Unformatted text preview: r ) such that the estimate  φ ( z )  < Me  z  /r holds for all z ∈ C . 4. (20 pts) Let S denote the strip S = n z ∈ C :π 2 < Im z < π 2 o . Show that the entire function e e z is bounded on the boundary of S ⊂ C , but it is not bounded on S . 5. (20 pts) Suppose that f ( z ) is continuous on { z :  z  ≤ 1 }  { 1 } and analytic on { z :  z  < 1 } . Prove that if f ( z ) is bounded on { z :  z  < 1 } , then  f ( z )  ≤ sup  ζ  =1 ,ζ 6 =1  f ( ζ )  for each z in { z :  z  < 1 } . Is this inequality valid if f is not assumed to be bounded?...
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 Spring '09
 Math, pts, Logarithm, unit disc D1

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