  We revisit generalized entropic formulations of the uncertainty principle for
an arbitrary pair of quantum observables in two-dimensional Hilbert space.
R\'enyi entropy is used as uncertainty measure associated with the distribution
probabilities corresponding to the outcomes of the observables. We derive a
general expression for the tight lower bound of the sum of R\'enyi entropies
for any couple of (positive) entropic indices (\alpha,\beta). Thus, we have
overcome the H\"older conjugacy constraint imposed on the entropic indices by
Riesz-Thorin theorem. In addition, we present an analytical expression for the
tight bound inside the square [0 , 1/2] x [0 , 1/2] in the \alpha-\beta plane,
and a semi-analytical expression on the line \beta = \alpha. It is seen that
previous results are included as particular cases. Moreover, we present an
analytical but suboptimal bound for any couple of indices. In all cases, we
provide the minimizing states.
