  We revisit entropic formulations of the uncertainty principle for an
arbitrary pair of positive operator-valued measures (POVM) $A$ and $B$, acting
on finite dimensional Hilbert space. Salicr\'u generalized
$(h,\phi)$-entropies, including R\'enyi and Tsallis ones among others, are used
as uncertainty measures associated with the distribution probabilities
corresponding to the outcomes of the observables. We obtain a nontrivial lower
bound for the sum of generalized entropies for any pair of entropic
functionals, which is valid for both pure and mixed states. The bound depends
on the overlap triplet $(c_A,c_B,c_{A,B})$ with $c_A$ (resp. $c_B$) being the
overlap between the elements of the POVM $A$ (resp. $B$) and $c_{A,B}$ the
overlap between the pair of POVM. Our approach is inspired by that of de
Vicente and S\'anchez-Ruiz [Phys.\ Rev.\ A \textbf{77}, 042110 (2008)] and
consists in a minimization of the entropy sum subject to the Landau-Pollak
inequality that links the maximum probabilities of both observables. We solve
the constrained optimization problem in a geometrical way and furthermore, when
dealing with R\'enyi or Tsallis entropic formulations of the uncertainty
principle, we overcome the H\"older conjugacy constraint imposed on the
entropic indices by the Riesz-Thorin theorem. In the case of nondegenerate
observables, we show that for given $c_{A,B} > \frac{1}{\sqrt2}$, the bound
obtained is optimal; and that, for R\'enyi entropies, our bound improves
Deutsch one, but Maassen-Uffink bound prevails when $c_{A,B} \leq\frac12$.
Finally, we illustrate by comparing our bound with known previous results in
particular cases of R\'enyi and Tsallis entropies.
