a-b-c-gt-0-Find-the-min-value-of-cyc-a-b-a-b-c-

Question Number 196049 by CrispyXYZ last updated on 17/Aug/23

a, b, c > 0 . Find the \min value of

\sum_{cyc} \sqrt{\frac{a + b}{a + b + c}} .

Commented by Frix last updated on 17/Aug/23

b = p a \land c = q a

Σ = \frac{\sqrt{p + 1} + \sqrt{p + q} + \sqrt{q + 1}}{\sqrt{p + q + 1}}

Σ = 2; q = \frac{1}{p} \land p \to \infty

Σ = \sqrt{6}; p = q = 1

\Rightarrow

2 < Σ ⩽ \sqrt{6}

Answered by mr W last updated on 17/Aug/23

S = \frac{\sqrt{a + b} + \sqrt{b + c} + \sqrt{c + a}}{\sqrt{a + b + c}}

= \frac{\sqrt{2} (\sqrt{a + b} + \sqrt{b + c} + \sqrt{c + a})}{\sqrt{2 (a + b + c)}}

= \frac{\sqrt{2} (\sqrt{a + b} + \sqrt{b + c} + \sqrt{c + a})}{\sqrt{(a + b) + (b + c) + (c + a)}}

= \frac{\sqrt{2} (x + y + z)}{\sqrt{x^{2} + y^{2} + z^{2}}} (w i t h x = \sqrt{a + b} > 0 \dots)

= \frac{\sqrt{6} (1 \times x + 1 \times y + 1 \times z)}{\sqrt{3} \times \sqrt{x^{2} + y^{2} + z^{2}}}

= \sqrt{6} \cos θ

θ = a n g l e b e t w e e n v e c t o r s (1, 1, 1) a n d (x, y, z)

0 ⩽ θ < \cos^{- 1} \sqrt{\frac{2}{3}}

\Rightarrow 2 < S ⩽ \sqrt{6}

m i n i n u m {d o e s n}^{'} t e x i s t .

Commented by mr W last updated on 18/Aug/23

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