  We show that, under the definiteness of holomorphic sectional curvature, the
spaces of some holomorphic tensor fields on compact Chern-K\"{a}hler-like
Hermitian manifolds are trivial. These can be viewed as counterparts to
Bochner's classical vanishing theorems on compact K\"{a}hler manifolds under
the definiteness of Ricci curvature or the existence of K\"{a}hler-Einstein
metrics. Our proof is inspired by and based on some ideas due to X. Yang and L.
Ni-F. Zheng.
