Hi it's related to the following conjecture An inequality for polynomials with positives coefficients :
We have the first conjecture :
Let $x,y>0$ then we have : $$(x+y)f\Big(\frac{x^2+y^2}{x+y}\Big)(f(x)+f(y))\geq 2(xf(x)+yf(y))f\Big(\frac{x+y}{2}\Big)$$ Where :$$f(x)=\sum_{i=0}^{n}a_ix^i$$ And $a_0 \ge a_1 \ge ... \ge a_n>0$ (Thanks to p4sch)
Second conjecture :
Let $x_i>0$ be $n$ real numbers then we have : $$(\sum_{i=1}^{n}x_i)f\Big(\frac{\sum_{i=1}^{n}x_i^2}{\sum_{i=1}^{n}x_i}\Big)(\sum_{i=1}^{n}f(x_i))\geq n(\sum_{i=1}^{n}x_if(x_i))f\Big(\frac{\sum_{i=1}^{n}x_i}{n}\Big)$$ Where :$$f(x)=\sum_{i=0}^{n}a_ix^i$$ And $a_0 \ge a_1 \ge ... \ge a_n>0$
It seems to be true in the case where $f(x)=\tan(x)$ see A sharp inequality for tangent (refinement of Jensen's inequality)
I have tried the approach of River Li in the case $n=3$ without success . I think it's really hard and interesting since at the beginning I have put together two applications of Jensen's inequality .
Putting $f(x)=e^x$ gives a nice refinement of Am-Gm (as applications).
If it's true I would like just one approach to prove it .
Any helps is greatly appreciated .
Thanks a lot for all your contributions .
Ps:Is it a good idea to put it on maths overflow ?
Reference :
Partial answer
I provide a partial proof for :
$$f(x)=1+x+x^2$$
$$Difference=(x-y)^4×\frac{positivepolynomial}{2(x+y)}$$
Now we can enlarge the proof using homogeneity or $y=aY,x=aX$
It's highly generalizable
A Conjecture:
Let the D be the difference between the two side above then :
There exists $C_n>0$ constants and $k,m$ positive integers and $n>2$ Then:
$$D=(x-y)^m\frac{positivepolynomial}{C_n(x+y)^k}$$
It seems $m=4$ if so we can start with the case where $a_i=1$