I ntern. J. ,h.
kh. Si.
(1982) 459-48
459
Vol. 5 No.
UNIVALENCE OF NORMALIZED SOLUTIONS OF
W"(x) + p (x)W(x) 0
R.K. BROWN
Department of Mathematical Sciences
Kent State University
Kent, Ohio 44242 U. S. A.
(Received August 27, 1981)
ABSTRACT.
Denote solutions of ’(z) + p(z)W(z)
W(z) z[1 +
zl
< i. We
3
b z
n-1
n
n]
where 0
KEY WORDS AND PfIRASES.
i80
1.
< aft) < 1/2 < afG)
p(z) so that
determine sufficient conditions on
[W(z) ]l/ are univalent
in
zl
W(z) z[l
and z2p(z) is
0 by
+
n=l
an zn]
and
holomorphic in
[W(z)]1/5
and
< 1.
Univalent, spirallike, starlike.
MATHEMATICS SUBJECT CLASSIFICATION CODES.
Primary
34A20; Secondary
30C45.
INTRODUCTION.
Consider the differential equation
W"(z) + p(z)W(z)= O,
is holomorphic for
z2p(z) P0 + Pl z +
zl < i.
+
(1.1)
n
where
Pn z
+
o’
(.)
The indicial equation associated with the regular singular point of the equation
(i.i) at the
origin is
12
-
+
P0
0
(1.3)
460
R.K. BROWN
and has roots which we designate by
and
,
1 and
+
where
We will also use the notation
+
i
Corresponding to the root
(1.4)
i + i2"
icz2’
(1.1)
there is always a unique solution of
of the
form
n
n=l
valid for
Izl
<
i.
We restrict our attention in this paper to those
then obtain a unique solution of
(1.1)
zl
<
We
of the form
W(z) z[l +
valid for
[} > O.
for which
5
b z
n
nl
n]
(1.6)
i.
F(z)
We define two normalizations
and
F(z)
(1.5)
of the solutions of
and
(1.6)
as follows
(z)
[w(z)]l
z +
F(z)
[W(z)]I/
z +
(1.7)
where we choose that branch of each function for which the derivative at the origin
is 1.
Next we consider the "comparison" equation
W"(z)
z
+
*
is non-constant and holomorphic for
P0 -< 1/4.
real axis
With these restrictions
(see [i]). We
singular point of
(Z
_> 1/2 _>
solution of
* +
PO + Pl z
(.8)
0, with
c(Z) c(z*(z) ) + 0’
2*
and where z p (z)
*
p(Z)Wc(Z
+
*n
Pn z
c > 0,
(1.9)
+
zl < i with Pi’ i 0,
on z2p * z) the solutions
i, 2,
of
(1.8)
are real on the
will designate the exponents associated with the regular
(1.8) at
the origin by
and
where
+
As in the case of equation (i.I) we obtain for any
(1.8)
real and
of the form
1 and
a unique
UNIVALENCE OF NORMALIZED SOLUTIONS
.
W
(z)
[I +
z
n=l
C ,C
zl
valid for
< i,
and for any
W
zl
valid for
In
tive to
Ill
.
6" > 0 a unique
an(C)zn]
(i.I0)
solution of the form
bn(C)zn
5
[i +
n=l
,C
461
(I.ii)
< 1.
Robertson determined fairly general sufficient conditions on p(z) rela-
p*(z)
under which
these results to
F(z)
F(z)
is univalent in
but only for real
zl
In [2] Brown extended
< 1.
satisfying 0
<
_< 1/2.
Theorem of this paper we present sharp sufficient conditions on
pc(Z)
under which the function
F(z)
In the Main
p(z) relative to
is univalent and spirallike in
zl
< l,
where
We then compare these results to those of Robertson for
may be complex valued.
F(z).
PRELIMINARIES
f(z)
S will denote the class of functions
disk D
zl
{z:
< l}
and normalized so that
We shall say that f(z) E
and some
,
0
<
holomorphic and univalent in the unit
f(0)
0,
F, if and only if for
f’(0)
1.
some real number
,
II
<
,
< l,
a zf’(z)
f(z) >
for all z 6
D. F
m F
,0
.
is the class of functions called spirallike in
Functions in the subclass S
()
F
the class of functions starlike in
in D
are called starlike of order
0
It follows that
D.
S*()
D, [3], [4]
c F c S;
S*(0)
is
(see [2]).
We will need the following result.
THEOREM
2.1.
Let z
pC(z), W. (z),
,C
and
(1.11)
respectively.
If for all
zl
g(z2p*(z)}
then for fixed C
,.C
w. ,C(Izl)
and
< I
_<
(z)
W.
,C
be defined by
(1.9), (1.10),
Izlp*(Izl)
is monotonic decreasing for all
zl
< n(, R .(C)),
462
R.K. Brown
and.
2,c-(I zl)
W
zl
is monotonic decreasing for all
< rain(l, R
.(C))
where
,C
R .(C) and R
a
(r)
W’.
.(C)
are the smallest positive zeros of the functions
follows from
(3.16)
ray 8
constant
#
this result is given on page
G
3.18
and Theorem
constant the equality
R[z2p*(z)]
of
Izl2p*(Izl)
z2p*(z)
G
3.
,C
and
For 13
the result
z2p*(z)
noting that if
cannot hold for all 0
_<
is non-
r
_<
r
I
on any
be nonconstant is necessary to ensure strict mono-
tonicity in the results above since if
. ,c(Izl)
[2] after
262 of [i].
0.
The condition that
G
and
respectively.
In the case of
W
(r)
W’.
a,C
w
z2p*(z)
is constant so are
. ,c(Izl)
,C
13
LEMMAS.
In this section we prove the lemmas used to obtain Theorem A and The Main
Theorem in section 4.
