Given an $n$-element complex vector $\mathbf{x}=[x_1,\ldots,x_n]\in\mathbb{C}^n$, where $\|\mathbf{x}\|_2^2=\sum_{i=1}^n|x_i|^2=1$, I am wondering if anything can be said about the product $A\bar{A}$ of the sum of its elements $A=\sum_{i=1}^nx_i$ and the complex conjugate $\bar{A}$.
The Cauchy-Schwartz inequality yields a trivial upper bound $A\bar{A}\leq n$, however, I am wondering if this can somehow be sharpened. Any ideas?
No, the bound is sharp. Consider the unit vector
$$\mathbf{x} = \left(\frac{1}{\sqrt{2}}, \frac{1}{\sqrt{2}}\right) \in \mathbb{C}^2$$
Since the elements are all real, $A = \overline{A} = \frac{2}{\sqrt{2}} = \sqrt{2}$.
So $A\overline{A} = 2 = n$. I think the real case is always the one to look at in situations like this, since the "interference" provided by complex multiplication is negative.