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Thread: inequality 2

  1. #1
    Member great_math's Avatar
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    inequality 2

    If a,b and c are non-negative reals such that $\displaystyle
    \frac{1}{1+a}+\frac{1}{1+b}+\frac{1}{1+c} \ge 2$, prove that


    $\displaystyle abc \le \frac{1}{8}$


    thank you.
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  2. #2
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    Quote Originally Posted by great_math View Post
    If a,b and c are non-negative reals such that $\displaystyle
    \frac{1}{1+a}+\frac{1}{1+b}+\frac{1}{1+c} \ge 2.$ prove that $\displaystyle abc \le \frac{1}{8}.$
    let $\displaystyle a=\tan^2x, \ b = \tan^2 y, \ c=\tan^2 z, \ \ 0 \leq x,y,z < \frac{\pi}{2}.$ so your inequality becomes: $\displaystyle \cos^2x + \cos^2y+\cos^2z \ge 2.$ hence:

    $\displaystyle \boxed{1} \ \ \cos^2x \ge 2 - \cos^2y - \cos^2z = \sin^2y + \sin^2z \ge 2\sin y \sin z.$ similarly:

    $\displaystyle \boxed{2} \ \ \cos^2 y \ge 2\sin x \sin z,$

    $\displaystyle \boxed{3} \ \ \cos^2 z \ge 2 \sin x \sin y.$

    multiplying $\displaystyle \boxed{1} \ , \boxed{2} \ , \boxed{3} \ ,$ together will give us: $\displaystyle \cos^2x \cos^2y \cos^2z \ge 8 \sin^2 x \sin^2y \sin^2 z.$ thus: $\displaystyle \tan^2x \tan^2y \tan^2z \le \frac{1}{8}. \ \ \ \Box$
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  3. #3
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    for those who are interested, the above inequality can be easily generalized, i.e. if $\displaystyle a_1, \cdots , a_n, \ n \geq 2,$ are non-negative numbers and $\displaystyle \frac{1}{1+a_1} + \frac{1}{1+a_2}+\cdots + \frac{1}{1+a_n} \geq n-1,$ then:

    $\displaystyle a_1a_2 \cdots a_n \leq \frac{1}{(n-1)^n}.$ the proof is the same and is based on AM-GM: let $\displaystyle a_j=\tan^2x_j, \ 0 \leq x_j < \frac{\pi}{2}, \ 1 \leq j \leq n.$ then the inequality becomes: $\displaystyle \cos^2x_1 + \cos^2x_2 + \cdots + \cos^2x_n \geq n-1.$ so:

    $\displaystyle \cos^2 x_j \geq n-1 - \sum_{i \neq j} \cos^2x_i=\sum_{i \neq j} \sin^2 x_i \geq (n-1) \left(\prod_{i \neq j} \sin^2x_i \right)^{\frac{1}{n-1}}, \ \ 1 \leq j \leq n.$ multiplying these inequalities together gives us: $\displaystyle \cos^2 x_1 \cos^2 x_2 \cdots \cos^2x_n \geq (n-1)^n \sin^2x_1 \sin^2x_2 \cdots \sin^2x_n.$

    hence: $\displaystyle a_1 a_2 \cdots a_n = \tan^2x_1 \tan^2x_2 \cdots \tan^2x_n \leq \frac{1}{(n-1)^n}. \ \ \ \Box$
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