Further Analysis | Part IB, 2002

(a) Let ff and gg be two analytic functions on a domain DD and let γD\gamma \subset D be a simple closed curve homotopic in DD to a point. If g(z)<f(z)|g(z)|<|f(z)| for every zz in γ\gamma, prove that γ\gamma encloses the same number of zeros of ff as of f+gf+g.

(b) Let gg be an analytic function on the disk z<1+ϵ|z|<1+\epsilon, for some ϵ>0\epsilon>0. Suppose that gg maps the closed unit disk into the open unit disk (both centred at 0 ). Prove that gg has exactly one fixed point in the open unit disk.

(c) Prove that, if a<1|a|<1, then

zm(za1aˉz)naz^{m}\left(\frac{z-a}{1-\bar{a} z}\right)^{n}-a

has m+nm+n zeros in z<1|z|<1.

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