At university I have got a problem set with lots of inequalities. Unfortunately there are no explanations given how to do them. In Highschool we only did very easy inequalities. Therefore I am looking for a resource for inequalities. Especially for more difficult inequalities like $$1 \leq z \overline {z} \leq 4 , |\Im(z)|<\Re (z),$$ where $z$ is a complex number.
I would be glad at any recommendations.
You need to translate this into something you can work with.
So in your example let $z=x+iy$. Then $\bar{z}=x-iy$. So you have $$1 \le (x+iy)(x-iy) \le 4 \text{ and } |y|\lt x$$ and you will find $(x+iy)(x-iy)= x^2+y^2$.