Since all of the results of this section are stated for
W(,z)
and W
.
(z)
we
6,C
adopt the following notational convention:
W
z13[i
W6(z)
W(z)
WC= Wc(Z
W
.
+
n=l
(z)=
z13
[1 +
13 ,C
bn zn]’
5
r-i
bn(C)zn].
It is important to note that all of the results of this section remain valid if
W and W
C
moreover,
are replaced by either
W(z)
and W
.
(z)
a ,C
or by
W(z)
and W
.
(z) and,
,C
the proofs are obtained by making corresponding changes in the proofs
given here.
In our lemmas we will investigate the rate of change of
rays issuing from the origin.
and use
(i.i) to
obtain
zW ’. and
R--)
For this reason we designate z by
zW ’.
---j on
i8
re
fix 8 vary
463
UNIVALENCE OF NORMALIZED SOLUTIONS
-r .w’
W
r
-zep(z)
dr
<.]2
zW’
+
zW’
--Q--
where W’ designates differentiation with respect to
Taking real and imaginary parts of
r
d
zW
a[-----}
a[
(3.2)
z2p(z)]
+
z.
we obtain
zW
2r--}
zW ’.
’.
ar---j
zW ’.
+
(3.3)
and
zW ’.
d
Also from
(1.8)
zW
and the fact that W
r
d
rW(r)
is real for real
C
2
-r
r(,Wc(r))
*
PC(r)
+
Our goal is to determine conditions on
ensure that on every ray
e
Then it will follow from
where
R(C)
for z
rW(r) rW(r) 2
Wc(r) (Wc(r))
z2p(z) relative to z2p( z)
>
0
which will
(3.6)
Wc(r
(1.7)
8zF(z)
F(z)
g[) > 0 (3.7) implies
that for
rW6(r)
Wc(r ->
F(z)
e
constant
0 for all 0
_<
r
i.
<
(3.7)
is univalent and spirallike in
is the smallest positive zero of
will show that C can be adjusted so that
in
z, we obtain
rW (r) > 0 for all 0_< r < i.
zW’
Since
zW ’.
constant
R[---}
[
zW
W(r)
R(C)
zl
< R(C),
or 1 whichever is the smaller.
1 and, therefore,
F(z)
We
is univalent
D.
In [i] the inequality (3.6) was obtained when W
WG(z)
and
Wc(r)
W
G
.
(r)
by
,C
a method that relied upon the inequality
r
obtained from
(3.3)
d
zW’.
-T R|T}
[i].
R[
zep(z)}
zW’.
+ R|V
by neglecting the term
zW’
2 IT
(3.6)
W(z)
is not sharp enough to yield
of
->
when W
In this paper we retain the term
zW’.
2.[--J
zW’.
g2.IT
Unfortunately this inequality
and
in
Wc(r
(3.3)
W
.
(r)
by the method
8,C
and derive estimates for
464
R.K. BROWN
its rate of growth relative to that
(3.6)
for W
we
W6(z)
Wc(r
and
W
.
These estimates enable us to establish
of----.
C
(r).
,C
introduce the following notation where z
<
satisfies the inequalities 0
r
-g[---}
M(r)
-j[)
r
is
constant,
and r
(3.8)
w ’c (r)
(3.9)
Wc(r
W6(r)
r
zW’
(3.o)
We (r).
zW’
r
[--W--)
W(r)
(3.11)
w(
T(r)
-[z2p(z))
(r) -=
R[z2p(z))
+ r
l(r)
[z2p(z)}
2
+ r
(r)
-[z2p(z)]
R(C)
e
w6()
Wc(
zw,
S(r)
-
ie
< 1.
zw’
T() -= g--q-}
N(r)
re
PC(r).
+ r
(3.)
PC(r).
-
(3.13)
Pc(r).
(3.1)
pC (r).
(3.15)
+ r
is the smallest positive zero of
W’(r)
C
min(l, R(C)).
R*
(3.16)
(3.17)
In terms of this notation our goal is to establish conditions under which
T(r) > T(O)
From
on every ray
(3.3), (3.4),
dT(r)
r
e
and
constant,
(3.5)
r(r)
+
zl
<R
we obtain the following relationS:
T(r)(l
dS(r)dr --;(r) + S(r)(l +
W(r)
w()
W(+
S2(r)
T2(r) + 2.zW’.
[--.
2.zW’.{__}
+
2(
We (r))2"
(3.18)
(3.19)
465
UNIVALENCE OF NORMALIZED SOLUTIONS
r
dM(r)
dr
(r)
+
M(r)
+ 2
r
dN(r)
(r)
+
N(r)
2
dr
zW’
zW’
zW’
zW’
R[---} [---}
R[----} [--W-
+
+
rW6(r)
2
rW(r)
2
(Wc(r))
(3.20)
(3.21)
Wc (r))
The proofs of most of the lennas in this section reflect a common simple theme
that is set forth formally in the following lemma.
LEMMA A.
G(r) >
G(r)
Let
_<
0 for all a
be a real-valued differentiable function on a
<
r
_<
0
b and
G(0
O.
Then it follows that
< r < b.
Let
G’(0) _< O.
It should be noted that from their definitions it follows that the functions
_
T(r), S(r), M(r)
segment
e
0_< r_<
r
2
constant,
< l,
LEMMA 1.
have
a) T(r)
c)
rlW
1
I21
T(O)
T(r)
that
Thus if, for example,
<
0_
r
2.1.
T(O)
k.
If for fixed
e
we
r_<
r
R
I <
*-I21,
*)’
for all 0
for which
dT(r)
dr
r=p
k for some r in
< l,
<
< E
r
Assume that the conclusion is false.
