I think it's not new however I think it's interesting :
Let $a,b,c>0$ and $x>0$ then the function :
$$f(x)=\frac{a^x}{b^x+c^x}+\frac{b^x}{a^x+c^x}+\frac{c^x}{b^x+a^x}$$
$f(x)$ is increasing
My first try :
The derivative of $f(x)$ is :
$$f'(x)=\sum_{cyc}\frac{a^x\ln(a)(b^x+c^x)-a^x(b^x\ln(b)+c^x\ln(c))}{(b^x+c^x)^2}$$
We multiply by $x$ it gives :
$$f'(x)=\frac{1}{x}\Big(\sum_{cyc}\frac{a^x\ln(a^x)(b^x+c^x)-a^x(b^x\ln(b^x)+c^x\ln(c^x))}{(b^x+c^x)^2}\Big)$$
$x$ is positive so we have to prove :
$$\sum_{cyc}\frac{a^x\ln(a^x)(b^x+c^x)-a^x(b^x\ln(b^x)+c^x\ln(c^x))}{(b^x+c^x)^2}\geq 0$$
We put $a^x=u$ , $b^x=v$ and $c^x=w$ so the inequality is equivalent to :
$$\sum_{cyc}\frac{u\ln(u)(v+w)-u(v\ln(v)+w\ln(w))}{(v+w)^2}\geq 0$$
Now with the concavity of $\ln(x)$ and the inequality of Jensen's we have :
$$\ln\Big(\frac{v^2+w^2}{v+w}\Big)\geq \frac{v\ln(v)+w\ln(w)}{v+w}$$
So we have :
$$\sum_{cyc}\frac{u\ln(u)(v+w)-u(v\ln(v)+w\ln(w))}{(v+w)^2}\geq\sum_{cyc}\frac{u\ln(u)-u\Big(\ln\Big(\frac{v^2+w^2}{v+w}\Big)\Big)}{(v+w)}\geq 0$$
Or :
$$\sum_{cyc}\frac{u\ln(u)(v+w)-u(v\ln(v)+w\ln(w))}{(v+w)^2}\geq\sum_{cyc}\frac{u\Big(\ln\Big(\frac{u(v+w)}{v^2+w^2}\Big)\Big)}{(v+w)}\geq 0$$
Now we have to prove :
$$\sum_{cyc}\frac{u\Big(\ln\Big(\frac{u(v+w)}{v^2+w^2}\Big)\Big)}{(v+w)}\geq 0$$
Unfortunatly I'm stuck here …
Second try :
For $a,b,c>0$ the function $g(x)=\frac{a^x}{b^x+c^x}$ is convex so the function $f(x)$ is convex as sum of convex functions .
Now we have with $a,b,c>0$ :
$$\sum_{cyc}\frac{a^2}{b^2+c^2}\geq \sum_{cyc}\frac{a}{b+c}$$
Fact wich have been proven by Michael Rozenberg (see my questions)
Remains to apply the three chord lemma to get the increase of the function $f(x)$
My question :
Can someone could find an alternative proof or prove this last inequality ?
There are generalizations of this facts ?
Best Answer
Lemma: [Smoothing of convex functions] Given a convex function $f(x)$, and variables $ a \leq b$, if $ \epsilon < b-a$, then $ f(a) + f(b) \geq f( a + \epsilon ) + f(b- \epsilon)$.
(This is "well-known", so I'm not going to prove it. If you're stuck, explain what you've tried.)
Corollary: Given a convex function $f(x)$, and variables $ a \leq b \leq c$, $ A \leq B \leq C$ such that $ a + b + c = A + B + C $, $ C \geq c$, and $A \leq a$, then $ f(A) + f(B) + f(C) \geq f(a) + f(b) + f(c)$.
Comments:
The proof follows by applying the above lemma (or see the hidden block).
I consider this "less well-known", though it's still often used for Olympiad inequalities.
The 4-variable analogue need not hold, because we don't have enough control over the middle terms.
I (personally) call this the "3-variable smoothing Jensens".
Corollary: The problem follows.
First, a simplification: To show that the function is increasing, we just need to show that for $ z = x/y > 1$, $$\sum \frac{ \frac{ p^z } {p^z + q^z + r^z } } { 1 - \frac{ p^z } {p^z + q^z + r^z } } \geq \frac{\frac{ p } { p + q + r }}{1 - \frac{ p } { p + q + r } } \quad (1) $$
then apply the substitution $ p = a^x, q = b^x, r = c^x$ to conclude that $f(y) \geq f(x)$.
Now, WLOG $ p = \max (p, q, r)$.
$ \frac{ p^z } {p^z + q^z + r^z } \geq \frac{ p } { p + q + r } \Leftrightarrow p^z q + p^z r \geq p q^z + p r^z$, which is true.
Likewise, WLOG $ r = \min (p,q,r)$ and $ \frac{ r^z } {p^z + q^z + r^z } \leq \frac{ r } { p + q + r }$
Hence, we can apply the previous corollary with $ f(x) = \frac{x}{1-x}$, $\{A, B, C \} = \{ \frac{ p^z } {p^z + q^z + r^z }, \frac{ q^z } {p^z + q^z + r^z }, \frac{ r^z } {p^z + q^z + r^z } \} $, $ \{ a, b, c \} = \{ \frac{ p } { p + q + r }, \frac{ q } { p + q + r }, \frac{ r } { p + q + r }\}$, so inequality (1) is true.