According to some references, following inequality constraint in $w \in \mathbb C^2$ is not convex.
\begin{equation}
\frac{|c^H w|^2}{\|Aw\|_2^2+\|Bw\|_2^2} \geq r_k, \qquad (1)
\end{equation}
where $A$ and $B$ are $2 \times 2$ matrices and $c$ is a $2$-vector
However, after some phase rotation, the absolute value of the complex number in the numerator can be written as
\begin{equation} |c^H w| = \mathbb{R}\{e^{-j \theta} c^H w \}, \text{ where } \theta = \arg\{c^H w\}. \qquad (2) \end{equation} Then, after some manipulation, eqn (1) can be expressed again in the form of \begin{equation} \|\Phi w + d\| \leq \mathbb{R}\{e^{-j \theta} c^H w \} \qquad (3) \end{equation} which becomes a convex constraint as a SOCP (second order cone programming) form.
Here, I am very wondering why eqn (1) is a non-convex constraint, and eqn (3) is a convex constraint (How can we prove their non-convexity and convexity of (1) and (3), respectively ?).