Inequality

Try this, it's really appealing....

a,b,c>0

Prove the following inequality:

ab(a+b)+bc(b+c)+ca(c+a)\ge \sum_{cyc}ab\sqrt{\frac{a}{b}\left(b+c \right)\left(c+a \right)}

1 Answers

consider (b+c)a and (a+c)b

on applying A.M G.M,

(b+c)a+ (a+c)b â‰¥ 2√(b+c)(a+c)ab

from this we can say,

(a+b)c + (b+c)a+ (a+c)b â‰¥ √(b+c)(a+c)ab ...( equality will not hold)

=> ab(a+b) + bc(b+c) + ac(a+c) â‰¥ ab√cab(b+c)(a+c)

...
=

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