# Upper bound for modulus of a complex integral

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 $$1\leq\lvert z\rvert\leq 2$$

• Use $|A| = |\bar A|$. Feb 23, 2023 at 0:30
• @aschepler do you mean something like $\left|\overline{\frac{1}{\overline z + i}}\right|=\left|\frac{1}{z- i}\right|$...but I don't see any boundary again Feb 23, 2023 at 0:40

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.