Chern number inequalities of deformed Hermitian-Yang-Mills metrics on four dimensional Kähler manifolds

In this paper, we give an affirmative answer to a conjecture of Collins-Yau [8]. We investigate the Chern number inequalities on 4-dimensional Kähler manifolds admitting the deformed Hermitian-Yang-Mills metrics under the assumption \({{\hat{\theta }}}\in (\pi ,2\pi )\) $\widehat{\theta }\in (\pi ,2\pi )$ .