Real solution to a complex equation

Real solution to a complex equation

I have some trouble solving this equation. Let $a\in\mathbb R$, $a>1$. I
want to show that there is a unique solution of
$ze^{a-z} = 1$,
with $|z|<1$ and that this solution is real and positive.
It is easy to prove the existence part of this problem. Indeed, taking log
on both side, we get $\log x + a -x = 0$. However, $\log 1 + a-1>0$ and
$\lim_{x\to 0} \log x + a- x < 0$. Thus, by the mean value theorem, there
is a positive real solution. How can one show that this is the unique
solution in the unit disk? Any help would be appreciated. Thanks!