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Math Help - Holomorphic function

  1. #1
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    Question Holomorphic function

    What are all entire holomorphic function for which is Re f(z)<=Im f(z)
    for every z from C?

    Hint: use Picard theorem
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  2. #2
    Super Member Rebesques's Avatar
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    Since it is entire, we don't need to add "holomorphic".


    Consider the function {\rm e}^{f}. This is also entire and

    |{\rm e}^{f}|=|{\rm e}^{Ref+{\rm i}Imf}|={\rm e}^{Ref}\leq {\rm e}^{Imf}, so the function {\rm e}^{-Imf}{\rm e}^{f}={\rm e}^{f-Imf} is bounded. So it must be constant; That is, f-Imf must be constant, or (Ref-Imf)+{\rm i}Imf is constant. From this we get Ref, Imf are constant.
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