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In this work, we evaluate quantum coherence using the l_1-norm andconvex-roof l_1-norm and obtain several new results. First, we provide some newgeneral triangle-like inequalities of quantum coherence, with results betterthan existing ones. Second, for some special three-dimensional quantum states,a method for calculating the convex-roof l_1-norm is presented. Lastly, weoffer distinct upper bounds in the l_1-norm measure of coherence based on thequantum state itself.

Notes on quantum coherence with l_1-norm and convex-roof l_-norm