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Vì abc = 1 và a, b, c >0 nên tồn tại x, y, z > 0 sao cho a = x/y , b = y/z , c = z/x
Thay vào BĐT cần chứng minh ta được
1/(ab + a + 2) + 1/(bc + b + 2) + 1/(ca + c + 2)
= yz/(xy + xz + 2yz) + xz/(yz + xy + 2xz) + xy/(xz + yz + 2xy)
= yz/[(xy + yz) + (xz + yz)] + xz/[(yz + xz) + (xy + xz)] + xy/[(xz + xy) + (yz + xy)]
Mặt khác, theo Cauchy thì:
a + b ≥ 2√(ab)
1/a + 1/b ≥ 2√(1/ab)
Từ đó: (a + b)(1/a + 1/b) ≥ 4.√(ab/ab) = 4
<=> 4/(a + b) ≤ 1/a + 1/b
hay 1/(a + b) ≤ (1/4).(1/a + 1/b)
Sử dụng BĐT trên thì ta có:
1/[(xy + yz) + (xz + yz)] ≤ (1/4).[1/(xy + yz) + 1/(xz + yz)]
Hay
yz/[(xy + yz) + (xz + yz)] ≤ (1/4).[yz/(xy + yz) + yz/(xz + yz)] ---- (1)
Tương tự với 2 bộ còn lại
xz/[(yz + xz) + (xy + xz)] ≤ (1/4).[xz/(yz + xz) + xz/(xy + xz)] ---- (2)
và
xy/[(xz + xy) + (yz + xy)] ≤ (1/4).[xy/(xz + xy) + xy/(yz + xy)] ---- (3)
Cộng Vế (1), (2), (3) và nhóm những đa thức có mẫu chung ta được
Vế trái ≤ (1/4).[ (xy + yz)/(xy + yz) + (yz + xz)/(zy + xz) + (xz + xy)/(xz + xy)] = 3/4
Như vậy bài toán đã được chứng minh
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