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I found the following exercise in a problem book (with no solutions):

Given $a,b,c>0$ such that $abc=1$ prove that $$\frac{1}{1+a}+\frac{1}{1+b}+\frac{1}{1+c}\leq\frac{a+b+c+3}{4}$$

I tried AM-GM for the fraction on the LHS but got stuck from there.

Albus Dumbledore
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John WK
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3 Answers3

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let $p=a+b+c,q=ab+bc+ca,r=abc$

Now it is easy to prove the following as $r=1$

  • $pq\ge 9r=9$
  • $p^2\ge 3q$

Now we have to prove $$4(3+2p+q)\le (p+3)(2+p+q)$$ or $$p^2+pq\ge 6+3p+q$$ $$p^2+9\ge 6+3p+q$$ using $q\le p^2/3$ $$\frac{2p^2}{3}-3p+3\ge 0$$ which is true as $p\ge 3$

Albus Dumbledore
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Let $f(x) = \frac{1}{1 + x} - \frac{x+1}{4} + \frac{1}{2}\ln x$. We have $f'(x) = \frac{(1-x)(x^2+x+2)}{4x(1+x)^2}$. Thus, $f'(x) > 0$ on $(0, 1)$, and $f'(x) < 0$ on $(1, \infty)$. Thus, $f(x)$ is strictly increasing on $(0, 1)$, and strictly decreasing on $(1, \infty)$. Also $f(1) = 0$. Thus, $f(x) \le 0$ on $(0, \infty)$.

Thus, $f(a) + f(b) + f(c) \le 0$ which results in $\frac{1}{1+a} + \frac{1}{1+b} + \frac{1}{1+c} \le \frac{a+b+c+3}{4}$ (using $\ln (abc) = 0$).

We are done.

River Li
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  • This is nice and straight forward much better than mine(+1) – Albus Dumbledore Dec 24 '20 at 14:42
  • @AlbusDumbledore Thanks. I think both are nice. – River Li Dec 24 '20 at 15:22
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    What made you think of $ + 1/2 \ln x$? – Calvin Lin Dec 25 '20 at 03:54
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    @CalvinLin We want to find the form of $f(x) = \frac{1}{1 + x} - \frac{x+1}{4} + m\ln x$ such that $f(x) \ge 0$ for all $x > 0$. Clearly $f(1) = 0$. By letting $f'(1) = 0$, we obtain $m = 1/2$. – River Li Dec 25 '20 at 08:07
  • lol, my point is that "of the form $ + m \ln x$" wasn't obvious to me. Was it experience that caused you to think of it? – Calvin Lin Dec 25 '20 at 14:12
  • @CalvinLin :-), it is a trick in the inequality of the form of $f(a) + f(b) + f(c) \ge 0$ subject to $abc = 1, a, b, c > 0$. The inequality is written as $\sum_{\mathrm{cyc}} (f(a) + m \ln a) \ge 0$ for any $m$. So if we can choose appropriate $m$ such that $f(x) + m\ln x\ge 0$ for all $x>0$, then we are done. But sometimes it does not hold for all $x > 0$ but for $x > x_0$ with some $x_0 > 0$. For example, https://math.stackexchange.com/questions/3953789/a2-b2-c2-6-ge-3a-b-c-abc-1/3954835#3954835 – River Li Dec 25 '20 at 15:29
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By Schur's inequality and AM-GM $$\begin{align*}x^3+y^3+z^3+3xyz\ge xy(x+y)+yz(y+z)+zx(x+z)&\\ \ge 2( {(xy)}^{3/2}+{(yz)}^{3/2}+{(zx)}^{3/2})\end{align*}\tag S$$

Now as $$\frac{1}{1+x}\le \frac{1}{2\sqrt{x}}$$ we have to prove $$a+b+c+3\sqrt[3]{abc}\ge 4\left(\frac{1}{2\sqrt{a}}+\frac{1}{2\sqrt{b}}+\frac{1}{2\sqrt{c}}\right)$$ or it suffices to prove $$a+b+c+3\sqrt[3]{abc}\ge 2\sqrt{ab}+2\sqrt{bc}+2\sqrt{ca}$$ which is just a direct consequence of inequality (S) and so we are done!

Note here $\sqrt[3]{abc}=\sqrt{abc}=1$ is used to normalize inequality

Albus Dumbledore
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