T(O) >
T(r)
0 in this interval for which
satisi’y the conditions of Theorem
for all
_< 2(1
< R
T(r)
0 for all 0
< O. We
<
T(O)
<
r
will
for all 0
<
r
Then there exists an r,
Let 0 be the smallest such r.
O.
0 it follows from Lemma
A,
with
G(r)
Then since
T(r)
show, however, that our hypotheses imply that
Thus there can be no roots of
T(r) > T(O)
< i.
r2
for all 0
I
T(r) > T(O)
r
_< r_<
z2p*(z)
Let
WC(rl)
PROOF.
r<
r
0
can assume a value at most a finite number of times on any
then there is a smallest r
b) T(r) > T(O)
then
N(r)
and
T(r)
T(O)
on r
I
<
r
< R
T(O),
dTIr)J
D(%
dr
r=p
and consequently
< R
From (3.18) and a) and b) of our hypotheses we have
r
dr
Ir=p
j
> (0)
+
T(O)(I-
w6(p)
wc()
2(o).
(3.22)
R.K. BRO
466
From Theorem 2.1 i follows that f(r)
r
I
<
r
.
<R
d)
dT(r)
of our hypotheses is
dT(r)l
dr
T(r) > T(O)
LEMMA 2.
<
for all 0
Let
O.
p*(z)
<R
r
satisfy the conditions of Theorem
the inequalities 0 < r < R
I
I satisfy
> S(O), and for all 0 < rl_< r_< r2 < R
and let r
S(rl)
2
62 m2(o
+
This is the desired contradiction from which it follows
O.
Ir=-p
26*)
T(O)(1
+
Jr=O
r
that
T2(O)
> T(O) + T(O)f(rl)
,drJr= 0
> "1"(0)
Thus
is monotonic increasing on
r)
Thus
r
which by
2rW(r)
_= i
2.1.
If for fixed
e
Let
61
we have
-
>0,
a) () > (o),
then
S(r) > S(O)
for all r
PROOF. From (3.19)
r
dS(r),
dr
Jr=p
l_< r_<
and
->(0)
+
a)
r2.
and
S(O)
+
b)
of our hypotheses we have
S2(O) + 2pW(o)
Wc(P
Now use the method of proof of Lemma 1 with
zero of
G(r)
on r
I_< r_<
r2,
2w6()
Wc(r
From Theorem 2.1 it follows that if
< r_< r2. Thus
from
G(r)
+
S(r)
pW6(p)
2(Wc(P))
2
S(O), p
2
2"
(3.23)
the smallest
and
f(r)--
on 0
S(O)
(3.22)
61
we have
S(O) + 2(
6 >
tWO(r)
WC (r))2.
0 then
f(r)
is monotonic increasing
467
UNIVALENCE OF NOP@ALIZED SOLUTIONS
r
dS(r)
Jr=
dr
0
>(0)
+
S(O)
+
$2(0)
13
f(O)
+
dS(r)
r
dr
Jr=O
2.1.
Let
O,
and the lemma follows as in the proof of Lemma 1.
LEMMA 3A.
Let
z2p*(z)
satisfy the conditions of Theorem
132 _< 131 13 and let rI satisfy the inequalities 0 < rI < R
S(rl) > S(O), N(rl) > N(O), and if for all 0 < rl_< r_< r2 < R we
0
<
a) (r) _>
If for a fixed
have
(0),
b) (r) _> (0),
c) o _<
zW’.
{-- _< 132,
then it follows that
N(r) > N(O)
PROOF. From Lemma
(3.9) we
r2.
S(r) > S(O)
2 it follows that
for all r
I
_<
_< r2,
r
and from
have
.zW’.
R{---} <
rW(r)
Wc(r)
S(O)
Using this inequality along with
r
r_<
for all rl_<
dN(r) > M(r)
dr
N(r)
+
131 +
(3.21)
and
(3.11)
rW(r)
Wc(r
c)
for all r l_<
r_<
r2.
of our hypotheses we have
rW
rW (r)
2(131 + 132
where we have used the definition
.
13
Wc( r) (N( r) +
in the third
Wc
rW (r)
r
+
(r)
2
WC( r)
term.
From (3.24) we obtain
(r)>dr M( r)+N( r) [1-2(131+
)]
+
2N(r)
tWO(r)
*) rW(r)
’Wc(r
WC(r -2(81 +
Now use the method of proof of Lemma 1 with
zero of
G(r)
in r l_<
r_< r2,
f(r)
Then from
(3.25),
r
N(r)
G(r)
+
3(
N(O),
rW(r))
Wc(r
2
.(3.25)
0 the smallest
and
2IN(O)
Theorem
2.1,
r)jr=0 > M(O)
rW(r)
(131 + 13. )] ’Wc(r
and
+
a)
and
b)
+
3(
rW(r) 2
Wc(r ))
of our hypotheses we have
N(O)[1- 2(1
+
)]
+
f(o).
(3.26)
R.K. BROWN
468
From Theorem 2.1 it follows that if
increasing on
0_< r
_<
r
2.
dN(r)
r
dr
(3.26)
Thus from
r=
0
62 _< i
f(r)
then
is monotonic
we have
> v(O)
+
N(O)[1-
v(0)
+
N(0)
2(81
-212
r(rr)r=0
+ 8* )] +
+
f(O)
.e
0.
The lemma now follows as in the proof of Lemma 1.
0
<
62 _< l
rle
iw
’(
and let r
rleie
> 0,
W(r le ie)
0
< rI _<
d)
r
_<
rW(r)
Wc(r
r2
.
zW’
To prove that
[--}
r I_< r_< 0
the interval
N(rl)
> N(0),
>
0 and
-zW’
J[---} >
N(r) > N(O)
in the interval r
to obtain
zW’
I_< r_<
.
and if for all
and
2
B
d)
.
