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Question Number 211605 by MrGaster last updated on 14/Sep/24

Let a_{1}, a_{2}, \dots a_{n}

Is n real numbers .

All fall in the interval (- 1, 1)

________________________

(1) Prove that :

\prod_{1 \leq i, j \leq n} \frac{1 + a_{i} a_{j}}{1 - a_{i} a_{j}} \geq 1

(2) Determine the necessary

andsufficient conditions forl

equaity in the inequality .

Commented by mnjuly1970 last updated on 15/Sep/24

m r h a r d m a t h ?

Answered by a.lgnaoui last updated on 14/Sep/24

E = 1 \Rightarrow 2 Π a_{i} a_{j} = 0

So; if one of a_{i} or a_{j} = 0 then Π \frac{1 + a_{i} a_{j}}{1 - a_{i} a_{j}} = 1

Commented by mr W last updated on 15/Sep/24

a_{i} o r a_{j} i s n o t a p a r t i c u l a r n u m b e r!

f o r e x a m p l e y o u c a n n o t s a y :

s u c h t h a t \underset{i = 1}{\sum^{n}} a_{i} = 0,

\Rightarrow a_{i} = 0 .

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