MHB Fractional linear transformation--conformal mapping

[Dustinsfl]

Find necessary and sufficient conditions on the real numbers $a$, $b$, $c$, and $d$ such that the fractional linear transformation
$$
f(z) = \frac{az + b}{cz + d}
$$
maps the upper half plane to itself.

I just need some guidance on starting this one since I am not sure on how to begin.

 

[Opalg]

Find necessary and sufficient conditions on the real numbers $a$, $b$, $c$, and $d$ such that the fractional linear transformation
$$
f(z) = \frac{az + b}{cz + d}
$$
maps the upper half plane to itself.

I just need some guidance on starting this one since I am not sure on how to begin.

Begin by multiplying top and bottom by the complex conjugate of the denominator: $$f(z) = \dfrac{az + b}{cz + d} = \dfrac{(az + b)(c\overline{z} + d)}{(cz + d)(c\overline{z} + d)}.$$

The denominator in that last fraction is real, so you need to find the imaginary part of the numerator and investigate what makes it positive whenever $z$ has positive imaginary part.

 

[Dustinsfl]

Begin by multiplying top and bottom by the complex conjugate of the denominator: $$f(z) = \dfrac{az + b}{cz + d} = \dfrac{(az + b)(c\overline{z} + d)}{(cz + d)(c\overline{z} + d)}.$$

The denominator in that last fraction is real, so you need to find the imaginary part of the numerator and investigate what makes it positive whenever $z$ has positive imaginary part.

So $ad > bc$

 

[Opalg]

So $ad > bc$

(Yes)