+
r_<
_<
I
r
_<
r2.
r2 note that if 0 is the
r
we can apply Lemma 3A on
I < < r2, then
N(r) > N(O)
S {---} >
(3.27)
for all r
0 for all rl_<
it follows that
for all r
If for a fixed
0),
2’
[----}
zW
> S(O),
< rI < R
Let
have
> max (
smallest zero of
satisfy the conditions of Theorem 2.1.
satisfy the inequalities 0
I
S(rl)
< R we
then it follows that
PROOF.
z2p*(z)
Let
COROLLARY 3A.
2
.
for all r
+
I
<_
r
rW(r)
_<
0.
Then from
3. ii)
(3.27)
Wc(r)
of our hypotheses give
rW(r) _>
Wc(r)
0 on
rl_< r_<
0.
(3.28)
e
469
UNIVALENCE OF NORM-LIZED SOLUTIONS
Then
(3.27)
and
(3.28)
imply that
eiew
W(0e
_
which contradicts the assumption on p.
from Lemma
3A
it follows directly that
LEMMA 3B.
I21
_< E1
8
and let r
> M(O),
M(rl)
we have
a) (r)
b) a(r)
z2p*(z)
Let
_zW’_
then it follows that
PROOF.
zW’.
{-- _>
N(r) > N(O)
0 for all r
for all r
> S(0),
and if for all 0
< rI _<
(3.21)
M(r) > M(O)
for all r
I
<_
and replace the condition 0
COROLLARY 3B.
l-x-’
Ii21 --< i1
0
leiSw
r
le
< rI _< r_<
zW
Let
< O,
r2
< R
[-V-}
-> 2"
d)
rW (r)
>
r2.
Let
< O,
2
r
2
< R
we have
r
_<
r2.
Start with (3.20)
< I2 _< I1
by
I121
dM
Jr= 0 >0o
satisfy the conditions of Theorem
I satisfy the inequalities 0
_< I 1
2.1.
Let
< r I < R* ." If for a
S(rl)
> S(0),
> M(O),
M(rl)
2
< 0,
fixed
and if for all
we have
M(r) > M(0)
for all r
I
_<
r
_<
r2.
The method of proof is the same as that of Corollary 3A using
N(r)
Now
max
then it follows that
place of
z2p*(z)
and let r
C)
Wc(r
r2.
i8
W(rleie)
PROOF.
_<
_<
0,
The lemma then follows by establishing the contradiction
r
r
r
If for a fixed 8
The method of proof is the same as that of Lemma 3A.
instead of
[
2.1.
the inequalities 0 < r < R
I satisfy
I
S(rl)
_<
_< r_<
I
satisfy the conditions of Theorem
I
(0),
_< a[--q-} _<
c)
Thus
throughout.
M(r)
in
O,
470
R.K. BROWN
LEMMA 4A.
2 -<
M(rl) > M(O),
0
<
Let
z2p*(z)
satisfy the conditions of Theorem
2.1. Let
I satisfy the inequalities 0 < r1 < R
and if for all 0 < r <_ r <_ r < R we have
2
I
’
and let r
If for a fixed
e
a) (r) _(0),
) (r) > (o),
.zW’.
C) J[---}
-> 2’
Wc(r) _> max(8
d)
then it follows that
From
PROOF.
r
M(r) > M(O)
(3.20), (3.8),
dM(r) > (0)
dr
O.
with equality for r
+
for all r
and
a)
I
and
_<
b)
M(r) + 2(T(O) +
Using definition
_<
r
r2.
of our hypotheses it follows that
rW6(r)
[--Q--}
Wc(r)) -zW’
(3.10)
+
rW6(r)
WC
2
r)
we rewrite this inequality in the
form
r
dM(r) >I/(0)
dr
+
M(r)(l- 2T(O)) -2(M(r)+ T(O))
Now use the method of proof of Lemma i with G(r)
zero of
G(r)
on r l_<
r_< r2,
+
From Theorem 2.1 it follows that if
<
r
r
_<
r2.
dr
Then from
Jr=
M(r)
M(O),
(3.29)
T(O))
rW(r)
rW(r)
2
kr)
(3.29)
0 the smallest
2
Wcr (.Wc(r))
2 -< (i /2
then
f(r)
is monotonic increasing
we have
,w6(),
WC---T)
_>/(0)
+
M(0)(I- 2T(0)) -2(M(0)
+
T(0))k
>(o)
+
(o)(- 2T(O)) -2((0)
+
T(O))
r
rW6(r),
"c"
and
f(r)-- -2(M(O)
on 0
rW6(r)wc(9)
M(r)J
dr
r=O
O,
and the lemma now follows as in the proof of Lemma i.
-
w6()
(WC(0)
471
UNIVALENCE OF NORMALIZED SOLUTIONS
*
I21 --< l 2
N(rl)
z2p*(z)
Let
LEMMA 4B.
and let r
>N(O), and if for
a) (r) _> (0),
b) T(r) _> T(O),
satisfy the inequalities 0
< rl_< r_< r2 < R
all 0
2.1.
< r I < R*
Let
rW6(r)
max(*- e21
Wc(r _>
If for a fixed
e
we have
PROOF.
o),
N(r) > N(O)
then it follows that
for all r
l_< r_<
r
2.
The proof proceeds precisely as that of Lemma 4A except that
(3.10),
used in place of
and the condition
z2p*(z)
Let
L4MA 5A.
I121
<
IB2 _<
replaces
2
satisfy the conditions of Theorem
*
<
(3.11)
is
2
2.1. Let
i
and let r satisfy the inequalities 0 < r < R*
2 --< 2
I
I
s(rl) > s(0), M(rl) > M(O), and if for all 0 < rl_< r _< r2 < R* we
0
B2 < O,
zW’.
c)
d)
I
satisfy the conditions of Theorem
If for a fixed
e
have
a) (r) _> (0),
b) (r) _>(0),
c) T(r) _> T(O),
-
zW’
e)
rW
--C
max
.
