Prove the following inequality I think it can be done using

Prove the following inequality: (I think it can be done using AM-GM)

Solution

Using the Arithmetic Mean-Geometric Mean Inequality on each factor of the LHS we get

((a²b + b²c + c²)/3) ((ab² + bc² + ca^2)/3) ? (a³b³c³)^(1/3)(a³b³c³)^(1/3)= a²b²c² .

Multiplying by 9 we get the desired inequality.

Another solution consists of using the Cauchy-Schwarz inequality:

(a²b + b²c + c²a)(ab² + bc² + ca² ) = ((a ? b)² +b ? c)²+ (c ? a)²) ( ? b c)² + (? c a)² + (? a b)² ? (abc + abc + abc)² = 9a² b² c².