inequality with three variables and condition If a , b and c positive real numbers

kixEffinsoj

kixEffinsoj

Answered question

2022-06-22

inequality with three variables and condition
If a, b and c positive real numbers such that a + b + c = 1, prove
b 2 a + b 2 + c 2 b + c 2 + a 2 c + a 2 3 4 . I have tried several methods to solve this,but can't get any result. Any idea?

Answer & Explanation

laure6237ma

laure6237ma

Beginner2022-06-23Added 27 answers

By C-S and Vasc we obtain c y c a 2 a 2 + c = c y c a 4 a 4 + a 2 c ( a + b + c ) ( a 2 + b 2 + c 2 ) 2 c y c ( a 4 + a 3 c + a 2 b 2 + a 2 b c )
( a 2 + b 2 + c 2 ) 2 c y c ( a 4 + a 2 b 2 + a 2 b c ) + ( a 2 + b 2 + c 2 ) 2 3 = 3 ( a 2 + b 2 + c 2 ) 2 c y c ( 4 a 4 + 5 a 2 b 2 + 3 a 2 b c )
Id est, it remains to prove that 4 ( a 2 + b 2 + c 2 ) 2 c y c ( 4 a 4 + 5 a 2 b 2 + 3 a 2 b c ), which is
c y c c 2 ( a b ) 2 0
Done!

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