Peak function

Remarks

Example: on the unit ball $B \subseteq \C^n$, let $e_1 = (1, 0, \dots, 0)$ and $f(z) = (1 + z_1)/2$. Then $f(e_1) = 1$, and $\abs{f(z)} \le (1 + \abs{z_1})/2 \le 1$ on $\overline{B}$, with equality only if $z_1 = 1$, which forces $z = e_1$. Non-example: on the bidisc $\Delta^2$, the point $p = (1, 0)$ is not a peak point. Suppose $f$ satisfies the three clauses. The function $w \mapsto f(1, w)$ is holomorphic on the open unit disc, as a locally uniform limit of the functions $w \mapsto f(r, w)$ for $r \to 1$. It has modulus at most $1$ and reaches the modulus $1$ at the interior point $w = 0$, so it is the constant $1$ by the maximum modulus principle, against clause 3. Every boundary point of a bounded strongly pseudoconvex domain with smooth boundary is a peak point.