(
then it follows that
2’
0),
S(r) > S(O)
for all r
I_< r_<
By Lemma 4A we have
PROOF.
rW6(r)
.zW’_
[--q-}_<-(O)- We()= 2
for all r
r
I
_<
dS(r
dr
r
_<
r2.
_>(r)
Thus from
+
S(r)(1
+
(3.19)
2tWO(r)
and
WC(,-))
/
(3.30)
$2 (r)
Now use the method of proof of Lemma 1 with
zero of
r2.
G(r)
on r l_<
r_<
r2 and
/
.
tWO(r)
wc()
>
(3.30)
it follows that
(-.(0)
G(r)
S(r)
tWO(r)
)
We
2
/
S(O), p
tWO(r) .)
e(-Wc().
the smallest
472
R.K. BROWN
()
(0))(
e(s(0)
From Theorem 2.1 it follows that if
increasing on 0
<
_<
r
r2.
rW(r)
wc(r ))
2 -<(I /2
(3.31)
Then from
tWO(r) 2
wc())
+
then
and
a)
through
d)
2
(o)
-(o)
+
f(r)
is monotonic
of our hypotheses
it follows that
-s()
Jr=p >(o)
dr
s(o)
+
rdS(r)
+ S
j r=O
dr
O,
I.
and our lemma follows as in the proof of Lemma
2
Let z p
LEMMA 5B.
I21
--<(l
2,
e, N(rl) > N(0),
(z)
and let r
S(rl)
(o)
satisfy the conditions of Theorem
2.1.
r
I satisfy the inequalities 0 < I <
> S(0),
and if for all 0
< r I _<
r
_<
r2
R*.
< R
Let
2 < O,
If for a fixed
we have
a) (r) > 9(0),
b) (r) > G(O),
c) T(r) > T(O),
d)
zW
J[---] _< IB2
e)
rW (r)
we(r)
>mx( #.
then it follows that
PROOF.
lel
S(r) > S(O)
LEMMA 6.
Let
the inequalities 0
<
I_< r_<
r
for all r
r2
4A, and the
z2p*(z)
<
r l_
< R we
<R
.
then
_<
r
_<
r2.
zW’.
2.f.----J
T(r) > T(O)
_>
condition
I21
5A except that Lemma 4B
_< (i
satisfy the conditions of Theorem
If for fixed 8
have
a) "r(’) _> "r(o),
b)
I
The proof proceeds precisely as that of Lemma
is used in place of Lemma
0
0)
2
for all rI_<
r_<
r2.
T(rl)
> T(0),
)/2
replaces
2.1.
Let r
I satisfy
and if for all
473
UNIVALENCE OF NORMALIZED SOLUTIONS
PROOF.
With
G(r)
and p defined as in Lemma i, we obtain from
(3.18)
and our
hypotheses
rdT<r)]
dr
r= 0
> r(O)
T2(O)
T(O)(I- 2*)
+
rdT(r)|
0
3r=p
dr
+
and the lemma follows as in the proof of Lemma 1.
__
LEMMA 7A.
0
8
0
2 --< i
<
< r I _<
_<
r
a) (r)
b) T(r)
c) M(r)
9"
e)
then
< R
r2
T(rl)
> T(O), [
re
i
iS
w ’(
rlei8
If for a fixed
> 0;
W( r i8
me
and if for all
we have
(0),
T(O),
M(O),
_
rW(r) _> max
Wc(r
T(r) > T(O)
+
Let
the inequalities 0 < r < R
I satisfy
I
> N(0),
N(rl)
2.1.
zW’.
PROOF.
(N(0)
satisfy the conditions of Theorem
and let r
> S(O),
S(rl)
z2p*(z)
Let
Wc(r))
r_<
0),
an r I _< r _< r2.
(3.11) and e) of our
0 for all r
rdTdr > T(r)
for all r l_<
2’
for
From
rW(r)
(
r
+
I
_<
r
T(r)(l- 2r
_<
hypotheses it follows that
r
2.
Then from
(3.18)
T2 (r)
(N(0)
W(r)
Wc(r))
+
and Corollary
rW(r)
3A we have
2
Wc(r))
+
2.
Now use the method of proof of Lemma 1 with
G(r)
and 0 defined as in Lemma 1
and with
f(r) =-2(T(0) -N(0))
From Theorem 2.1 it follows that if
increasing on 0
<
(3.3)
r
_<
r2.
rW (r)
WC(r
2 -< i
+
rW(r)
2
(WC(r))
then
f(r)
is monotonic
Thus, from (3.32) and b) of our hypotheses we have
474
R.K. BROWN
rdT(r)
dr
Jr=
0
T2(O)
> (0)
+
T(O)
(0)
+
T(O)(I-
rdT(r)
dr
r=
+ N
2
(0)
2*) T2(O)
f(O)
+
(N(O)
+
+
*)
0
0
and our lemma follows as in the proof of Lemma i.
LEMMA 7B.
0
> S(O),
M(rl)
< rI < r <
> M(O),
< R we
r
2
satisfy the condition of Theorem
< r I < R and
the inequalities 0
S(rl)
z2p*(z)
Let
T(rl)
let
< O,
2
> T(O),
2.1. Let r I
I21 < i
satisfy
If for a fixed
[rlei@W(r.leie )]
W(rlee)
e
< 0, and if for all
have
_.
a) (r) _> (
b) T(r) _> r(O),
c) (r) _>
then
d)
zW
[} _> I2,
e)
rW(r)
Wc(r
T(r) > T(O)
PROOF.
