<s>[INST] If $x,y,z\gt 0$ and $xyz=1$ Then minimum value of $\frac{x^2}{y+z}+\frac{y^2}{z+x}+\frac{z^2}{x+y}$ 
If $x,y,z\gt 0$ and $xyz=1$ Then find the minimum value of $\displaystyle \frac{x^2}{y+z}+\frac{y^2}{z+x}+\frac{z^2}{x+y}$

$\bf{My\; Try::}$Using Titu's Lemma $$\frac{x^2}{y+z}+\frac{y^2}{z+x}+\frac{z^2}{x+y}\ge \frac{(x+y+z)^2}{2(x+y+z)} = \frac{x+y+z}{2}\ge 3\frac{\sqrt[3]{xyz}}{2} = \frac{3}{2}$$
and equality holds when $$x=y=z=1$$
My question is how can we solve it without the above lemma, like using Jensen's Inequality or other inequality.
Please explain me.
Thanks
 [/INST] This answer shows how Nesbitt's inequality can used with other proof techniques to prove the given inequality.

Nesbitt's inequality
$$\frac{x}{y+z}+\frac{y}{z+x}+\frac{z}{x+y} \geq \frac{3}{2}\tag{$*$}$$


$(1a)$ Observe that
$$\sum_{cyc}\left(\frac{x^2}{y+z}\right)+(x+y+z)=(x+y+z)\left(\frac{x}{y+z}+\frac{y}{z+x}+\frac{z}{x+y}\right)\tag{A}$$
Using $(*)$ and (A), we get
$$\sum_{cyc}\left(\frac{x^2}{y+z}\right) \geq \frac{(x+y+z)}{2}\tag{B}$$
Now, using AM-GM inequality, we get
$$\frac{(x+y+z)}{2} \geq  \frac{3\sqrt[3]{xyz}}{2}=\frac{3}{2}\tag{C}$$
Using (B) and (C), we get the desired result.
$(1b)$ Observe that
$$\sum_{cyc}\left(\frac{x^2}{y+z}\right)=(x+y+z)\left[\left(\sum_{cyc}\frac{x}{y+z}\right)-1\right]\tag{D}$$
Using $(*)$ and (C) in (D), we get
$$\sum_{cyc}\left(\frac{x^2}{y+z}\right) \geq 3 \cdot \left(\frac{3}{2}-1\right)=\frac{3}{2}$$

$(2)$
Without loss of generality, let $x \geq y \geq z$
Therefore, $$\frac{x}{y+z} \geq \frac{y}{x+z} \geq \frac{z}{x+y}$$
Now, using the Chebyshev inequality for these two increasing sequences, we get
$$3 \sum_{cyc}\left(\frac{x^2}{y+z}\right) \geq (x+y+z)\sum_{cyc}\left(\frac{x}{y+z}\right)\tag{E}$$
Using $(*)$ and (C) in (E), we get the required result.

$(3)$ Let $f(x)=x^2$
Now, we can write
$$\sum_{cyc}\left(\frac{x^2}{y+z}\right)=\sum_{cyc}\left(\frac{f(x)}{y+z}\right)$$
Note, that $f''(x)=2 > 0$, so the function is convex.
Now, using the Weighted-Jensen inequality, we get
$$\sum_{cyc}\left(\frac{f(x)}{y+z}\right) \geq 3f(M)\tag{F}$$
where using $(*)$, we get
$$3M=\sum_{cyc}\left(\frac{x}{y+z}\right) \geq \frac{3}{2} \Longleftrightarrow M \geq \frac{1}{2}\tag{G}$$
Using (G) in (F), we get the desired result.
Note: Notice that this method is independent of the constraint $xyz=1$

Finally, note that equality is indeed attained when $\boxed{x=y=z=1}$
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