If a function is holomorphic on a bounded domain and continuous on its closure, the maximum of its absolute value on the closed domain is attained on the boundary, never in the interior, unless the function is constant. It is a basic structural fact about holomorphic functions used throughout complex analysis.
Facts
StatementIf a function is holomorphic on a bounded, connected open region of the complex plane and continuous up to the region's boundary, then the maximum of its modulus over the closed region is attained on the boundary, never at an interior point, unless the function is constant throughout the region. 1 Connections
Sources
1. Maximum modulus principle (Wikipedia)
Wikimedia FoundationPhysical interpretation sectionQuote, Physical interpretation section
A physical interpretation of this principle comes from the heat equation.
Maximum Modulus Principle (MathWorld)
Theorem statementQuote, Theorem statement
An analytic function attaining an interior maximum of its absolute value is constant.
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