I ntern. J. Math. & Math. S ci.
Vol. 5 No. 2 (1982)351-356
351
ON THE DIFFERENTIABILITY OF T(r,f)
DOUGLAS W. TOWNSEND
Department of Mathematical Sciences
Indiana University-Purdue University at Fort
Fort Wayne, Indiana 46805
Wayne
U.S.A.
(Received June 16, 1980)
ABSTRACT.
It is well known that T(r,f) is differentiable at least for r
>
r
0.
We show that, in fact, T(r,f) is differentiable for all but at most one value of
r, and if T(r,f) fails to have a derivative for some value of r, then f
is a con-
stant times a quotient of finite Blaschke products.
KEY WORDS AND PHRASES.
Nevanlinna Theory, Nevanlinna Characteristic Function.
1980 MATHEMATICS SUBJECT CLASSIFICATION CODE: 30A70
i.
INTRODUCTION.
In this paper we will discuss the differentlability of the Nevanlinna characteristic function,
T(r,f), for
f meromorphic in
Izl
< R <
.
As early as 1929
Cartan stated in [13 that T(r,f) is differentiable, and gave a formula for
dT(r,f)/d log r, but
did not indicate whether the derivative may fall to exist
for some values of r. We will show that T(r,f) is differentiable for all but at
most one value of r, and if the derivative fails to exist for some value of r,
then f is a constant times a quotient of finite Blaschke products.
2.
DEFINITIONS AND STATEMENT OF THE THEOP.
We begin by defining the standard Nevanlinna functionals.
D.W. TOWNSEND
352
zl
Let f be a meromorphic function in
DEFINITION.
R
<
(R).
Izl
<
r, and
<
Then for
r<R,
1,
n(r,f)
n(r,(R),f)
the number of poles of f in
n(r, f 1
n(r,a,f)
..a)
the number of solutions to the equation
f(z)
r
2.
-
n(0,f
t
r.
<
dt + n(O,f)log r, and
0
1
N(r,f a).
N(r,a,f)
+
1
m(r,f)
m(r,a,f)
4.
n(t,f)
N(r,(R),f)
N(r,f)
Izl
a in
f(re+/-e’)
i
and
de
0
1___)
m(r, f
m(r,f) + N(r,f), and
T(r,f)
T(r,a,f)
m(r, f 1 a
T(r,
+ N(r, f-. 1.
)
For the purposes of this paper we define the additional functional
5.
{the number of solutions to f(z)
n(r,a,f)
n-(r,a,f)
a on
Izl
r).
We note that the derivative of N(r,a,f) from the right with respect to log r
is
n(r,a,f)
n(0,a,f), and the
n-(r,a,f)
n(0,a,f)
f(z) # a on
Izl
r.
d
Thus,
derivative from the left with
N(r,a,f)
d log r
N(r,a,f)
Since
n(r,a,f)
respec to
log r is
n(0,a,f) provided
is continuous and
n(r,a,f)
n(0,a,f)
is
monotonically increasing, N(r,a,f) is an increasing convex function of log r.
The following lemma, which we state without proof, gives a characterization
of
T(r,f)
which has proved to be of great importance in the development of
Nevanlinna theory.
We will base our discussion of the differentiability of
T(r,f) largely on this lemma.
CARTAN’S LEMMA.
If f is meromorphic in
Izl
<
R and 0
<
r < R, then
2n
T(r,f)
=
i
N(r,ele ,f)
d +
log+if(0)l.
0
Since
N(r,a,f)
is an increasing convex function of log
r, it follows from
Cartan’s Lemma that T(r,f) is also a convex, increasing function of log r.
By
DIFFERENTIABILITY OF T(r, f)
a well known theorem concerning convex functions,
353
T(r,f)
has derivatives from
the right and from the left for all r > 0, and is differentlable for all but at
most countably many values of r.
We give an example
to show that
T(r,f) need
not be differentiable for all values of r.
Let f(x)
EXAMPLE.
8,
extended plane,
for all real
onto
1 + iz/2
i + 2iz
2
8,.and
The function f is a one-to-one map of the
takes the unit circle onto the unit circle.
Thus,
e,
=01 lif
r < i
n(r’eie’f)
if r >
Also,
n(t
N(r,eie,f)
dt
t
flog
r if r
>
1
and by Cartan’ s Lemma
T(r,f)
1
log 2
N(r,eie,f)
de
0(
if r < 1
og r if r > 1
0
Thus, T(r,f) is not differentiable at r
1.
That this example is representative of the class
T(r,f)
fails to have a derivative for some value of
f
functions for which
r, is evident from the
proof of the following theorem.
THEORg.
T(r,f)
Let f be a nonconstant meromorphic function in
is differentiable for all but
at most one value of r
<
Izl
<
R.
If
R
Then
(R).
T(r,f)
r0, then f is a constant times a quotient of
fails to have a derivative at r
finite Blaschke products.
PROOF OF THE TkOREM.
3.
STEP i.
In this part of the proof we will use the fact that for r
is uniformly bounded for all a e C by a finite
n(r,a,f)
<
r
I
<
R
constant depending
only on r I.
Suppose that 0
r’
<
r
k
<
<
r’
< r
0
<
r"
<
R, and consider a sequence {rk} satisfying
r", rk # r 0 for all k and lim rk
r
O.
D.W. TOWNSEND
354
By Cartan’ s Lemma we have
21
T(rk’f) T(ro’f)
llm
k
r0
rk_
N(rk,eie,f) N(r0,eie,f
rl’ r0
i
lim
2
k
m
de. (B.1)
0
n(r,a,f)
Since
(3. I)
integral in
r
K(r") for all a
<
a C and all r <
the integrand of the
r"
is
I
k
n(t,eie,f)t -I
dt
0
r0
n(t,eie,f)t-1
r0)-i
dt)(rk
0
r
k
-rk_ r.0
r
K(r’,r")
where
log r
n(t, e le ,f) t -I dt
i
is a
<
K( r"
log
k
r0
<
r0
rk
K(r’,r")
0
constant depending only on r’ and r".
