Applying Maximum Modulus

Complex Analysis · Axiom Academy

EXAMPLE Applying Maximum Modulus Find where the modulus of an analytic function is largest by working a problem entirely on the boundary. Find the maximum of |z^2 + 1| over the closed square with vertices at (the square , ). Nicely done. You used the Maximum Modulus Principle to turn a search over a whole 2D region into a search along its 1D boundary — then pinned the answer with a little real-variable calculus. The principle: if f is analytic on a closed bounded region and non-constant, |f| attains its maximum on the boundary , never strictly inside. The strategy: identify the boundary, parameterize each piece (corners, edges, arcs), maximize |f| on each piece, then compare to find the global max. This result: , attained at all four corners — bigger than the value 2 at the edge midpoints and the value 1 at the center z=0 . It generalizes: the same idea gives at z=2 (push as large as the disk allows), and — applied to f(z)/z — it is the engine behind the Schwarz Lemma. Analyticity is rigid: it forces the "biggest" behavior of |f| out to the edge. That single fact powers bounds, uniqueness theorems, and Liouville's theorem across complex analysis.

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