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Given $\frac{a}{x-1} \leq \frac{b}{y-1} \leq \frac{c}{z-1}$ with $a,b,c > 0$ and $x,y,z > 1$, I want to show that $$\frac{(\frac{a}{a+b})^{x-1}(\frac{b}{a+b})^{y-1}}{B(x,y)\cdot (x+y-1)} + \frac{(\frac{a}{a+c})^{x-1}(\frac{c}{a+c})^{z-1}}{B(x,z)\cdot (x+z-1)} + \frac{(\frac{b}{b+c})^{y-1}(\frac{c}{b+c})^{z-1}}{B(y,z)\cdot (y+z-1)}$$ is smaller than $$\frac{1}{x}\left(\frac{a(y-1)}{x\cdot b}\right)^{x-1} +\frac{1}{x}\left(\frac{a(z-1)}{x\cdot c}\right)^{x-1}+\frac{1}{y}\left(\frac{b(z-1)}{y\cdot c}\right)^{y-1}+2$$

I already know that the first formula is bounded above by 3, and each of the individual terms in both formulas is between 0 and 1. Further, I ran a lot of simulations with different values for $a,b,c,x,y,z$ and the bound seems tight only when $a$ approaches $\infty$ and $x=a+1$, while $b,c$ approach 0 and $y = b+1$ and $z=c+1$. The problem is that I cannot show it formally, because I don't know how to simplify or bound the functions involving the beta functions.

Thanks in advance,


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