关于复数模的降序性质不等式正确性的技术咨询

Hey there! Let’s clear up this confusion right away—your proposed inequality chain has a key mix-up with the $||z_2| - |z_1||$ term, so let’s break it down step by step.

First, the Critical Error in Your Original Chain #

The part $|z_1 + z_2| \leq ||z_2| - |z_1||$ is not true in general. This is where your confusion is coming from—you’ve placed the $||z_2| - |z_1||$ term in the wrong position relative to $|z_1 + z_2|$.

The Correct Inequality Chain #

Let’s start with the core triangle inequalities for complex numbers, then build out the full valid chain:

• The fundamental reverse triangle inequality tells us: $||z_1| - |z_2|| \leq |z_1 + z_2|$ (this gives a lower bound for $|z_1 + z_2|$)
• The standard triangle inequality gives the upper bound: $|z_1 + z_2| \leq |z_1| + |z_2|$
• Plus, since any real number is less than or equal to its absolute value, $|z_2| - |z_1| \leq ||z_2| - |z_1||$ (and since $||z_2| - |z_1|| = ||z_1| - |z_2||$, we can use either form)

Putting it all together, the valid full chain is:
$$|z_2| - |z_1| \leq ||z_1| - |z_2|| \leq |z_1 + z_2| \leq |z_1| + |z_2|$$

A Quick Example to Prove the Error #

Let’s pick simple complex numbers to test your original chain: take $z_1 = 1$ (a real number, which is a valid complex number) and $z_2 = 2$.

• $|z_2| - |z_1| = 2 - 1 = 1$
• $||z_2| - |z_1|| = |2 - 1| = 1$
• $|z_1 + z_2| = |3| = 3$
• $|z_1| + |z_2| = 1 + 2 = 3$

Your original chain claims $3 \leq 1$, which is obviously false. Swap the positions of $||z_2| - |z_1||$ and $|z_1 + z_2|$, and everything checks out: $1 \leq 1 \leq 3 \leq 3$.

Final Note #

The $||z_2| - |z_1||$ term is totally valid—it’s just a lower bound for $|z_1 + z_2|$, not an upper bound. Your intuition that most of the pieces are correct was right; you just had that one term in the wrong spot!

内容的提问来源于stack exchange，提问作者moony