Let lambda be real and lambda > 1, Show that the equation
ze^lambda−z = 1
has exactly one solution in the disc |z| = 1, which is real and positive.
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As stated, your problem seems to be trivial:
z e^lambda-z=1 <=> z=1/(e^lambda-1)
which is real and the positive. Since lambda>1, e^lambda-1 is bigger than 1, so 1/(e^lambda-1) is less than 1 and the solution is unique and inside the disc of radius one. In fact, I see no reason to bring complex variables into the mix.