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did i do this one correctly?
Well i would use something like:
$\displaystyle \left| \int_{\delta R} \frac{u(z)}{(z-z_0)^2} dz \right| \leq CR\int_{0}^{2\pi} \left| \frac{1}{(Re^{it}-z_0)^2} dt \right| \leq \frac{2\pi CR}{||R|-|z_0||^2} $
the last part goes to 0 as R grows
What do u think?
I think there is something wrong with your attempt, your bound must not depend on t.
Well i thouht using triangle inequality and then just squaring it up:
$\displaystyle ||Re^{it}|-|z_0||\leq | Re^{it}- z_0| \Rightarrow \frac{1}{| Re^{it}- z_0| } \leq \frac{1}{||Re^{it}|-|z_0||}=\frac{1}{||R|-|z_0||}$
where obviously $\displaystyle |Re^{it}|=|R|=R$
What do you think?