I ntn. J. Math.
Math. Sci.
Vol. 4 No. 4 (1981) 641-647
641
ON HOLOMORPHIC FUNCTIONS WITH CERTAIN EXTREMAL
PROPERTIES OF ITS ABSOLUTE VALUES
DIETER SCHMERSAU
Mathematisches Institut der Freien Universitt Berlin
FB 19, WE I, HOttenweg 9
1OOO Berlin 33
(Received March 4, 1981)
ABSTRACT.
This paper is concerned with a ’special class of holomorphic
functions with extremal properties of its absolute values on arbitrary
closed line segments in the complex plane.
metrical
and
characterization of the functions
(az+b) +i8
z
The main result is a geo-
with
a,b
z
e
az+b
z
(az+b)
n
n E Z
,8 E IR
KEY WORDS AND PHRASES. Maximum respectively minimum of the absolute
value
fl is taken on at one of the endpoints of every closed line
segment.
1880 MATHEMATICS SUBJECT CLASSIFICATION CODE.
30C45
INTRODUCTI ON
The present work is closely related to the following problem
raised by Rubel [I
closed line segment
Find all entire functions f
in the complex plane, wherever located an in
L
whatever direction, the maximum of
the two endpoints of
such that for every
fl
on
L
is taken on at one of
L
As a secondary result of the solution for this problem we will obaz+b
tain a simple characterization of the entire function z
e
Sup-
pose
f
G
is a complex function holomorphic in the region
G
642
D. SCHMERSAU
Then for all
z=x+iyG
f(z)
with
the partial derivatives of
O
#
first and second order of the function
If(z)
w(x,y)
are given by
[2]:
f’
(Z))
If(z)IRef(Z)
W
x
f(z)l
Wxx
Im2[
yy
y
(z)
<f(z)
(1.1)
(z)
J
Im((z)
If(z)
Wxy
If (z) Im[.f.i.Z)
W
Re(’f’(z
(z)
f(z)
Moreover the formula of Taylor implies:
w(x,y) + hwx(x,y) +
w(x+h,y+k)
+
kWy(X,y)
I {h2Wxx(X,y)+ 2hkWxy(X,y)+ k2Wyy(X,y)}
(1.2)
+ o (h2+k 2)
:= h+ik
Introducing the variable
lf(z+;)i
If(z)]
[f’ (z))
Re,,.f(z
lf(z)
we deduce from (I. I) and (1.2)
+
f (Z)) +
Im2,,f(z)
[f"(z)...2’}
Re.f(-&
)
(1.3)
+ o(1!
2
By means of this equation we prove the following Lemma.
LEMMA I.
with
f(z)
#
Let
0
f:G
f’ (z)
Reif(z)f"(z)
f’2(z)
be holomorphic in the region
z E G
G
and
0
>
respectively
Then there exists a line segment
L
through
Re[ f
z
f’2(z)(z)’)
-z) f"
such that
<
fl
does
not reach its maximum respectively minimum at one of the endpoints of
643
HOLOMORPHIC FUNCTIONS WITH EXTREMAL PROPERTIES
L
and
(,z) f" (z))
Re(f f’2(z)
t
For real
z
Suppose
PROOF.
G
with
f(z)
$
0
f’ (z)
$
0
>
in a sufficiently small neighbourhood of zero, we define
fz)
:= i f (z) t
Then we have:
f’(z)
)
Re[{(z)
Is(f’f(z)(z) [,
0
f"() 2)
Re[f(Z
t
2
t
f(z.)f_"(z).)
Re(f’2(z)
From these equations it follows by means of (I .3)
f[z+i
.}z)(z) t)
to
Hence there exists a
f(z+i f!z)(z,,)to)l
fz_i
f(z)
f’
(z)to)l
<
If(z)
<
If (z)
Re[ f’f,(z)f’’(z))
2
The case
If(z)
2
R
I[1-Re[f(z)f"(z)’))
f’2(z)
+
(t2)
(1.4)
such sthat:
and
<
is treated in the same
way.
(z)
The next Lemma is an immediate consequence of the well known theorem
of Picard [3], "Let
plane.
g
be a meromorphic function in the whole complex
If there exist three different numbers not belonging to the
range of
g
constant".
then
g
is
Let
f
be a meromorphic function in the whole complex
LEMMA 2.
plane, which is not constant.
f f,,
Then the function g :=
is either a constant or there exist
---,
z O, z
{
with
D. SCHMERSAU
644
Re(g(z O)) >
PROOF.
f f,,
g
also the function
and not constant,
is meromorphic in all of
f
Since
<
Re(g(zl
and
is meromorphic in all of
f,
Then our Lemma immediately follows from the theorem of Picard.
Collecting the results obtained so far we end up with the following
theorem:
THEOREM I.
Suppose
is a non constant function meromorphic in
f
ff,,
the whole complex plane such that also
g
Then there exist two lime segments
and
maximum of
f[
LO
on
L
is not a constant.
such that neither the
L
nor the minimum of
on
f
L
is taken on
at the endpoints of these segments.
Next we consider the case that the expression
f f"
is a constant on
f,
LEMMA 3.
