Proposed solution of problem J611

Let a, b, c be positive real numbers. Prove that 3(ab + bc + ca) ≤ a\sqrt{b^{2} + 8c^{2}} + b\sqrt{c^{2} + 8a^{2}} + c\sqrt{a^{2} + 8b^{2}} ≤ (a + b + c)^{2}