Which of the inequality holds?

Question

$f(x,y) = _{\theta,\phi}^{{sup}}$ $ ||e^{i\theta}x - e^{i\phi} y||_{2}$ ,where $x,y \in \mathbb{C^2}$ and $\theta,\phi \in \mathbb{R}$

Which of the following holds ?

1. $f(x,y) \leq ||x||^2 + ||y||^2 - 2Re \mid \langle x,y \rangle \mid$
2. $f(x,y) \leq ||x||^2 + ||y||^2 + 2Re \mid \langle x,y\rangle \mid$
3. $f(x,y) \leq ||x||^2 + ||y||^2 - 2 \mid \langle x,y\rangle \mid$
4. $f(x,y) \geq ||x||^2 + ||y||^2 - 2Re \mid \langle x,y\rangle \mid$

My claim :

Take n=2. let $x= (x_1 ,x_2), y = (y_1,y_2)$

then $||e^{i\theta}x - e^{i\phi}||_{2} = \mid \mid (e^{i\theta}x_1-e^{i\phi} y_1,e^{i\theta}x_1-e^{i\phi} y_2) \mid \mid _{2}.$

Is that any two norms in $\mathbb{C^2}$ are also equivalent ?

Now we have to find the partial derivative w.r.t $\theta$ and $ \phi$ of both the co-ordinates and equate to zero and find the critical points to find the maximum.

Is that a correct way to proceed ?