Monge-Ampère type equation for the Nakano positive curvature tensor of holomorphic vector bundles

For any Hermitian holomorphic vector bundle with Nakano positive curvature tensor, Demailly introduced a Monge-Ampère type equation. When the rank of the bundle is $1$, it becomes the usual Monge-Ampère equation. In this paper, we solve this equation in the conformal class of a Nakano positive Hermitian metric.