If u is a non-constant real-valued harmonic function in a region , then u cannot attain...

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If u is a non-constant real-valued harmonic function in a region , then u cannot attain a maximum (or a minimum) in 22. Suppose that is a region with compact closure . If u is harmonic in and continuous in , then sup |u(2) sup |u(z)|. ZEN ΞΕΩ-Ω [Hint: To prove the first part, assume that u attains a local maximum at zo. Let f be holomorphic near zo with u = Re(f), and show that f is not open. The second part follows directly from the first.] If u is a non-constant real-valued harmonic function in a region , then u cannot attain a maximum (or a minimum) in 22. Suppose that is a region with compact closure . If u is harmonic in and continuous in , then sup |u(2) sup |u(z)|. ZEN ΞΕΩ-Ω [Hint: To prove the first part, assume that u attains a local maximum at zo. Let f be holomorphic near zo with u = Re(f), and show that f is not open. The second part follows directly from the first.]

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