I have the following optimization problem: \begin{align}\max_x&\quad\log_2(1+|a+bx|^2+cx^2)\\\text{s.t.}&\quad0\le x\le1\\&\quad(1-x^2)\ge\text{constant}\end{align} where $a$ and $b$ are two complex numbers, and $c$ is a real positive number.
Does anyone have any idea regarding how this objective function can be converted into a convex one?
Is it true to say that the optimal value of $x$ is equal to $\sqrt{1-\text{constant}}$?