# Conjecture $\frac{a}{a^r+b^r}+\frac{b}{b^r+c^r}+\frac{c}{c^r+a^r}\geq \frac{a}{a^r+c^r}+\frac{c}{c^r+b^r}+\frac{b}{b^r+a^r}$

following this kind of inequality One of my old inequality (very sharp) I propose this because I don't see it on the forum :

Let $$a,b,c>0$$ and $$a+b+c=1$$ with $$r\in(\frac{1}{2},1)$$ and $$a\geq b \geq c$$ then we have : $$\frac{a}{a^r+b^r}+\frac{b}{b^r+c^r}+\frac{c}{c^r+a^r}\geq \frac{a}{a^r+c^r}+\frac{c}{c^r+b^r}+\frac{b}{b^r+a^r}$$

First of all it's a conjecture where I don't find counter-examples . Secondly when $$r\in(0,\frac{1}{2})$$ the inequality is reversed .I use Pari-gp for that .Furthermore (if it's true) I think it's really not new so I add the tag reference request.We have an equality case when $$r=0.5$$ whenever $$a,b,c>0$$.

So if you have idea to prove it or disprove it...

Thanks a lot .

If $$\prod\limits_{cyc}(a-b)=0$$, so it's obvious.
Let $$a>b>c.$$
Thus, we need to prove that: $$\sum_{cyc}\left(\frac{a}{a^r+b^r}-\frac{a}{a^r+c^r}\right)\geq0$$ or $$\sum_{cyc}\frac{a(c^r-b^r)}{(a^r+b^r)(a^r+c^r)}\geq0$$ or $$\sum_{cyc}a(c^r-b^r)(b^r+c^r)\geq0$$ or $$\sum_{cyc}a(c^{2r}-b^{2r})\geq0$$ or $$a^{2r}(b-c)+c^{2r}(a-b)-b^{2r}(a-b+b-c)\geq0$$ or $$\frac{a^{2r}-b^{2r}}{a-b}\geq\frac{b^{2r}-c^{2r}}{b-c}.$$ Now, use the Lagrange's mean value theorem for $$f(x)=x^{2r}$$ and that $$f'$$ increases.