  We obtain uncertainty and certainty relations of state-independent form for
the three Pauli observables with use of the R\'enyi entropies of order
$\alpha\in(0;1]$. It is shown that these entropic bounds are tight in the sense
that they are always reached with certain pure states. A new result is the
conditions for equality in R\'enyi-entropy uncertainty relations for the Pauli
observables. Upper entropic bounds in the pure-state case are also novel.
Combining the presented bounds leads to a band, in which the rescaled average
R\'enyi $\alpha$-entropy ranges for a pure measured state. A width of this band
is compared with the Tsallis formulation derived previously.
