Maximum Modulus Principle

Complex Analysis · Axiom Academy

LESSON The Maximum Modulus Principle An analytic function can never have a peak hiding inside — its largest size always lives on the boundary. 1. The Maximum Lives on the Boundary Let f be analytic on a bounded domain D and continuous on its closure . Then |f(z)| attains its maximum on the boundary — never strictly inside (unless f is constant). Watch the probe roam the interior: its size readout keeps falling short of the bright boundary ring. The interior max equals the boundary max An interior local max forces f to be constant 2. Why? The Mean‑Value Property For an analytic f , Cauchy's formula on a small circle |z-z_0|=r inside D says the value at the center is the average of the values around the circle. Sample the circle as it spins: the running average of f closes in on the center value f(z_0) — exactly. From there the rest is the triangle inequality. f(z_0) is the mean of f over the circle — the average of analytic values is the center value. The size of an average is at most the average of the sizes: . Then on the circle. With the line above, that forces |f| equal to |f(z_0)| all around. Every small circle is pinned the same way, so |f| is constant on connected D — hence f is constant. For f(z)=e^ z centered at z_0=0 we have f(0)=1 , and the mean of e^ z around any circle is exactly 1 — yet the mean of |e^ z | is strictly larger than 1 . That strict gap is precisely why the center cannot be a maximum. 3. A Concrete Disk — and What Follows

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