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Could I please have help solving this inequality. I have tried but haven't made meaningful progress.

If x,y and z are real numbers prove that: $$\frac 1x + \frac 1y + \frac1z \ge \frac{9}{x+y+z}$$

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  • $\begingroup$ it's called AM-HM, Aritmetic mean - Harmonic Mean sites.math.washington.edu/~dumitriu/Inequalities $\endgroup$
    – Exodd
    Apr 29 '17 at 12:59
  • $\begingroup$ Two naturally imposed assumptions missed, i.e. the conditions that $x,y,z \neq 0$ and that $x+y+z \neq 0$. $\endgroup$
    – Megadeth
    Apr 29 '17 at 12:59
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    $\begingroup$ also $x,y,z>0$. In fact $x=y=1$, $z=-1$ is a counterexample $\endgroup$
    – Exodd
    Apr 29 '17 at 13:01
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Multiply your inequality by $xyz(x+y+z)$, you get : $$y^2z+yz^2+x^2z+xz^2+x^2y+xy^2\ge 6xyz$$ Now redivide by $xyz$ (maybe not a good idea to multiply initially :-) : $$\frac yx+\frac zx+\frac xy+\frac zy+\frac xz+\frac yz \ge 6$$ As it is well know that for all $u>0$, $u+\frac1u\ge2$, you get your inequality for $x,y,z>0$.

Now for the case $x,y,z$ non null reals, I don't know...

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  • $\begingroup$ i do not understand how you get from the last line of working (all the fractions) proves it though? how can we see that it is greater than 9/(x+y+z) ? thanks $\endgroup$
    – John Hon
    Apr 30 '17 at 6:21
  • $\begingroup$ Let $u=\frac yx$, then $u+\frac1u=\frac yx+\frac xy$. But $u+\frac1u-2=\frac{u^2-2u+1}{u}=\frac{(u-1)^2}{u}\ge0$, so $u+\frac1u\ge2$. Apply this three times :-) $\endgroup$ May 2 '17 at 22:02
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Obviously we must assume that the numbers are positive, as otherwise the inequality doesn't always hold. Nevertheless you can prove this inequality by using Cauchy-Schwartz Inequality. We have:

$$\left( \frac 1x + \frac 1y + \frac 1z \right)(x+y+z) \ge \left( \sqrt{\frac xx} + \sqrt{\frac yy} + \sqrt{\frac zz}\right)^2 = 9$$

Now dividing by $(x+y+z)$ we get the wanted inequality.

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The claim is a striaghtforward consequence of Titu's lemma:

$$ \frac{1^2}{x}+\frac{1^2}{y}+\frac{1^2}{z} \geq \frac{(1+1+1)^2}{x+y+z}.$$

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it must be $$x,y,z>0$$ and this inequality is equivalent to $$\left(\frac{x}{y}+\frac{y}{x}\right)+\left(\frac{y}{z}+\frac{z}{y}\right)+\left(\frac{x}{z}+\frac{z}{x}\right)\geq 6$$ which is true. or we use $AM-GM$ twice: $$x+y+z\ge 3\sqrt[3]{xyz}$$ $$\frac{1}{x}+\frac{1}{y}+\frac{1}{z}\geq 3\sqrt[3]{\frac{1}{xyz}}$$ and multiplying both we get the statement

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