  An uncertainty relation for the R\'enyi entropies of conjugate quantum
observables is used to obtain a strong Heisenberg limit of the form ${\rm RMSE}
\geq f(\alpha)/(\langle N\rangle+\frac12)$, bounding the root mean square error
of any estimate of a random optical phase shift in terms of average photon
number, where $f(\alpha)$ is maximised for non-Shannon entropies. Related
simple yet strong uncertainty relations linking phase uncertainty to the photon
number distribution, such as $\Delta\Phi\geq \max_n p_n$, are also obtained.
These results are significantly strengthened via upper and lower bounds on the
R\'enyi mutual information of quantum communication channels, related to
asymmetry and convolution, and applied to the estimation (with prior
information) of unitary shift parameters such as rotation angle and time, and
to obtain strong bounds on measures of coherence. Sharper R\'enyi entropic
uncertainty relations are also obtained, including time-energy uncertainty
relations for Hamiltonians with discrete spectra. In the latter case
almost-periodic R\'enyi entropies are introduced for nonperiodic systems.
