math185-hw5sol - MATH 185 COMPLEX ANALYSIS FALL 2007\/08 PROBLEM SET 5 SOLUTIONS Notations D(0 1 ={z C | |z| < 1 D(0 1 ={z C | |z| = 1 N ={1 2 3 f g

math185-hw5sol - MATH 185 COMPLEX ANALYSIS FALL 2007/08...

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MATH 185: COMPLEX ANALYSIS FALL 2007/08 PROBLEM SET 5 SOLUTIONS Notations: D (0 , 1) = { z C | | z | < 1 } ; ∂D (0 , 1) = { z C | | z | = 1 } ; N = { 1 , 2 , 3 , . . . } ; f g denotes the composition of f and g and is defined by f g ( z ) = f ( g ( z )). 1. Let f : C C be an entire function. Let a R be an arbitrary constant. (a) Show that if Re f ( z ) a for all z C , then f is constant. (b) Show that if Re f ( z ) a for all z C , then f is constant. (c) Show that if [Re f ( z )] 2 [Im f ( z )] 2 for all z C , then f is constant. (d) Show that if [Re f ( z )] 2 [Im f ( z )] 2 for all z C , then f is constant. (e) Suppose h is another entire functions and suppose there exists an a R , a > 0, such that Re f ( z ) a Re h ( z ) for all z C . Show that there exist α, β C such that f ( z ) = αh ( z ) + β for all z C . [Hint: if f and g are both entire, then so are f g and g f ; find an appropriate g so that you may apply Liouville’s theorem.] Solution. Note that e x is a monotone increasing function on R . For (a), we choose g ( z ) = e z and note that | e f ( z ) | = | e Re f ( z ) e i Im f ( z ) | = e Re f ( z ) e a . For (b), we choose g ( z ) = e - z and note that | e - f ( z ) | = | e - Re f ( z ) e - i Im f ( z ) | = e - Re f ( z ) e - a . For (c), we choose g ( z ) = e z 2 and note that | e f ( z ) 2 | = | e [Re f ( z )] 2 - [Im f ( z )] 2 e 2 i Re f ( z ) Im f ( z ) | = e [Re f ( z )] 2 - [Im f ( z )] 2 e 0 = 1.
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