  The aim of the work is to give the explicit proofs of the Renyi-entropy
uncertainty relations presented in the previous work [A. Rastegin,
arXiv:0805.1777]. The relations with both the state-dependent and
state-independent entropic bounds are proved. For a pair of POVM measurements
the two relations are obtained. The first of them is generalization of the
known results, whereas the second is quite alternative. It is shown that both
these relations are meaty. The important case of POVM's with one-rank elements
is extra discussed. The measurements designed for distinction between two
non-orthogonal quantum states are considered as examples.
