*Problem:*

If it holds that , then find the value of

.

*1st solution:*

From the Power-Mean Inequality we have that

.

So, multiplying by we have that

.

Taking now in hand the left hand side, we know that

.

So, we know that . Doing the manipulations on both sides we get that

, hence ,* Q.E.D.*

*2nd solution (An idea by Vo Quoc Ba Can):*

The inequality is symmetric on . So, we only need to find the maximum of those two constants for the values of which

, that is . S

o, plugging on the above inequality the value

we get the desired maximum result, *Q.E.D.*

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You have a great website! 😀 I’ll definitely be visiting again!

Thank you Smitha for your nice words. I’m trying my best for the inequalities!

I think you need something more in the above solution. Ok you proved that , but now you have to prove that in case the inequality holds.

Yes silouan you are right. But can you please give me a hint of how to prove this inequality for because i can’t think of something to show that. Thanks in advance!