Prove this inequality: b + c </mrow> <msqrt> a

Wayne Steele

Wayne Steele

Answered question

2022-05-20

Prove this inequality: b + c a + c + a b + a + b c a + b + c + 3
I tried various methods. But, couldn't solve it. It'd be great if anyone can help.

Answer & Explanation

Syllingbs

Syllingbs

Beginner2022-05-21Added 11 answers

By AM-GM
b + c a + c + a b + a + b c 2 b c a + c a b + 2 a b c = 2 a + 2 b + 2 c
For the last equality a b c = 1 was used.
Then again by AM-GM;
1 a + 1 b + 1 c 3 a b c 3 = 3
and again by a well known inequality
1 a + 1 b + 1 c 1 b c + 1 c a + 1 a b = a + b + c
For the last equality a b c = 1 was used. Combine the last two inequalities to get the desired result.
raulgallerjv

raulgallerjv

Beginner2022-05-22Added 2 answers

You already have a proof using AM-GM and Muirhead. For a pure AM-GM proof, note that 1 2 ( a b + b a ) 1 and so on, so half the LHS takes care of the 3 on RHS.
For the rest, note you can cyclically sum the (weighted) AM-GMs:
5 18 ( a b + a c ) + 2 18 ( b a + b c ) + 2 18 ( c a + c b ) a 3 b c 6 = a

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