Help solve simultaneous inequality that has $\leq$ and $\geq$ in it 
I only have problems determining the values of $\alpha$ and $\beta$, so I will only show the solution used to derive their values:

So, I know how to get the inequalities at ★ and † but I don't know how use them to deduce $\alpha$ and $\beta$.
All I know is $\alpha \geq -1$ and I got this from ★ because if $\alpha=-1$ then ★ is 0 and if $\alpha > -1$, ★ is greater than 0.
Please show me the next steps (I don't understand the reasoning the solution provided).

Necessary conditions:
if $y=1$ is fixed, $\alpha+1>1$
if $x=1$ is fixed, $\beta-1>1$
Sufficient conditions :
if $\alpha>0$ and $\beta>-2$, $f$ is the product of 2 convex functions
