Can \max(\alpha_i, \beta) be less than \beta? It cannot.

Can \min(\max(\alpha_i, \beta)) be less than \beta. It cannot since all terms are \ge \beta.

Can there exist an \alpha_i\le\beta such that \min(\max(\alpha_i, \beta))\gt\beta . It cannot since \max(\alpha_i, \beta) is \beta for at least one i and therefore the minimum is \le \beta.

If all \alpha_i>\beta then \min(\max(\alpha_i, \beta))=\min(\alpha_i). This is because if \alpha_i>\beta then \max(\alpha_i, \beta)=\alpha_i.

Therefore we can deduce our answer is \min(\alpha_i) if \min(\alpha_i)>\beta, otherwise it’s \beta, which is max(min(\alpha_i), \beta). Therefore the 2 equations are equivalent.