I want to ask a question regarding the zeros of the complex function:
Show that for any complex number $\lambda$ and for $n\ge1$, the function $(z-1)^n e^z-\lambda$ has $n$ zeros satisfying $|z-1|<1$ and no other zeros in the right plane.
I think the question has some flaws. In my opinion, if $|z-1|<1$, then $|(z-1)^n e^z|\le e^2$ (since $Re(z)<2$ for all $|z-1|<1$), which means $(z-1)^n e^z$ is bounded on the region $|z-1|<1$. Is it possible for $|\lambda|$ large enough, $(z-1)^n e^z-\lambda$ has no zeros in $|z-1|<1$? If the question is true, how to prove it?