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Showing the inequality $|\alpha + \beta|^p \leq 2^{p-1}(|\alpha|^p + |\beta|^p)$

We can use Young's inequality to show that $(a+b)^2 \leq 2a^2 + 2b^2$.

Does a similar result hold for the n-th power as well? That is, do we have $(a + b)^n \leq c_1 a^n + c_2 b^n$ ?

If so, what are the values for $c_1$ and $c_2$?

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marked as duplicate by Marvis, Pedro Tamaroff, Willie Wong May 15 '12 at 15:25

This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.

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up vote 3 down vote accepted

Symmetry demands that $c_1 = c_2$. $a=b$ gives you the desired values of $c_1,c_2$ namely, $c_1 = c_2 = 2^{n-1}$. You might want to throw in the $\lvert \cdot \rvert$ into the equation i.e. $$\lvert a+ b \rvert^{n} \leq 2^{n-1} \lvert a \rvert^n + 2^{n-1} \lvert b \rvert^n$$ A very similar question and the proof for the above can be found here.

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Thanks a lot! I wasn't able to find the original question you've linked to. –  mkolar May 15 '12 at 1:42
    
@mkolar math.stackexchange.com/questions/143173/… –  user17762 May 15 '12 at 1:43

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