Radical layer length and syzygy-finite algebras

Let $\Lambda$ be an artin algebra. We obtain that $\Lambda$ is syzygy-finite when the radical layer length of $\Lambda$ is at most two; as two consequences, we give a new upper bound for the dimension of the bounded derived category of the category $\mod \Lambda$ of finitely generated right $\Lambda$-modules in terms of the projective of certain class of simple right $\Lambda$-modules and also get the left big finitistic dimension conjecture holds.