Inconsistent inequalities I want to prove that the following two inequalities cannot hold simultaneously 
$\beta_0^2(\beta_1-1)\geq \beta_1$
and  
$\beta_1^2(\beta_0-1)\leq \beta_0$, 
where $1<\beta_0<\beta_1<2$

That is a statement rather than a question...
Suppose that they do hold 'simultaneously' and I will use $a:=\beta_0$ and $b:=\beta_1$.
Add them together to get
$$\begin{align}
b+ab^2-b^2& \leq a^2b-a^2+a
\\\Rightarrow a^2b-ab^2-a^2+b^2-b+a&\geq0
\\ \Rightarrow ab(a-b)-(a-b)(a+b)+1(a-b)&\geq 0
\\ \Rightarrow (a-b)(ab-a-b+1)&\geq 0
\\ \Rightarrow (a-b)(a-1)(b-1)\geq 0
\end{align}$$
which is a contradiction because $a<b\Rightarrow a-b<0$ and $a,b>1\Rightarrow (a-1),(b-1)>0$.
