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7 . Suppose f is continuous on D and holomorphic on D- (a Show that |fl attains its maximum 0n the boundary €D. That iS, show that there is w € €D...

Question

7 . Suppose f is continuous on D and holomorphic on D- (a Show that |fl attains its maximum 0n the boundary €D. That iS, show that there is w € €D such that |f(2)l < |f (w)l for every 2 € D. (Hint: The assertion is clear if f is constant_ Assume f is not constant Since |f | is continuous on D which is a closed and bounded set) , by the extreme value theorem, |fl has a maximum 0n D. Can this maximum be in D? See the maximum modulus principle (Theorem 4.5, page 92 of Stein-Shakarchi).)

7 . Suppose f is continuous on D and holomorphic on D- (a Show that |fl attains its maximum 0n the boundary €D. That iS, show that there is w € €D such that |f(2)l < |f (w)l for every 2 € D. (Hint: The assertion is clear if f is constant_ Assume f is not constant Since |f | is continuous on D which is a closed and bounded set) , by the extreme value theorem, |fl has a maximum 0n D. Can this maximum be in D? See the maximum modulus principle (Theorem 4.5, page 92 of Stein-Shakarchi).)



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Suppose $f$ is a continuous odd function such that $f(a)$ is a relative maximum. What can be said about $f(-a) ?$


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