Find the regions that dilate and contract _Find the regions of the complex plane that dilate and contract for_ $$f(z)=\frac{z+i}{iz+3}$$ So we know that when $|f'(z)| < 1$ we have contraction and when it's $>1$ we have dilatation. I have computed $f'(z)=\frac{4}{(iz+3)^2}$. Now I need to compute the modulus of $f'(z)$ so I can make it equal to $1$. $$|f'(z)|=\frac{|4|}{|iz+3|^2}$$ $|iz+3|^2=4$ is the equation I have to solve in order to find these regions in the complex plane that neither dilate nor contract, is that right? If so I am quite stuck at computing the modulus. Thanks in advance!

Hint: Your argument is fine. For the last part, write $iz+3=i(x+iy)+3=ix + (3-y)$ and calculate the (square of the) modulus. You should come up with a circle.