I'm trying to absorb a perplexing proof of Young's inequality I've found. Young's inequality states that if [itex]A,B \geq 0[/itex] and [itex]0 \leq \theta \leq 1[/itex], then [itex]A^\theta B^{1-\theta} \leq \theta A + (1-\theta)B[/itex].(adsbygoogle = window.adsbygoogle || []).push({});

The first step they take is the following: We can assume [itex]B \neq 0[/itex]. (I get that.) But then they say we can just replace [itex]A[/itex] with [itex]AB[/itex], in which case it suffices to prove [itex]A^\theta \leq \theta A - (1-\theta)[/itex].

I understand why the replacement of A with AB in Young's inequality gets you to the new inequality in only A. But I don't see why you get to do that. Why is proving [itex]A^\theta \leq \theta A - (1-\theta)[/itex] logically equivalent to proving [itex]A^\theta B^{1-\theta} \leq \theta A + (1-\theta)B[/itex]?

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# Strange proof of Young's inequality

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