Let
c
7+i
Then
be an arbitrary complex number.
the solutions of the differential equation
ff,,
f,
(1.5)
c
are giben by:
az+b
e
for
c
(1.6)
f(z)
(az+b)
f,,
-r
it may easily be integrated [4].
The result is (1.6) with
8
by
for
c
#
Rewriting the differential equation (1.5) in the form
PROOF.
In the case
-
1-c
c
u + i8 :=
y+i
I---
c
f,
a 6 {o}
(1.7)
and
b E
the introduction of new variables
leads to the relations
u
and
645
HOLOMORPHIC FUNCTIONS WITH EXTREMAL PROPERTIES
(1.8)
and thereby
(I .9)
e<Ol <y
c,>Owy<
(,=Oy=l
which will be needed later on.
f(z)
The investigation of the functions
z
e
z
f(z)
respectively
with regard to the extremal properties of their absolute values causes
no difficulties.
z
Since the simple similarity transformation
az+b
maps line segments into line segments there directly follows:
THEOREM 2.
If
is a non-constant, entire function on
f
that on every line segment
maximum at one of the endpoints of
f(z)
e
az+b
or by
f(z)
]fl
its absolute value
L
(az+b)
L
n
takes on its
is given either by
f
then
n
such
l
Theorem 2 completely solves the problem of Rubel mentioned at the be-
In view of the equation
ginning.
f(z)
a further con-
f(z)------T
sequence of Theorem 2 is:
THEOREM 3.
If
is a non constant function meromorphic in the
f
entire complex plane such that on every line segment
value
fl
L
its absolute
reaches its minimum at one of the endpoints of
az+b
L
then
n 6 lq
n
(az+b)
The combination of theorem 2 and Theorem 3 leads to a simple characf
f(z)
is given either by
e
or by
f(z)
terization of the exponential function:
THEOREM 4.
Let
on every line segment
be a non-constant entire function such that
f
L
fl
the absolute value
as well as its minimum at the endpoints of
nential function of the form
f(z)
e
L
reaches its maximum
then
f
is an expo-
az+b
In view of Lemma 3 it seems to be interesting to investigate
the gen-
646
D. SCHMERSAU
eral power function
z e+i8
f(z)
with regard to the
Z
e+i8
extremal properties of its absolute value in the region
{z
G :=
{o}/-
6
< arg z <
re i
z
Introducing polar coordinates
z}
(1.10)
the absolute value of
f
reads:
If(z)
r
e
e -8
(I. 11)
const, the behaviour of
On the half-lines with
fl
is obvious.
In the case of straight lines not running through the origin we have
to consider separately those cutting the negative real axis.
in view of (1.11) it suffices to investigate
If[
Finally
on straight lines
cutting the positive real axis vertically respectively on half-lines
cutting the negative real axis vertically.
A straight line of the first kind is given in polar coordinate by:
__2
r
For
e > O
cos
it follows
p > O
Similarly for
f
on (I 12) given by:
(1.13)
e
there exists exactly one maximum of
e < O
(1.12) also fixed by (1.13).
Lemma
(1.12)
<
by means of elementary analysis that there
exists exactly one minimum of
tan
<
and equation (I .9).
fl
on
These results are in accordance with
Moreover the result (1.13) may be easily
derived also via geometrical arguments by considering the geometry of
the set of curves
For
> e,,e
O
which is fixed by:
r
e
e -8
const.
there occur two turning points, the position of
HOLOMORPHIC FUNCTIONS WITH EXTREMAL PROPERTIES
+ 1
8_
tan 2 ,3
e
647
/e2+82
(I .14)
On the half-lines mentioned above the arguments have to be slightly
modified because of the limits:
re
If(z)
lim
e -8
re eS
If(z)
lim
arg z-
arg z
Our final result reads:
THEOREM 5.
Let
holomorphic and non-constant in
f
of all functions
ff,.
g
further property that
plane.
be the region defined by (I. 10), K
G
Every function
fl
its absolute value
e
such that on any line segment
K
L c G
reaches its maximum (respectively minimum)
L
in one of the endpoints of
f(z)
with the
is meromorphic in the entire complex
,2
f
G
the class
az+b
or
(respectively
is given by
f(z)=(az+b) e+i8
f(z)
e
az+b
In passing it should be mentioned that.
or
with
e > O
f(z)=(az+b)e+i8
Ullrich [4
with e <0)
in his paper
"Betragflchen mit ausgezeichnetem KrHmmungsverhalten" ends up with
the same functions which I have discussed in my paper [5], too.
REFERENCES
1. RUBEL, L.A.
2. ZAAT, J.
Problem 6279, Am. Math. Monthly, Vol.
8__6, No. 9, 1979.
Differentialgeometrie der Betragflchen analytischer
Funktionen, Mitt. d. Math. Sere. d. Univ. GieBen,
3. CARATH.ODORY, C.
4. ULLRICH, E.
Funktionentheorie II, 2. Aufl.
Basel, 1961
Betragflchen mit ausgezeichnetem Kr[immungsverhalten,
Math. Zeitschr.,
5. SCHMERSAU, D.
3_O, 1944.
5__4, 1951.
Geometrische Untersuchungen der Betragflchen
holomorpher Funktionen, Diss., Berlin, 1977.