Therefore, by the
Lebesgue Dominated Convergence Theorem
2w
T(rk’f) T(ro’f)
lira
r
k
k
1
lim
k
r0
N(rk,e ie,f)
rk
N(r
r
0,eie,f)
d8,
0
0
__
provided the limit in the integrand above exists off a set of Lebesge measure
From the definition of N(r,a,f) we have that
zero.
lim
k
provided
N(rk,eiO,f)- N(r0,e ie,f) n(r0,eie,f)
rk
f(r0 ei)
r
0
W e ie for all 0
r0
27.
Therefore, if
f(roel)
i only on
a set of measure zero then, since {rk} is an arbitrary sequence converging to
r0,
we have
2
r0
dT(.rf)
l__
dr
2
r=r0
STEP 2.
lim
J
to
Cj
zl
line.
0"
ie
f) de.
0
i
Suppose that for some r0 < R,
1,2,3... and
)I 1 for
Let L be a linear fractional transformation mapping the real line
1
If(r0e
r0 and
Let g
n(r0,e
a linear fractional transformation mapping
L2
f
L1.
zl
1 to the real
Then g is meromorphic in a neighborhood of the real
axis and there exists a finite x
0 which is a limit point of {x
g(x)
is real}.
DIFFERENTIABILITY OF T(r,f)
From elementary power
355
series considerations it follows that g is
Thus, for an interval of
e
[0,2)
values in
n(r0,eie,f) n-(ro,eie,f)
If(r0ei)
Hence,
valued everywhere on the real axis.
(extended) real-
1 for all 0
27.
having positive length
+ 1.
(3.2)
then as in Step 1 above
If r
k+r0,
2
lim
k-
(rk’f)
(r0’f)
r
r
k
1
N(rk
llm
k
0
eie,f)
N(r 0 ,e ie ,f)
r0
rk
0
1
Similarly, if
lim
k+
then
rk+r0,
T(rk’f) T(r0’f)
r
n-(r0,eie,f) r0-1 de
r
k
1
n(r
0
0
,ei8 f)r81
(3 4)
de
0
By (3.2) the limits
(3.3) and (3.4) are not equal and hence T(r,f) fails to
in
have a derivative at r
r
0.
To summarize steps 1 and 2, either
T(r,f) possesses
in which case
on
Izl
to,
STEP 3.
f(z)l
Suppose
{b 1,b 2
b
N},
Izl
Define
lajl
r0
zal 2
j=IH
The function P(z)
z
1
i-
lal
ro
a and a pole at z
r()
zr
-I
f(z) and (z) are holomorphic in
f-7
jzr
za -I
2
i
1
zl
1 everywhere on
respectively.
M
B(z)
If(z)l
r 0, or
<
r
/
0
be
N
H
j=l
r
0.
r
0.
Let the zeros and
a
(al,a 2
M and
Ibl
1
r0
I
zb 1 2
b% zr
is a Blaschke factor having a zero at
and having modulus one on
zl
r 0,
1 everywhere
fails to have a derivative at r
poles of f, counting multiplicity, in
Izl
1 finitely often on
a derivative at r
T(r,f)
in which case
If(z)l
Izl
r0
r 0, have no zeros or poles in
Thus
zl
r 0,
D.W. TOWNSEND
356
and have modulus one on
zl
that f(z)=
II
aB(z), where
r
It follows from the maximum modulus theorem
O.
1.
If(z)ol
It remains only to show that
1 on
Noe that from the above argument, if
of r.
has as many zeros
f(z)l
in
zl
in
r
ro,
zl
1 for
(poles)
(poles)
<
r8
>
Izl
<
r
0
Izl
r for at most one value
If(z)
1 for
Izl
ro,
as it has poles (zeros) in
Izl
then f(z)
>
r
If
O.
then by the same argument f(z) has as many zeros
as it has poles (zeros) in
zl
Izl <
r.
It follows readily
r.
that f(z) must have no zeros or poles in r n <
Therefore, f(z) and
1
are analytic in r 0 < zl <
and both have modulus one on
r 0 and
r
r.
zl
By the Maximum
zl
Modulus Theorem, f(z) must be a constant, which
contradicts one of the hypotheses.
Hence,
If(z)l
1 on
Izl
r for at most
one value of r, which completes the proof of the theorem.
If we let (r,f) be the number of solutions of the equation
zl
r, then we have shown that (r,f)
Questions concerning the growth of
<
If(z)l
1 for
for all but at most one value of r.
(r,f) have been posed
in
[2] and [3], and
these questions have been investigated by the author and J. Miles in
[4] and
[5].
REFERENCES
log r de
Sur la drive par rapport
T(r,f), Comptes Rendus, 189 (1929), p. 525-527.
la fonction de croissance
i.
Cartan, H.
2.
Clunie, J. and W. K. Hayman, editors, Symposium on complex analysis,
London Mathematical Society Lecture Note Series 12, 1973.
3.
Hellerstein, S. The distribution of values of a meromorphic function
and a theorem of H. S. Will, Duke Math, J_. 32 (1965), p. 749-764
4.
Miles, J. and D. Townsend, Imaginary values of meromorphic functions,
Indiana University Mathematics Journal, Vol. 27, No. 3 (1978),
p.
5.
491-503.
Townsend, D.
submitted.
Imaginary values of meromorphic functions in the disk,