Help proving inequality $\alpha^\alpha\left(1 - \alpha \right)^{1-\alpha} \geq \beta^\alpha \left(1 - \beta\right)^{1-\alpha}$
Proving inequality $\left( \frac {x} {y}\right) ^{x}\left( \frac {1-x} {1-y}\right) ^{1-x}\geqslant 1$