I am trying to show that $ f(z) = ze^{\lambda - z} - 1$, $\lambda > 1$ has a real root inside the disk. I have already showed, using Rouche's Theorem, that there is exactly one root inside the disk. But I am not sure how to show that this root is real.
I have considered the real-valued analog: $xe^{\lambda - x} - 1$ and tried appealing to calculus. I was able to show that this function is increasing on the interval $(-\infty, 1)$. So If I could find two values in this interval such that $f(x) < 0$ and $f(x) > 0$, I would be done by appealing to the intermediate value theorem.
Any tips?
Note that I have already looked at Show $z e^{\lambda-z}-1$ has only one real root in the unit disk. but that post does not discuss how to show the root is real.