I21,
),
I_< r _<
r
max
for all r
2.
The proof proceeds precisely as that of Lemma
7A except
that
Corollary 3B is used in place of Corollary 3A.
LEMMA 8.
If for all
e,
0
_< e _< 2,
and for all 0
< r < R
a) (r) _> ;(0), b) T(r) _> T(O), c) (r) _>(0), d) v(r) _> v(0)
e)
dS
where a
=
through
h)
> O, f)
e,
f, c
b
l>
if
same techniques.
For 0
g, and d
<
r
.
<R
Jr=O > 0
h.
Moreover,
and
h)
rrJ
then it follows that
r:O
> 0
strict inequality holds in
e)
> O.
We prove that b
PROOF.
g) dM
_j r=o > O,
we have
we have
=
f.
All four implications can be proved using the
475
UNIVALENCE OF NORMALIZED SOLUTIONS
zW’
W
8 +
zW(z)
-Pl
and c*
28
I
I
n
+ cn z
+
*
+
Wc(z)
where c
ClZ
ClZ+
-CPl with
28*
+
+ c* z
n
n
and
Pl
Pl
(3 33)
and
+
*
.4
defined by
(1.2)
and
(1.9)
respectively.
Now T(r) >
if
f(r) _> (0)
(-(pl z)
+
T(O)
if and only if
for all
Crp) _> 0
e,
0_<
e _<
-g(z2p(z) po
2, and
2
+ r
*
PC(r)
0_< r < R
all
PO ->
O.
Therefore,
it follows that
for sufficiently small r and for all
8.
Thus we must have
_[Pleie
Now from
(3.33)
and
(3.34)
+ Cp*
I_>
e.
(3.35)
CP 1
(3.36)
0 for all
we have
Jr=o
g[cleiS)
-Ple
Cl
CP *1
iS
28*"
Then if we write
8
in the form
18 le i,
we have
(e
Jr=0
Thus
dT
-jr=O->
for all
0 if and only if
*
-Pll #1’
)
CP
-[pl ei(e’)
1
o
+--->
(3.37)
e.
*/181 -< 1 and it follows from (3.35)
that
’-Jr=O -> 0 with strict inequality if 131 > 8*.
However,
since
8l_>
and
(3.37)
4.
THEOREM A AND THE
MAI
> O, we
have 0
<
THEOREM.
We have designated the first result of this section as Theorem A since it is
our analog of Theorem A of [i] when y
0.
476
R.K. BROWN
W"(z)
+
p(z)W(z)
W2(Z) Z2[I +
W(Z)
Let W
THEOREM A.
zl
is holomorphic in
W"(z)
+ z
2
< i,
and
.
(z)
Wc(Z -=
pc* (z)W(z)
z
W
+
P0
2
+
+
n=l
< 1
()
*
*
<
+
* n +
Pn z
"’’’
Izl
r
let
z2p(z)
z2p(z)
and
z() oI
< R
21 _< 1/2.
be the unique solution of
* < 1/4,
PO
and real on the real axis; and where C > 0 and 0
ll(lzl)
y2
_=
W(r),
0
< r < i,
Lemma 8.
2*_<
1/2.
if such exists.
Po"
it follows that
> o
min(R(C), i).
We first note that (i) of our hsrpotheses ensures that
conditions of Theorem
<
satisfy the inequalities
w()
zw’.
_>
a-qwc(
for all
<
p] + PO. with
Po + Pl z+
I21 -<(-
Then for
be the unique solution of
n
n]
b*(C)z
n
Let R(C) be the smallest positive root of
zl
n]
0 where
z2p * (z)
For
Pn
with 0
z 2 [i +
2,C
pc(Z)= C[z2p*(z)
zl
Pl z
i + i2
2 *
ooc
n
0 where
2
z p(z)
Let WC
b z
n:l
2.1.
z2p*(z)
satisfies the
In addition (ii) implies inequalities a) through c) of
For example, inequality
arz(z)}
a)
of Lemma
+
zlp(Izl) _> a o} + Po’
8
is valid if and only if
and this is true if and only if
R[
z2P(z)
which in turn follows from
(ii).
po] _>
[I zl2p(l zl)
po
UNIVALENCE OF NORMALIZED SOLUTIONS
477
We will obtain the result of Theorem A by proving that T(r) > T(O) on any ray
constant,
< r < R
0
.
To establish this fact we will need Lemmas 1 through 7
zW ’.
whose hypotheses require us to know whether |--Q-}
we introduce on each ray @
constant,
0
<
r
< R
_>
82
zW
S[--Q-) _< 82.
or
the points
_zW’
0i < 0i+l are consecutive values of r at which [---]
show that T(r) > T(O) on every interval O _< r _<
i
0i+l.
82
<
2
Case 2.
0
<
i
82 < ----, e
<
--’
2
constant,
zW’
d[ S[--W-]r=o
constant,
r[[--} Jr=0 <
,
l
,
0,
1121 < l
8 constant,
Case 4"
2
< 0,
I2!
@ constant,
For
0 the result of Theorem A was established by Robertson [i] for
W(z)
2
and W
W
C
.
(z)
and by Brown
rW(r)
--Wc
(r) >
.
I21
I,
’(p]
<
[2]
zW’
d[ S[--}
]r=O
> O,
zW’
d
[[V]jr=0 <
W(z),
for W
,C
w()
r
> O,
< O,
<
for all
0_<
Pi+l
We then
2
We will assume that
0_<
-
and
cases.
zW’.
d
Pi
"
Case
W
l
0
where
changes sign.
The proof requires consideration of the following four
Case I:
0i
Therefore,
W
W
C
< R
r
.
0.
.
(z).
