#### Vol. 15, No. 1, 1965

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### Robert Franklin Jolly

Vol. 15 (1965), No. 1, 159–169
##### Abstract

The author investigates those subsets $M$ of the complex plane with the group property that $M$ is closed with respect to complex multiplication. In particular if $M$ is closed, bounded and has for its boundary a curve given in polar form by $\rho \left(\theta \right)=r\left(\theta \right)exp\left(i\theta \right)$ where $r$ is a positive continuous function with period $2\pi$, then $r$ is characterized by these requirements, together with the additional condition that $r$ be submultiplicative. If $f\left(x\right)=-logr\left(x\right)$, the corresponding conditions on $f$ are: $f$ is a continuous nonnegative subadditive function with period $2\pi$.

Some relations between the roots (zeros) and periods of subadditive functions are discussed and in particular, it is shown that: if $f$ is a continuous subadditive function not identically zero, with period 1 and with a root $c$ (i.e., $f\left(c\right)=0$), then $c$ is a rational number $m∕n$ (in lowest terms), $f\left(0\right)=0$ and $f$ has period $1∕n$.

For each positive number $c$ and function $f$ on the set of all numbers, a type of polygonal approximation $P\left(c,f\right)$ is defined such that if $f$ is continuous, $limP\left(c,f\right)=f$ uniformly over every bounded number set as $c\to 0$. If $f$ is subadditive, $P\left(c,f\right)$ is subadditive. The subadditive $P\left(c,f\right)$ 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 $2\pi$. For each positive integer $n$, the collection ${K}_{n}$ of all functions $P\left(1∕n,f\right)$ for all continuous subadditive functions $f$ with period 1, is shown to have a finite basis. In fact, ${K}_{n}$ 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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