I solved this question with a Arithmetic mean ≥ Harmonic mean inequality. If there is another solution, please show me.

Question

$x,y,z>0$. Prove that:$$\frac{2xy}{x+y} + \frac{2yz}{y+z} + \frac{2xz}{x+z} ≤ x+y+z $$ My solution:$$\frac{x+y}{2}≥\frac{2xy}{x+y}$$ $$\frac{y+z}{2}≥\frac{2yz}{y+z}$$ $$\frac{x+z}{2}≥\frac{2xz}{x+z}$$ Summing up these inequalities $$\frac{2xy}{x+y} + \frac{2yz}{y+z} + \frac{2xz}{x+z} ≤ x+y+z $$I solved this question with a Arithmetic mean ≥ Harmonic mean inequality. If there is another solution, please show me.

Answer 1

This can even be shown without the help of any theorems.

Note that $4 x y = (x+y)^2 - (x-y)^2$, likewise for the other terms. Substituting this into the question gives

$$ \frac{(x+y)^2 - (x-y)^2}{2 (x+y)} + \frac{(y+z)^2 - (y-z)^2}{2 (y+z)} + \frac{(x+z)^2 - (x-z)^2}{2 (x+z)} \\ ≤ \frac12 ((x+y) + (y+z) + (x+z)) $$ or $$ - \frac{(x-y)^2}{2 (x+y)} - \frac{(y-z)^2}{2 (y+z)} - \frac{ (x-z)^2}{2 (x+z)} ≤ 0 $$ which is obviously true. It also shows that equality will only be obtained for $x=y=z$. $\qquad \Box$

Answer 2

Well I think using means inequalities $HM\le GM\le AM$ is pretty much the way to go, and it all boils down in the end to $(x-y)^2\ge 0$ if you decide to prove it directly.

Indeed the $AM\ge HM$ inequality is just

$AM-HM=\dfrac{x+y}2-\dfrac 2{\frac 1x+\frac 1y}=\dfrac{x+y}2-\dfrac{2xy}{x+y}=\dfrac{(x+y)^2-4xy}{x+y}=\dfrac{(x-y)^2}{x+y}\ge 0$

Even using $AM\ge GM$ inequality leads to the result:

$\dfrac{2xy}{x+y}+\dfrac{2yz}{y+z}+\dfrac{2xz}{x+z}\le \dfrac{2(\frac{x+y}2)^2}{x+y}+\dfrac{2(\frac{y+z}2)^2}{y+z}+\dfrac{2(\frac{x+z}2)^2}{x+z}=\dfrac{x+y}2+\dfrac{y+z}2+\dfrac{x+z}2$