Proving Boundedness of Entire Functions with Harmonic Components

[physicsjock]

Hey,

I'm trying to prove that uv=>0 is bounded so I can state that an entire function is constant when f = u + iv, when f is entire.

I have worked out the rest but I'm struggling to prove that its bounded,

Can you say u=>0, v=>0 then u + v => 0, and that bounded from below?

 

[Eynstone]

Hey,

I'm trying to prove that uv=>0 is bounded so I can state that an entire function is constant when f = u + iv, when f is entire.

I'm not sure what you are trying to prove - is it given that u,v are harmonic & uv>=0 ?

 

[physicsjock]

They gave

f = u + iv is an entire function,

that means u and v are harmonic right?

and it asks to show f is constant if uv=>0 everywhere

I think i have done that part just using louivilles theorem, but i realized i was just assuming u and v were bounded