hopefully my last question for proofing (1 Viewer)

AbstractBlade

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This is hopefully my last question for proofing

if x,y,z real numbers and
i) prove
ii) if
 

CM_Tutor

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Both of these are standard questions following from AMGM for n=3

[edit] - actually, easier still... see below
 
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CM_Tutor

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Or also:



Putting and into (*) yields

Putting and into (*) yields

Putting and into (*) yields

Summing these three inequalities proves the first result
 

CM_Tutor

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Then, noting that

and using this with in the above and the previous result should get the answer needed
 

CM_Tutor

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Or also:



Putting and into (*) yields

Putting and into (*) yields

Putting and into (*) yields

Summing these three inequalities proves the first result
Then, noting that

and using this with in the above and the previous result should get the answer needed
Note that the AM-GM-HM inequalities yield equality only if all terms are the same.

Thus, if I know (say) that then I can start with as the LHS can only be zero if .

So, in both questions here, applying the restriction that makes the inequalities strict.
 

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