  Analytic representation of both position as well as momentum waveforms of the
two-dimensional (2D) circular quantum dots with the Dirichlet and Neumann
boundary conditions (BCs) allowed an efficient computation in either space of
Shannon $S$, R\'{e}nyi $R(\alpha)$ and Tsallis $T(\alpha)$ entropies, Onicescu
energies $O$ and Fisher informations $I$. It is shown that a transition to the
2D geometry lifts the 1D degeneracy of the position components $S_\rho$,
$O_\rho$, $R_\rho(\alpha)$. Among many other findings, it is established that
the lower limit $\alpha_{TH}$ of the semi-infinite range of the dimensionless
R\'{e}nyi/Tsallis coefficient where one-parameter momentum entropies exist is
equal to 2/5 for the Dirichlet requirement and 2/3 for the Neumann one. Since
their 1D counterparts are $1/4$ and $1/2$, respectively, this simultaneously
reveals that this critical value crucially depends not only on the position BC
but the dimensionality of the structure too. As the 2D Neumann threshold
$\alpha_{TH}^N$ is greater than one half, its R\'{e}nyi uncertainty relation
for the sum of the position and wave vector components
$R_\rho(\alpha)+R_\gamma\left(\frac{\alpha}{2\alpha-1}\right)$ is valid in the
range $[1/2,2)$ only with its logarithmic divergence at the right edge whereas
for all other systems it is defined at any coefficient $\alpha$ not smaller
than one half. For both configurations, the lowest-energy level at $\alpha=1/2$
does saturate R\'{e}nyi and Tsallis entropic inequalities. Other properties are
discussed and analyzed from the mathematical and physical points of view.
