Let-a-b-c-0-and-a-b-c-1-Prove-that-1-a-2-1-1-b-2-1-1-c-2-1-5-2-

Question Number 142577 by loveineq last updated on 02/Jun/21

Let a, b, c ⩾ 0 and a + b + c = 1 . Prove that

\frac{1}{a^{2} + 1} + \frac{1}{b^{2} + 1} + \frac{1}{c^{2} + 1} ⩾ \frac{5}{2}

Answered by ajfour last updated on 02/Jun/21

l . h . s = \frac{1 / a}{a + \frac{1}{a}} + \frac{1 / b}{b + \frac{1}{b}} + \frac{1 / c}{c + \frac{1}{c}}

= 3 - (\frac{a}{a + \frac{1}{a}} + \frac{b}{b + \frac{1}{b}} + \frac{c}{c + \frac{1}{c}})

⩾ 3 - \frac{1}{2} (a + b + c) ⩾ \frac{5}{2}

a s a + \frac{1}{a} ⩾ 2 i f a > 0 .

Commented by loveineq last updated on 02/Jun/21

But this can not explain that equality holds iff (a, b, c) = (0, 0, 1)

Answered by 1549442205PVT last updated on 03/Jun/21

i) if (a, b, c) \in A = {(0, 0, 1), (0, 1, 0), (1, 0, 0)}

we have equality occurs

if one of three numbers equal to zero

for example c = 0 then we need prove

for a + b = 1 : \frac{1}{a^{2} + 1} + \frac{1}{b^{2} + 1} ⩾ \frac{3}{2} \Leftrightarrow

2 (a^{2} + b^{2} + 2) ⩾ 3 (a^{2} b^{2} + a^{2} + b^{2} + 1) \Leftrightarrow

{3 a}^{2} b^{2} + a^{2} + b^{2} ⩽ 1 = {(a + b)}^{2} = a^{2} + b^{2} + 2 ab

\Leftrightarrow {3 a}^{2} b^{2} - 2 ab ⩽ 0 \Leftrightarrow ab (3 ab - 2) ⩽ 0 that

is true because 3 ab ⩽ 4 ab ⩽ {(a + b)}^{2} = 1 < 2

ii) Consider a, b, c - \neq 0 . Then

\frac{1}{a^{2} + 1} + \frac{1}{b^{2} + 1} + \frac{1}{c^{2} + 1} ⩾ \frac{5}{2} \Leftrightarrow

- (\frac{1}{a^{2} + 1} + \frac{1}{b^{2} + 1} + \frac{1}{c^{2} + 1}) ⩽ - \frac{5}{2} \Leftrightarrow

3 - (\frac{1}{a^{2} + 1} + \frac{1}{b^{2} + 1} + \frac{1}{c^{2} + 1}) ⩽ 3 - \frac{5}{2} \Leftrightarrow

\frac{a^{2}}{a^{2} + 1} + \frac{b^{2}}{b^{2} + 1} + \frac{c^{2}}{c^{2} + 1} ⩽ \frac{1}{2} (2)

L . H . S (2) ⩽ \frac{a^{2}}{2 a} + \frac{b^{2}}{2 b} + \frac{c^{2}}{2 c} = \frac{a + b + c}{2} = \frac{1}{2}

\Leftrightarrow \frac{1}{a^{2} + 1} + \frac{1}{b^{2} + 1} + \frac{1}{c^{2} + 1} ⩾ \frac{5}{2} (q . e . d)

The equality occurs if and if (a, b, c) \in A