Boundedness of weak Fano pairs with alpha-invariants and volumes bounded below

We show that fixed-dimensional klt weak Fano pairs with alpha-invariants and volumes bounded away from 0 and the coefficients of the boundaries belonging to the set of hyperstandard multiplicities $\Phi(\mathscr R)$ associated to a fixed finite set $\mathscr R$ form a bounded family. We also show $\alpha(X, B)^{d-1}\mathrm{vol}\,(-(K_X+B))$ are bounded from above for all klt weak Fano pairs $(X,B)$ of a fixed dimension $d$.