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Math Help - Metric

  1. #1
    Senior Member slevvio's Avatar
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    Metric

    Hello there, I have this question about metrics, but the first part asks me to prove this horrible identity which I am having difficulty with. I would appreciate if somebody could point me in the right direction before I cover all my writing supplies with algebra.

    Let a,b,c be non-negative real numbers such that a \le b+c. Show that \frac{b}{1+b} + \frac{c}{1+c} - \frac{a}{1+a} \ge 0 .

    Thanks again.
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  2. #2
    MHF Contributor

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    Quote Originally Posted by slevvio View Post
    Hello there, I have this question about metrics, but the first part asks me to prove this horrible identity which I am having difficulty with. I would appreciate if somebody could point me in the right direction before I cover all my writing supplies with algebra.

    Let a,b,c be non-negative real numbers such that a \le b+c. Show that \frac{b}{1+b} + \frac{c}{1+c} - \frac{a}{1+a} \ge 0 .

    Thanks again.
    f:x\mapsto\frac{x}{1+x} is increasing on \mathbb{R}_+, thus f(a)\leq f(b+c), and the question becomes: f(b+c)\leq f(b)+f(c)? i.e. \frac{b+c}{1+b+c}\leq \frac{b+c+2bc}{1+b+c+bc}? If \frac{u}{v}\leq \frac{x}{y}, then \frac{u}{v}\leq\frac{u+x}{v+y}. With suitable u,v,x,y, this concludes.
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  3. #3
    Senior Member Shanks's Avatar
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    Notice that the function Laurent has defined is increasing on [0, infinity).
    thus f(b)+f(c)>=f(b+c)>=f(a) since b+c>=a.
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  4. #4
    Senior Member slevvio's Avatar
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    Thanks very much but how do i show that ?
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  5. #5
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    Lemma:
    \begin{gathered}<br />
  0 \leqslant x \leqslant y \Rightarrow \quad x+xy \leqslant y + xy \hfill \\<br />
   \Rightarrow \quad x(1 + y) \leqslant y(1 + x) \Rightarrow \quad \frac{x}<br />
{{1 + x}} \leqslant \frac{y}{{1 + y}} \hfill \\ <br />
\end{gathered}

    Using that lemma, we know that a\le b+c so

    \frac{a}{1 +a } \leqslant \frac{b+c}{1+b+c} \leqslant \frac{b}{1+b}+ \frac{c}{1+c}
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  6. #6
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    nb: The fact that x\mapsto \frac{x}{1+x}=1-\frac{1}{1+x} is increasing is almost obvious with second writing (composition of decreasing maps => increasing)
    Quote Originally Posted by slevvio View Post
    Thanks very much but how do i show that ?
    Just multiply by v(v+y) (i.e. get the denominators away) and simplify. But this step is somewhat direct (btw it corresponds to the last inequality in Plato's post, that comes with no further argument), my trick was just a way to avoid any computation by reducing to an elementary inequality.
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