nguoivn wrote:
Given $ a, b, c > 0$. Prove that: $ (1 + a + b + c)(1 + ab + bc + ca) \ge 4\sqrt {2}\sqrt {(a + bc)(b + ca)(c + ab)}$
Note that
$ (1 + a + b + c)(1 + ab + bc + ca) = (1 + a)(1 + b)(1 + c) + (a + b)(b + c)(c + a) \ge 2\sqrt {(1 + a)(1 + b)(1 + c)(a + b)(b + c)(c + a)}.$
Therefore, it suffices to prove that
$ 8(a + bc)(b + ca)(c + ab) \le (1 + a)(1 + b)(1 + c)(a + b)(b + c)(c + a).$
By the AM-GM Inequality, we have
$ 2\sqrt {(a + bc)(b + ca)} \le a + bc + b + ca = (c+ 1)(a + b),$
$ 2\sqrt {(b + ca)(c + ab)} \le (a + 1)(b + c),$
$ 2\sqrt {(c + ab)(a + bc)} \le (b + 1)(c + a).$
Multiplying these inequalities, we get the result.