Re: [中學] 求證不等式

[Sfly (topos)]

※ 引述《ej001 (  )》之銘言：
: a, b are all positive real number
: please prove
: {1 + a } b+1   { a }b
: {------}     > {---}
: {1 + b }       { b }
: thank you!

First of all, '>' is wrong, as two sides are equal when a=b.
So we are proving the case of '>=' instead.

       b(1+a)  (b+1)
<=>  (--------)      >= b/a
       a(1+b)

<=>  ( 1 + (b-a)/(ab+a))^(b+1)  >= b/a.

note that (b-a)/(ab+a) > -a/(ab+a) > -1.

Thus, the result follows from the Bernoulli inequality.

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