It's entirely possible these inequalities have been solved before but I can't easily find them by searching. This is Exercise 10 from page 83 of Analysis I by Amann and Escher.
Exercise:
My attempt:
I've made almost no progress with these inequalities. For $(a)$, I was looking for a slick way to do it in one go, but couldn't find anything, so I thought maybe I have to consider 8 cases separately. The case when the arguments for all the absolute values are non-negative seems simple. Specifically, in that case we have
\begin{align*} \frac{ \lvert a + b \rvert }{1 + \lvert a + b \rvert} &= \frac{ a + b }{1 + a + b }\\ &= \frac{a}{1 + a + b} + \frac{b}{1 + a + b}\\ &\leq \frac{a}{1 + a} + \frac{b}{1 + b} \end{align*}
but I'm looking at some basic properties for ordered fields and I'm not even sure I can justify this result.
The next case I considered was $\lvert a + b \rvert = -(a + b), \lvert a \rvert = a, \lvert b \rvert = b$, but I don't think this is possible.
The third case I considered was $\lvert a + b \rvert = -(a + b), \lvert a \rvert = -a, \lvert b \rvert = b$ but I don't know how to deal with this.
For $(b)$, I am lost. For $(c)$, I see that this is equivalent to $\lvert ab^{-1} + a^{-1}b \rvert \geq 2$. I was trying to use (ix) in the properties of fields from the text (picture below) but I don't know how to use it. I can't make the assumption that $a > b > 0$. I could assume that $\lvert a \rvert \geq \lvert b \rvert > 0$, and if I'm right, that would give me $\lvert a \rvert \lvert b \rvert ^{-1} \geq 1$, but I'm not even sure this approach would work, due to the triangle inequality issue.
I appreciate any help.
Properties of fields:
