Let $\gamma$ be a close piecewise smooth contour consisting of the three straight lines from $1$ to $2$, from $2$ to $1+i$ and from $1+i$ to $1$.
I have to show that $$\left|\int_{\gamma}\frac{1}{\overline z +i}\,dz\right|\leq 2+\sqrt{2}$$
I know that I can use the estimation lemma to bound $$\left|\frac{1}{\overline z +i}\right|\leq M$$ and I think that must be $M=1$, because $\lvert\gamma\rvert=2+\sqrt2$, but I can't find a suitable upper bound. Any hint?
Maybe $$\left|\frac{1}{\overline z +i}\right|= \frac{1}{\lvert\overline z +i\rvert}\leq\frac{1}{\lvert\overline z\rvert - \lvert i\rvert}=\frac{1}{\lvert z\rvert - 1}$$ but this does not give me any further information since $1\leq\lvert z\rvert\leq 2$
Find the points where $\left|\frac1{\overline z+i}\right|\le1$: $$|\overline z+i|\le1\iff\overline z\text{ is in unit circle centred on }-i$$ $$\iff z\text{ is in unit circle centred on }i$$ Since all parts of the curve lie on or outside the unit circle centred on $i$, $\left|\frac1{\overline z+i}\right|\le1$ on all of $\gamma$ and we get the desired result.