
a
positive constant
K
such that,
for all
sufficiently large
r,
f |/(re^)|2Rei?>e*V0
<
eA{r),
JX(r,K)
where
X{r,K)
=
[<?,<
Proof.
By
Lemma
2.7,
there exists e(r), such that e(r)
> 0, e(r)
=
o(l),
e(r)(l—
r)
->
00
as
r
-> 1, and
f |/(re^)|2Re^(re^)^
=
o(l)A(r), (2.14)
JYM

AN AREA THEOREM
FOR
STARLIKE FUNCTIONS
743
where
Y(r)
=
[<p,9
+
2nl\ U
(<Pk-e(r),<pk
+
e(r)).
k=l
We
now
show that,
if
8
>
0, there
is a
positive constant
K
such that,
for
sufficiently large
r,
f \f(reid)\2R,eF{rei0)dd
Jx{r,K)\Y(r)
where
C
is
an
absolute constant,
and
(2.14) then shows that this is sufficient
to prove
the
lemma.
For
r
such that e(r)(l
— r)
>
K (K
>
0
fixed), write
X(r,K)\Y(r)=
U
(Ik(r)u
Jk(r)),
fc=i
where
4(0
=
(<Pk-*{r),<pk-K{l-r)),
Jk(r) =
(<pk
+
K{l-r),<pk
+
e(r)).
We
now
consider
the
behaviour
of
\f(rei0)
\
on the
intervals
Ik(r) and
Jk{r).
Let
0 < a
<
b,
and 6
e
(<pk
—
b(l
—
r),
<pk
—
a(l
—
r)).
Then,
for
sufficiently
large
r,
k
—
9) arsina(l—r)
^
ocra
and
so, by
Lemma
2.2 (i),
»)
=
°^
sin(y>fc
-
0)
o(
)
- +
Thus
we
have shown that,
for
sufficiently large
r,
-lmF(rei6)>0
(9
e
(<pk-b(l-r),<pk-a(l-r))).
(2.15)
One
can
similarly show that,
for
sufficiently large
r,
-ImF(reie)
<
0
{6
e
{<pk
+
a{l-r),<pk
+
b(l-r))). (2.16)
Since
Yd\f(rei9)\
=
-\f(re«)\ImF{re«),
(2.17)
we deduce from (2.15)
and
(2.16) that,
if
pk(r) =
<pk-K(l-r),
yk(r)
=
<pk
—
K(l—r)
and
ift
>
K,
then,
for
sufficiently large
r,
\f(re™)\
<

744
R. R.
LONDON
AND D. K.
THOMAS
It follows therefore that there exists «/r(r), tending
to
infinity
as
r -»
1,
such
that,
for
sufficiently large
r, K <
ifj(r)
<
e(r)/(l
—
r)
and
l|/('W)|
(0 ( Z(l)
^()(l)))
Now
put
and
w(r)
= sup
\f(reie)\.
deS(r)
Then,
by
Lemma
2.6 (i), we see
that
\f(rei0)\
=
o(l)m(r), uniformly when
tfj(r)(l
— r)
<
\9 — <pk\
<
e(r).
Since,
by
Lemma
2.6 (ii),
|/(re^*(r))
|
>
cm{r)
and
|/(re^*(r))|
>
cm{r)
for some
c = c(K) > 0, it is
clear that (2.18)
can be
extended
to the
inequalities
(»eJk(r))
for sufficiently large
r.
Thus,
for any
such
r,
f
\f(rei0)\2ReF(reie)d6
Jlk(r)UJk[r)
l
k
(r)UJ
k
[r)
f
f
ReF(reie)d8.
lkir)
k
(2.19)
We
now
consider
the
integrals
in
(2.19). First note that,
for
sufficiently
large
r,
elr)
dt re(r) dt 2 r00 du
-r)|l-re-«|a ^ Wr) (l-r)2 +
|<2
< 1
whilst
Jo |l-re-«|a> Jo (l-r)2 + «2-IZ7j0 1+^' ( • 0)
Hence, if
8
> 0, then
if K is sufficiently large. Now let
S
> 0 be given; then
de
sP dd
andf ^ r«+X(l-r) rffl
|lre^-^|2'

