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(Az^4 + Bz^3 + Cz^2 + Dz + E) on some annulus k1 < |z-z0| < k2

So, I've tried applying the theorem by the version |f - g| < |f|, and I've chosen g as different terms in the polynomial, starting from z^4, and working down until the inequity is satisfied.

Basically what I can't figure out, is how to use the triangle inequity to evaluate the function on the contour. Also, doing the triangle-inequity-approximation, if I find it to be 3 zeros in the bigger circle, how can I know that the inequity approximation didn't ignore one of the zeros, and that there actually is 4 zeroes there? I mean, the |f - g| < |f| inequity may be satisfied even if the triangle equity says it aint? Or not?