Hi it's a supposed refinement of Prove or disprove $\sum\limits_{1\le i < j \le n} \frac{x_ix_j}{1-x_i-x_j} \le \frac18$ for $\sum\limits_{i=1}^n x_i = \frac12$($x_i\ge 0, \forall i$) :
Problem :
Let $n\geq 5$ and $x_i>0$ such that $\sum_{i=1}^{n}x_i=1/2$ and $x_i\in[0, 0.5/n+0.5/n^2]$,$\min(x_i)+\max(x_i)\leq 0.5/n+0.5/n^2$ then it seems we have :
$$\sum_{1\le i < j \le n}f\left(2\sqrt{x_ix_j}+\frac{1}{3x_ix_j}\left(\exp\left(\frac{\ln\left(x_ix_j\left(x_i-x_j\right)^{2}+1\right)}{x_ix_j\left(x_i-x_j\right)^{2}+1}\right)-1\right)\right)\le1/8$$
Where :
$$\frac{\frac{x^{2}}{4}}{1-x}=f(x)$$
Some tought :
For the lower bound see the inequality here (Dis)prove $f(2\sqrt{xy}+\frac{1}{3xy}(\exp(\frac{\ln(xy(x-y)^{2}+1)}{xy(x-y)^{2}+1})-1))-\frac{xy}{1-x-y}\ge 0$ :
Let us consider the inequality for $x,y>0$ and $x+y\leq 0.5$ : $$G\left(x,y\right)=f\left(2\sqrt{xy}+\frac{1}{3xy}\left(\exp\left(\frac{\ln\left(xy\left(x-y\right)^{2}+1\right)}{xy\left(x-y\right)^{2}+1}\right)-1\right)\right)-\frac{xy}{1-x-y}\ge 0 \text{ where } f\left(x\right)=\frac{\frac{x^{2}}{4}}{1-x}.$$
Next as $$f'''(x)>0$$ for $x\in(0,0.5]$ I have tried to apply the equal variable method (Vasile Cirtoaje) without success .
Question :
How to (dis)prove it properly ?