AN AREA THEOREM FOR STARLIKE FUNCTIONS
745
for some
K.
Now for this
K
we have, from Lemma 2.1 (i) and Lemma 2.2
(ii),
that
and similarly
Thus,
from (2.21),
I
Uttr)
J*t-ini-r)2|l-reW-»
since, by (2.20),
and so
f
"ReFire*0)dd
<
(8
+ o(l))T* ReF(rei0)dd, (2.23)
where we have used Lemma 2.1 (i). Similarly one sees from (2.22) that
T&eF(rei0)dd
<
(S
+
o(l))
l&eF(rei0)d6.
(2.24)
We return now to (2.19), and observe that (2.23) and (2.24) give
<(8 +
o(l))[|,
+ |/(re^i(r))|2f *
B,eF(rei0)dd
.
(2.25)
Now,
if
K> 1, (2.15), (2.16), and (2.17) give,
for
sufficiently large
r,
\f(rei0)\:
Also,
by
Lemma
2.6 (ii),
\f(rei0)\
>
cm(r) i£\6
—
<pk\
<l
—
r,
where
c > 0 is

746
R. R.
LONDON AND
D. K.
THOMAS
an absolute constant. Thus, again for sufficiently large
r,
where C~2
=
min(l,c). Therefore, from (2.25) we have
\f(reid)\*~ReF{reie)d6
(Pk
'
>I
k
(r)VJ
k
{r)
'
"<Pk
'<p
k
-K(l-r)
from Lemma 2.4 (i). Finally, since
X{r,K)\Y(r)= VVk(r)uJk{r)),
we have
f \f(reie)\2~ReF{rei6)dd
JX{r,K)\Y(r)
and this
is
what we sought to prove.
3.
Proof of the result
Case (i):
a = 0.
From Lemma 2.4 (ii),
rA'(r)
=
(n\f(rei0)F(rei6)\2dd
Jo
r2ff
r
^
Jo Jo
',F)\
\f'(peid)\*pdPdd
Jo
Jo
/•r
/«2
+ 2Jo Jo
,
say, (3.1)
on using (2.1). Now Lemma 2.1 (iii) and Lemma 2.4 (i) give
2M(r,F)I{r) < o(l)^.
(3.2)

AN AREA THEOREM FOR STARLIKE FUNCTIONS 747
Let e > 0 be given. Then, by Lemma 2.1 (iii),
by Lemma 2.1 (v). Thus, again by Lemma 2.4 (i),
From (3.2) and (3.3), together with (3.1), we see that C^-
and this is the result.
(3.3)
A'lr)
o(!)
Case (ii): a > 0.
Let a,
<p,
and
<p
v
<p2,
...,
<pN
be defined as in Lemma 2.7. Then, by
Lemma 2.7 (i) and Lemma 2.3 (i), there exists e(r) positive and o(l), such
that
f \f(re
ie
)F(re
ie
)
|2
dd
=
o{
1)
^-
and
f \f(re
i0
)F{re
ie
)
|2
dd
< OC
+
Q(1) f
|/(
re
^)
|a
Re ^(re^) d«,
Jt7(r) 1—»" Jl7(r)
where
T(r)
=
[<p,<p +
2nJ\U{r)
and J7(r) = U
(<P,c
-
«(r),
p*
+
«(r)).
A;=l
Thus,
by Lemma 2.4 (ii),
rA'(r)= f
|/(re^)i^(re^)|
2
^
^C7(r)Ur(r)
a + 0(l) f . ..
l-r
Hence
(3.4)

748 AN AREA THEOREM FOR STARLIKE FUNCTIONS
Now, by Lemma 2.8 and Lemma 2.3 (ii),
if
e
is given so that
0 < e
<
a,
there is
a
positive constant
K
such that
f \f(rei0)\2ReF(rei0)dd < eA(r)
JXir.K)
and
! \f(rei0)F{rew)\*d9
~ -^- f
\f(rei9)\2ReF{rei0)dd,
JW(r.K) l-rJW(r.K)
whereX(r>K)
and
W(r,K)= \J(<pk-K(l-r),<pk
k=l
Thus,
again from Lemma 2.4 (ii),
if
r is sufficiently large,
rA'(r)=
\
\f(reid)F(reie)\2dd
>
f
\f(reie)F{reie)\2d6
JW(r,K)
>
£Zf f
\f(reie)
|2
Re F(reie)
dd
l—rjW(r,K)
= ^^-T U(r)
~
i
[ l/(^) |2ReF(re*)
dd)
Hence
Uminf(l-r)4rr^ 2a,
and (1.6) follows from this and (3.4).
REFERENCES
1.
W. K. HAYMAN, 'On functions with positive real part', •/. London Math.
Soc.
36
(1961) 35-48.
2.
F. HOLLAND and D. K. THOMAS, 'The area theorem for starlike functions', ibid.
(2) 1 (1969) 127-34.
3.
'On the order of a starlike function', submitted for publication.
4. CH. POMMERENKE,
'On starlike and convex functions',
J.
London Math. Soc.
37
(1962) 209-24.
5.
'On starlike and close-to-convex functions', Proc. London Math.
Soc.
(3) 50
(1963) 290-304.
University
College
Swansea