maximum principle

Maximum principle

Let $f:U\to \Reals$ (where $U\subseteq \Reals^d$ ) be a harmonic function. Then $f$ attains its extremal values on any compact $K\subseteq U$ on the boundary $\partial K$ of $K$ . If $f$ attains an extremal value anywhere in the interior of $K$ , then it is constant.

Maximal modulus principle

Let $f:U\to\Complex$ (where $U\subseteq \Complex$ ) be a holomorphic function. Then $|f|$ attains its maximal value on any compact $K\subseteq U$ on the boundary $\partial K$ of $K$ . If $|f|$ attains its maximal value anywhere on the interior of $K$ , then it is constant.