<s>[INST] Inequality $\frac{x^3}{x^2+y^2}+\frac{y^3}{y^2+z^2}+\frac{z^3}{z^2+x^2} \geqslant \frac{x+y+z}{2}$ Help to prove this Inequality:
If  x,y,z are postive real numbers then:
$\dfrac{x^3}{x^2+y^2}+\dfrac{y^3}{y^2+z^2}+\dfrac{z^3}{z^2+x^2} \geqslant \dfrac{x+y+z}{2}$
I tied to use analytic method with convex function but no result:
Since $f(x)=\frac{1}{x}$ is a convex function, by Jensen we obtain:
$$\frac{1}{x+y+z}\sum_{cyc}\frac{x^3}{x^2+y^2}=\sum_{cyc}\left(\frac{x}{x+y+z}\cdot\frac{1}{\frac{x^2+y^2}{x^2}}\right)\geq$$
$$\geq\frac{1}{\sum\limits_{cyc}\left(\frac{x}{x+y+z}\cdot\frac{x^2+y^2}{x^2}\right)}=\frac{x+y+z}{\sum\limits_{cyc}\left(x+\frac{y^2}{x}\right)}.$$ 
Thus, it's enough to prove that
$$\frac{x+y+z}{\sum\limits_{cyc}\left(x+\frac{y^2}{x}\right)}\geq\frac{1}{2}$$ or
$$x+y+z\geq\sum_{cyc}\frac{y^2}{x},$$ which is wrong.
thanks
 [/INST] Hint: We have $$ \frac {x^3}{x^2 + y^2}=x-\frac{xy^2}{x^2+y^2}\ge x-\frac{y}{2}$$ because by AM-GM $x^2+y^2\geq 2xy$ so that $$\frac{xy}{x^2+y^2}\le\frac12$$
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