I am struggling with this problem
Let $z_1,z_2\in\mathbb{C}$ prove that $$\left|1+z_1\right| +\left|1+z_2 \right| + \left|1+z_1z_2\right|\geq 2$$
I know a similar question has been solved: if $|z_i|=1$ prove $|z_1+1|+|z_2+1|+|z_1z_2+1|\ge 2$ .
However, I don't have the $|z_1|=|z_2|=1$ condition, so I am not sure if the question is wrong or the last condition is implicit. This is the work I have done:
\begin{align*}
|1+z_1|+|1+z_2|+|1+z_1 z_2| &= |1+z_1| + |1+z_2|+|-(1+z_1z_2)|\\
&\geq |1+z_1| + \left|(1+z_2)-(1+z_1z_2)\right|\\
&= |1+z_1|+ |z_2-z_1z_2|\\
&\geq |(1+z_1)+(z_2-z_1z_2)|
\end{align*}
I'll appreciate your help. I am not sure what I am missing here.
Best Answer
If $|z_2|\geq 1$, then \begin{align*} |1+z_1|+|1+z_2|+|1+z_1z_2| &\geq |1+z_1|+|(1+z_2)-(1+z_1z_2)|\\ &=|1+z_1|+|z_2||1-z_1|\\ &\geq |1+z_1|+|1-z_1|\geq 2, \end{align*} where the last step uses the triangle inequality. If $|z_2|\leq 1$, then \begin{align*} |1+z_1|+|1+z_2|+|1+z_1z_2| &\geq|z_2||1+z_1|+|1+z_2|+|1+z_1z_2|\\ &\geq|(1+z_2)-(z_2+z_1z_2)|+|1+z_1z_2|\\ &=|1-z_1z_2|+|1+z_1z_2|\geq 2. \end{align*}