As an important resource, quantum coherence is one of the fundamental features of the quantum world. It is essential to understand the relationship between the resource and the mixedness of an open quantum system. We present complementary relations between the l p ( 1 ⩽ p < 2 ) norm of coherence and the normalized linear entropy mixedness. We then study the geometry of the complementary relations for Bell-diagonal states, as well as the evolutions of Werner and isotropic states under bit flip, phase flip, bit-phase flip, depolarizing, generalized amplitude damping and amplitude damping channels. Our results include some existing conclusions as special cases.

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