Let U ⊆ C be a region containing D(0; 1) and let f be a meromorphic function on U, which has no zeros and no poles on ∂D(0; 1). If f has a zero at 0 and if Re f(z) > 0 for every z ∈ ∂D(0; 1), show that f has a pole in D(0; 1).