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Quantum coherence and entanglement are two key features in quantum mechanicsand play important roles in quantum information processing and quantumcomputation. We provide a general triangle-like inequality satisfied by the$l_1$-norm measure of coherence for convex combination of arbitrary $n$ purestates of a quantum state $\rho$. Furthermore, we present triangle-likeinequality for the convex-roof extended negativity for any states of rank 2,which gives a positive answer to a conjecture raised in [Phys. Rev. A 96,062308 (2017)]. Detailed examples are given to illustrate the relationscharacterized by the triangle-like inequalities.

Triangle-like inequalities related to coherence and entanglement negativity