,C
since if for some
we can restrict our attention to the interval
0 and then use Lemma i on the interval 0
<
< R
r
PROOF OF CASE i.
1.
From Lemma 8 it follows that there exists a
all 9
S(r) > S(O), T(r) > T(O), M(r) > M(O)
<
< 01 such that
0
0
and
N(r) > N(O)
0
for all
0<r< 0
2.
Now fix
e
and apply Lemma 6 to the interval p
T(r) > T(O)
}.
on 0
_<
r
N(01)
> N(O).
rWd(r)
r
_<
p to obtain
_< 01.
From definitions (}.10) and
monotonicity of
_<
(3.11), the
on 0 < r < R
definition of the
it follows that
0i
M(01)
and the
> M(O)
and
for
R.K. BROWN
478
5A applied to the
_<
r
_<
01 we have S(01)
4.
Then by Lemma
5.
By Lemma 7A it then follows that T(r) > T(O) on the interval
6. By
6, T(r) > T(O)
Lemma
8. As
in
3
above we obtain
Then by Lemma
9.
02 _<
on the interval
S(02
> S(O).
> M(O),
N(02)
By 4 above and Lemma 2 we have
7.
interval 0
M(02)
5A we have
01 _<
> S(O).
_< 02.
r
_< 03.
r
> N(O)
and
N(03)
> N(O).
> S(O).
S(03)
From 8, 9 and Lemma 7A it follows that T(r) > T(O) on the interval
10.
03 _< r _< 04.
By successive iterations of steps 6 through lO it follows that if
T(r) > T(O) on 0i <
_< Oi+l
then
T(r) > T(O)
on
proof actually demonstrates that
T(r) > T(O)
on any interval of the ray 8
O<r<R
r
on which either
T(r) > T(O)
zW’
[--
2 ->
zW’
_< 0i+2. Moreover,
r
2 -<
[---]
0 or
-> 0 for all 0 < r < R
l
0i+l _<
on any ray
e
O.
the
constant,
Thus it follows that
constant.
PROOF OF CASE 2
As in step i of Case i we have that there exists a 0
i.
e,
that for all
aii 0 _< r _< p
s(r) > s(0), T(r) > T(O), M(r) > M(O)
2.
By Lemma 7A it then
follows that
T(r) > T(O)
3.
By Lemma 6 we have T(r) > T(O).on
l_< r_< 2"
4.
By definition (3.10) and the monotonicity of
(0)
By i above and Lemma 2 we have
6.
Then by Lemma
5A
7.
By definition
(3.11)
8.
on the
rW6(r)
Wc(r
< p <
such
01
N(r) > N(O)
and
for
ilterval
.
on 0
<
r
< R we
on 0
<
r
< R we have
have
> (0).
5.
N(02
0
> S(O).
S(01
it follows that
S(02)
> S(O).
and the monotonicity
rW6(r)
of.c(r
.
> N(O).
Then by 3,
6, 7,
and Lemma
7A
we have
T(r) > T(O)
on the interval
s2/r/03.
By successive iteration of steps 3 through 8 (omitting the reference to step 1
5)
in step
0i+i_<
r
it follows that if
< 0i+2.
T(r) > T(O)
on
Then as in Case 1 we obtain
0i _<
r
_< 0i+l
T(r) > T(O)
then
i
T(r) > T(O)
_>
on
0 for all
UNIVALENCE OF NORMALIZED SOLUTIONS
0
<
< R*
r
479
constant.
on any ray 8
PROOFS OF CASE 3 AND CASE 4.
The proofs of Case 3 and Case 4 are identical to
those of Case 2 and Case 1 respectively except that all A lemmas are replaced by
corresponding B lemmas.
COROLLARY A.
W(z)
.
and W
Theorem A remains true if
(z)
or by
a ,C
PROOF.
W(z)
.
and W
W(z)
Wc(Z)
and
are replaced by either
(z).
,C
The result follows from the fact that all of the lemmas used in the
proof of Theorem A remain valid under the indicated substitutions.
Our Main Theorem will be derived from Theorem A precisely as the Main Theorem
of
[1] was
[1].
derived from Theorem A of
We will not reproduce Robertson’s proofs
Wc(Z)
but simply mention that his methods apply equally well to
Wc(Z)
W
.
(z),
(z)
and
,C
and then summarize the needed results in the following lemma.
Let
fixed, 0 < R < 1.
zp*(z)
W(R](R
0 and
z2p*(z)
we have
lim_C(R)
A,
satisfy the conditions of Theorem
C(R) >
Then there exists a C
we have
>
W(R(r
-_- A(p*)
0 for all 0
>
< r < R.
A is finite and
0 there exists
8
2.1
0 such that when
W(r)
r( ),
and let R be
p(z) _-- PC(R)(Z)
Moreover,
>
called the universal constant corresponding to
the sense that for any e
W(r(e))
.
,C
LEMMA .l.
The value
W
for fixed
0 for all 0
z2p*(z.),
0
< r(e) < i,
*
n
< r < 1.
is largest in
such that
A+ (r(e)) < 0.
0 and W’
THE MAIN THEOREM.
Let
z2p* (z)
*
* +
PO + Pl z
zl
be nonconstant and holomorphic in
<
+
Pn Z
+
1 and real on the real axis with
P0 -< 1/4.
Let
a[z2p*(z)} _< Iz!2p*<Izl)for Izl
Let z
2
*
pA (z)
A( z2p* (z)
corresponding to
z2p*(z).
* + * where A
P0
p0
is the universal
Let
WA(Z)--W. ,A (z)
be the unique solution of
A(p*)
z
[+
D bn(C)]
n=l
(4.1)
< 1.
Izl
< i,
constant
480
R.K. BROWN
W’(z)
pA (z)W(z)
+
0
corresponding to the smaller root of the indicial equation.
