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MathematicalPhysicist

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|z|<=1, Let f be analytic function s.t f(0)=1, |f(z)|>3/2 in |z|=1, show that in |z|<1 f has at least 1 root, and and one fixed point, i.e s.t that f(z0)=z0.

now here what I did, let's define g(z)=f(z)-z, and we first need to show that the number of roots of f(z) is at least as z (i.e 1), by roche theorem, if |f(z)|<|z-f(z)| for every |z|<1 then the number of roots of f(z) are the same as those of (z-f(z))+f(z), now the pesky task to show this.

|z-f(z)|<=|z|+|f(z)|<1+|f(z)|

here is where I'm stuck, don't know how to follow.