How is symmetry in an inequality determined? I was reading a book about inequalities, in that I found that a/b+b/c+c/a>1 is a symmetric inequality in a,b,c. But if I change the order from (a,b,c) to (a,c,b), I am not getting the same inequality which is the basic definition of symmetric inequality. What am I doing wrong?

miniliv4 2022-10-11 Answered
How is symmetry in an inequality determined?
I was reading a book about inequalities, in that I found that
(1) a b + b c + c a > 1
is a symmetric inequality in a,b,c. But if I change the order from (a,b,c) to (a,c,b), I am not getting the same inequality which is the basic definition of symmetric inequality. What am I doing wrong?
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Answers (1)

Zara Pratt
Answered 2022-10-12 Author has 12 answers
It's not symmetric inequality because the permutation ( a , b , c ) ( a , c , b ) gives another inequality:
a c + b a + c b > 1.
By the way, the inequality
a + b + c > a b c
is symmetric.
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