  The R\'enyi and Shannon entropies are information-theoretic measures which
have enabled to formulate the position-momentum uncertainty principle in a much
more adequate and stringent way than the (variance-based) Heisenberg-like
relation. Moreover, they are closely related to various energetic
density-functionals of quantum systems. Here we find sharp upper bounds to
these quantities in terms of the second order moment $\langle r^2\rangle$ for
general spherically symmetric potentials, which substantially improve previous
results of this type, by means of the R\'enyi maximization procedure with a
covariance constraint due to Costa, Hero and Vignat \cite{CosHer03}. The
contributions to these bounds coming from the radial and angular parts of the
physical wavefunctions are explicitly given.
