Recently I had use for the following inequality, which was proven in a rather inelegant way:

For all $y \in [0,1]$ and all integers $0\leq i \leq s$, the following inequality holds: \begin{align*}\left(2+2y^i-4y^s+y^{2s-i}\right)^s \leq \left(2-y^s\right)^{2s-i}. \end{align*}

I now wonder:

1. Is this a special case of some known inequality?

2. Does anyone know some *nice* proof for this and similar inequalities in one or more variables?