Quantum coherence is an important physical resource in quantum information science, and also as one of the most fundamental and striking features in quantum physics. In this paper, we obtain a symmetry-like relation of relative entropy measure $C_r(\rho)$ of coherence for $n$-partite quantum states $\rho$, which gives lower and upper bounds for $C_r(\rho)$. Meanwhile, we discuss the conjecture about the validity of the inequality $C_r(\rho)\leq C_{\ell_1}(\rho)$ for any state $\rho$. We observe that every mixture $\eta$ of a state $\rho$ satisfying $C_r(\rho)\leq C_{\ell_1}(\rho)$ and any incoherent state $\sigma$ also satisfies the conjecture. We also note that if the von Neumann entropy is defined by the natural logarithm $\ln$ instead of $\log_2$, then the reduced relative entropy measure of coherence $\bar{C}_r(\rho)=-\rho_{\rm{diag}}\ln \rho_{\rm{diag}}+\rho\ln \rho$ satisfies the inequality ${\bar{C}}_r(\rho)\leq C_{\ell_1}(\rho)$ for any mixed state $\rho$.