The author investigates those subsets
of the complex plane with the group property that
is closed with respect to complex multiplication. In particular if
is
closed, bounded and has for its boundary a curve given in polar form by
where
is a positive continuous
function with period
,
then
is characterized by these requirements, together with the additional condition that
be submultiplicative.
If
, the corresponding
conditions on
are:
is a continuous nonnegative subadditive function with period
.
Some relations between the roots (zeros) and periods of subadditive
functions are discussed and in particular, it is shown that: if
is a
continuous subadditive function not identically zero, with period 1 and with a root
(i.e.,
), then
is a rational
number
(in
lowest terms),
and
has
period
.
For each positive number
and function
on the set of all numbers, a type of polygonal approximation
is defined such that
if
is continuous,
uniformly over every
bounded number set as
.
If
is subadditive,
is subadditive.
The subadditive
are characterized in terms of their slopes. Since a change of scale does not affect the
subadditive property, the author studies functions with period 1 rather than those with period
. For each positive
integer
, the collection
of all functions
for all continuous
subadditive functions
with period 1, is shown to have a finite basis. In fact,
forms
a function cone with finitely many extremal elements (the basis). While an explicit
representation is not given, the proof shows how these extremal elements may be
constructed.
Several examples are given to illustrate some pathological cases. The methods of
this paper may easily be applied to the solution of certain other functional
inequalities with corresponding restrictions.
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