Find-all-possible-value-a-a-b-d-b-a-b-c-c-b-c-d-d-a-c-d-when-a-b-c-d-vary-over-positive-reals-

Question Number 199847 by cortano12 last updated on 10/Nov/23

Find all possible value

\frac{a}{a + b + d} + \frac{b}{a + b + c} + \frac{c}{b + c + d} + \frac{d}{a + c + d}

when a, b, c, d vary over positive

reals

Commented by Frix last updated on 10/Nov/23

No problem! I^{'} m also trying \dots

I think 1 ⩽ f is true but f < 2 ?

Commented by witcher3 last updated on 10/Nov/23

i didnt want delate Y re coment

Commented by witcher3 last updated on 10/Nov/23

sorry frix my solution is false

Commented by witcher3 last updated on 12/Nov/23

f < 2 i ? think also

Commented by witcher3 last updated on 12/Nov/23

if a + b + c > d or evry sum of three ⩾ forthe

we get f ⩽ \frac{4}{3}

Commented by Frix last updated on 12/Nov/23

f (a, b, c, d) = \frac{a}{a + b + 0 c + d} + \frac{b}{a + b + c + 0 d} + \frac{c}{0 a + b + c + d} + \frac{d}{a + 0 b + c + d}

f (x, x, y, y) = \frac{2 (x^{2} + 4 x y + y^{2})}{(x + 2 y) (2 x + y)} \Rightarrow 1 ⩽ f ⩽ \frac{4}{3}

f (x, y, x, y) = \frac{4 (x^{2} + x y + y^{2})}{(x + 2 y) (2 x + y)} \Rightarrow \frac{4}{3} ⩽ f ⩽ 2

This is still no proof \dots

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