FA(z)
-_-
Then the function
[WA(z) ]i/
zl
< i,
holomorphic and univalent in any larger circle whenever A
> O.
is a holomorphic function, univalent and starlike in
Let
for all
z2p(z)
zl
zl
be holomorphic in
i + i2’ i
x bn zn] zl
W(z) z[l + n=l
> O, be
,
corresponding to the root
I21 --<
Then if
part, of the indicial equation.
the function
2
[W(z)]l/
z +
function, univalent and spirallike in
is the largest possible
PROOF.
+ p(z)W(z): 0
with smaller real
F(z)
is a holomorphic
zl
< i.
The constant
one.
From Theorem A we have
ar
Now if we choose
and from Theorem
3.23
z)
F(z)]
z2p(’z)
of
zwW(z’)}
< _>
WA(IZl
pc (z)
2
z)
--a[
_=
z
then
O,
Wc(II) Izl
(4.4)
>0,
W(z)
[2] we have
Izlw <Izl)
Thus from
< 1
the unique solution of
W"(z)
A(p*)
1 with
< 1. Let
W(z)
A
<
and is not both
and the definition of A we have
< (
Izl
=_
W
.
(z)
,C
_=
Wc(Z)
UNIVALENCE OF NOPMALIZED SOLUTIONS
g[
(4.5)
from
it follows that
since equality holds in
R,
some
FA+(z)
0
< R < l,
> i__
FA(Z)}
From (4.3) it follows that
F(z)
FA(Z
(4.4)
1621
whenever
G
_<
zl
<
.
and
A+ (R)
,
0 for
it is clear that
we
take.
Thus
The proof that the radius of univalence of
[i] page
265
and will not be reproduced here.
W(z)
G*
W(z)
and also when W
2
< i,
< I. Moreover,
1 no matter how small a positive
The Main Theorem is true when W
I
zl
when z is real and positive, and since W’
is precisely I is contained in
G
zl
is univalent and starlike in
for arbitrarily small positive values of
is not univalent in
COROLLARY B.
(4.5)
is univalent and spirallike in
the constant A is the largest possible.
FA(Z)
481
Wc(z)
and
c(z)
and W
W
.
(z)
W
6
.
(z)
,C
whenever
,C
I
The proof of this corollary is immediate since all of our previous results
remain valid under the indicated
ubstitutions for W(z) and
Wc(z).
5. REMARKS.
We will now indicate how the results of our Main Theorem and Corollary B
compare to or extend those of Robertson’s Main Theorem in
In
Ill
Ill
for
0.
the condition
zl)
_<
replaces our condition
refer to the case W
Our condition
(4.2),
W(Z)
(4.2)
there is no explicit bound on
and W
C
W
6
.
IG21,
and the results
(z).
,C
(4.6).
is independent of condition
When
_> 1/2
our
Corollary B requires that
(4.7)
while
(4.6)
implies that
2
2
*
2
*
so that
I%1 -<i-"1
(4.8)
R.K. BROWN
82
We refer now to the "exponent plane" of Figure i in which
measured horizontally and G2 and
62 are measured vertically.
61
are
I and
Our conclusions are
G
the following:
i.
In region I only our Main Theorem applies.
2. In
3.
regions II only Robertson’s Main Theorem applies.
In regions III U IV Robertson’s Main Theorem applies and our Corollary B
WG(z)
applies with W
4.
and W
W
C
.
(z).
6 ,C
In region IV Robertson’s Main Theorem applies and our Corollary B applies
W(z)
with W
and W
C
W
G
.
(z).
,C
[2].
Thus it is in region I that we have an extension of the results of [i] and
Figure 1
We note that we can obtain Robertson’s Theorem A with y
WG(z)
W
and W
Theorem A of
Gl
>
G
then
follows that
C
Ill
W
G
.
(z).
imply that
with
This follows by noting first that the hypotheses of
G1 -> G
and that
if G
G
T’(O) > O. Now
(3.18)
(3.18)
,C
T(O) > O,while
we obtain from
0 from
1
T(r) _>
then G2
0 for all
0_< r < 1.
Also, if
0 and as in the proof of Lemma
if we let p be the smallest zero of
T(r)
on 0
<
r
<
8
it
1
483
UNIVALENCE OF NOPMALIZED SOLUTIONS
0dT(o)
(0)
dr
Thus either
dT(0) >
do
+
2[ 0eiew(O
eie
Wp(p e i@)
0 or both terms in the right member of
(4 9)
vanish
The
former conclusion yields an immediate contradiction to the definition of 0.
If we
assume the latter conclusion then an examination of the successive derivatives of
(3.18)
shows that the first non-vanishing derivative of
of even order.
T(r) >
Thus
0 for all 0
Finally we point out that for V
O,
T(r) at p
is positive and
< r < R
IYI
< /2,
Robertson’s Theorem A can be
obtained by applying the same reasoning as above to the following analog of
where W
G(z)
W
rdT
dr
and W
W
C
T(r)
G
+
.
(3.18)
(z).
,C
T(r)(1-
2rW&(r)
We(r)
sec
T2(r)
+ sec
zW’
2 [---).
REFERENCES
ii
ROBERTSON, M.S. "Schlicht
254-275, 1954.
2. BROWN, R.K.
PAEK,
4.
’
thorie
"Univalent Solutions of
Lad.
PBst.
Solutions of W" + pW
"Contribution
la
Mat, Fys. 62, 12-19,
POMMERENKE, CHR.
+ pW
0" Trans. Amer. Math. Soc,
0" Can J. Math. 14, 69-78,
76,
1962.
des fonctions univalentes" Casopis
1936.
Univalent Fun_ctions, Vandenhoeck & Ruprecht, Gttingen,
1975.