Extension of the Triangle Inequality

AI Thread Summary
The discussion centers on the extension of the triangle inequality, specifically the assertion that |a + b|^p ≤ 2^p (|a|^p + |b|^p) for p > 1. Participants express uncertainty about the validity of this inequality and share attempts at proving it, including using induction and the binomial theorem. One contributor notes that the inequality can be derived from the convexity of the function x^p for p > 1. The conversation highlights the complexity of proving the inequality and seeks confirmation of its truth. The conclusion remains that further exploration is needed to definitively establish the inequality's validity.
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Pretty much knows the triangle inequality.
\left| a + b \right| \le \left| a \right| + \left| b \right|

I was reading a source which asserted the following extension of the triangle inequality:
\left| a + b \right|^p \le 2^p \left(\left| a \right|^p + \left| b \right|^p\right)

This is bugging me because I can't figure out how to prove or disprove it. It's sensible enough that it might actually be true... but want to know for sure whether it's true.

I tried proving it by induction, but it got really messy...

Is this inequality true? Or, is it wrong? Thanks!
 
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when you tried induction, did you try the binomial theorem?
 
It follows from convexity of xp for p > 1.
<br /> (\frac{1}{2}|a|+\frac{1}{2}|b|)^p\leq \frac{1}{2}|a|^p+\frac{1}{2}|b|^p<br />
<br /> (|a|+|b|)^p\leq 2^{p-1}(|a|^p+|b|^p)<br />
 
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