A,b,c are three distinct positive numbers. Prove that $b a^{2} + a b^{2} + b c^{2} + c b^{2} + c a^{2} + a c^{2} < 2 (a^{3} + b^{3} + c^{3})$ $ba^2+ab^2+bc^2+cb^2+ca^2+ac^2<2(a^3+b^3+c^3)$ ?

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A,b,c are three distinct positive numbers. Prove that $b a^{2} + a b^{2} + b c^{2} + c b^{2} + c a^{2} + a c^{2} < 2 (a^{3} + b^{3} + c^{3})$ $ba^2+ab^2+bc^2+cb^2+ca^2+ac^2<2(a^3+b^3+c^3)$ ?

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Answer 1 · 2017-11-01T14:49:16

See below.

Explanation:

Considering

$f (a, b, c) = 2 (a^{3} + b^{3} + c^{3}) - (b a^{2} + a b^{2} + b c^{2} + c b^{2} + c a^{2} + a c^{2})$ $f(a,b,c) =2(a^3+b^3+c^3)-( ba^2+ab^2+bc^2+cb^2+ca^2+ac^2)$

as we can verify $f (a, b, c)$ $f(a,b,c)$ is symmetric regarding the order of it's arguments. We should prove that

$f (a, b, c) \geq 0$ $f(a,b,c) ge 0$ for $a > 0, b > 0, c > 0$ $a > 0,b > 0,c > 0$

Due to symmetry, $min f(a,b,c) = min f(x_0,x_0,x_0) = 0$

Supposing now $epsilon$ such that $x_0+epsilon > 0$

$f(x_0,x_0,x_0+epsilon) = 2epsilon^2(2x_0+epsilon) > 0$ so

$f(a,b,c)$ has minimum $0$ value for $a=b=c$ and is positive for any other choice of $a > 0, b > 0, c > 0$ so this demonstrates that

$2(a^3+b^3+c^3) ge ( ba^2+ab^2+bc^2+cb^2+ca^2+ac^2)$

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A,b,c are three distinct positive numbers. Prove that $b a^{2} + a b^{2} + b c^{2} + c b^{2} + c a^{2} + a c^{2} < 2 (a^{3} + b^{3} + c^{3})$ $ba^2+ab^2+bc^2+cb^2+ca^2+ac^2<2(a^3+b^3+c^3)$ ?

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Explanation:

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A,b,c are three distinct positive numbers. Prove that ba^2+ab^2+bc^2+cb^2+ca^2+ac^2<2(a^3+b^3+c^3)ba2+ab2+bc2+cb2+ca2+ac2<2(a3+b3+c3)ba^2+ab^2+bc^2+cb^2+ca^2+ac^2<2(a^3+b^3+c^3) ?

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Explanation:

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A,b,c are three distinct positive numbers. Prove that $b a^{2} + a b^{2} + b c^{2} + c b^{2} + c a^{2} + a c^{2} < 2 (a^{3} + b^{3} + c^{3})$ $ba^2+ab^2+bc^2+cb^2+ca^2+ac^2<2(a^3+b^3+c^3)$ ?