Let $X=\{X_k\}^{N-1}_{k=0} $ be the Discrete Fourier Transform of $\{ x\} = \{ x_n \}^{N-1}_{n=0} $
what is the relationship between $X$ and Discrete Fourier Transform of $\overline{x}$
Definition Discrete Fourier
Let $a$ discrete function of $N$ samples $\forall n = 0,1, \dots , N-1, f[n].$
The DFT of $f$ denotes $\widehat{f} [x]$ is defined by $$ \widehat{f}[x] = \sum^{N-1}_{n=0}f[x] e^{-2i \pi \frac{nk}{N}}$$
Inversion Discrete Fourier
Given discrete fourier transfomr $\widehat{f}[x]$ of a discrete function $f[n]$ the inverse of $DFT$ is given by
$$f[n]=\frac{1}{N} \sum^{N-1}_{k=0} \widehat{f}[k] e^{2 i \pi \frac{nk}{N}} $$
Scratchwork
$$X = \sum^{N-1}_{n=0} x_n e^{e i \pi \frac{nk}{N}}$$
now finding the relationship
$$ \sum^{N-1}_{n=0} \overline{x_n} e^{-2i \pi \frac{nk}{N}} =\sum^{N-1}_{n=0} \overline{x_n e^{2i \pi \frac{nk}{N}}} = \overline {\sum^{N-1}_{n=0}x_n e^{2i \pi \frac{nk}{N}}} $$
and we have that
$$x= \frac{1}{N} \sum^{n-1}_{N=0} X_n e^{2i \pi \frac{nk}{N}} $$
Not sure where to go from there besides double summation
The DFT of an $f:{\mathbb Z}_N\to{\mathbb C}$ is defined as $$\widehat f(k):=\sum_{n=0}^{N-1}f(n)e^{-2\pi i n k/N}\ .$$ Consider now the function $\bar f$ defined by $\bar f(n):=\overline{f(n)}$. It is a simple calculation involving complex conjugates only to show that $$\widehat{\bar f}(k)=\overline{\widehat f(-k)}\qquad(k\in{\mathbb Z}_N)\ .$$