Let-a-b-0-Prove-that-1-4-2-a-2-b-1-a-1-b-4-a-b-4-a-b-

Question Number 140959 by loveineq last updated on 14/May/21

Let a, b ⩾ 0 . Prove that

\frac{1}{4} \cdot \frac{(2 + a) (2 + b)}{(1 + a) (1 + b)} ⩾ \frac{4 - a - b}{4 + a + b}

Answered by MJS_new last updated on 14/May/21

this is easy

a, b ⩾ 0 \Rightarrow 1 + a > 0 \land 1 + b > 0 \land 4 + a + b > 0

\Rightarrow

(2 + a) (2 + b) (4 + a + b) ⩾ 4 (4 - a - b) (1 + a) (1 + b)

\Rightarrow

5 a^{2} b + 6 a^{2} + 5 {a b}^{2} + 6 b^{2} ⩾ 0

which is true since a, b ⩾ 0

Facebook Tweet Pin