Question

Find two different linear mappings which map imz>3 onto rez>3

Answer 1

$\varphi \left( z \right) =-iz:\\z=x+iy,y>3\\\varphi \left( z \right) =-i\left( x+iy \right) =y-ix\Rightarrow \mathrm{Re}\varphi \left( z \right) >3\\\varphi \left( z \right) =-iz+i:\\z=x+iy,y>3\\\varphi \left( z \right) =-i\left( x+iy \right) +i=y-i\left( x-1 \right) \Rightarrow \mathrm{Re}\varphi \left( z \right) >3$