Prove that b^2c^2+c^2a^2+a^2b^2gtabc(a+b...

Prove that `b^2c^2+c^2a^2+a^2b^2gtabc(a+b+c)`, where `a,b,c gt 0` .

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Using A.M. `gt` G.M., we have
`(b^(2) c^(2) + c^(2) a^(2))/(2) gt sqrt((b^(2) c^(2))(c^(2) a^(2)))`
`implies b^(2) c^(2) + c^(2) a^(2) gt 2 abc^(2)`
similarly,
`c^(2) a^(2) + a^(2) b^(2) gt 2 a^(2) bc`
`a^(2) b^(2) + b^(2) c^(2) gt 2 ab^(2)c`
Adding (1),(2 and (3), we get
`2b^(2) c^(2) + 2 c^(2) a^(2) + 2 a^(2) b^(2) gt 2 abc^(2) + 2 bca^(2) + 2cab^(2)`
`implies b^(2) c^(2) + c^(2) a^(2) + a^(2) b^(2) gt abc (a + b + c)`

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