  The resource theories of quantum coherence attract a lot of attention in
recent years. Especially, the monotonicity property plays a crucial role here.
In this paper we investigate the monotonicity property for the coherence
measures induced by the R\'{e}nyi $\alpha$-relative entropy which present in
[Phys. Rev. A 94, 052336, 2016]. We show that the R\'{e}nyi $\alpha$-relative
entropy of coherence does not in general satisfy the monotonicity requirement
under the subselection of measurements condition and it also does not satisfy
the extension of monotonicity requirement which presents in [Phys. Rev. A 93,
032136, 2016]. Due to the R\'{e}nyi $\alpha$-relative entropy of coherence can
act as a coherence monotone quantifier, we examine the trade-off relations
between coherence and mixedness. Finally, some properties for the single qubit
of R\'{e}nyi $2$-relative entropy of coherence are derived.
