  The coupled cluster method is implemented at high orders of approximation to
investigate the zero-temperature $(T=0)$ phase diagram of the frustrated
spin-$s$ $J_{1}$--$J_{2}$--$J_{3}$ antiferromagnet on the honeycomb lattice.
The system has isotropic Heisenberg interactions of strength $J_{1}>0$,
$J_{2}>0$ and $J_{3}>0$ between nearest-neighbour, next-nearest-neighbour and
next-next-nearest-neighbour pairs of spins, respectively. We study it in the
case $J_{3}=J_{2}\equiv \kappa J_{1}$, in the window $0 \leq \kappa \leq 1$
that contains the classical tricritical point (at $\kappa_{{\rm
cl}}=\frac{1}{2}$) of maximal frustration, appropriate to the limiting value $s
\to \infty$ of the spin quantum number. We present results for the magnetic
order parameter $M$, the triplet spin gap $\Delta$, the spin stiffness
$\rho_{s}$ and the zero-field transverse magnetic susceptibility $\chi$ for the
two collinear quasiclassical antiferromagnetic (AFM) phases with N\'{e}el and
striped order, respectively. Results for $M$ and $\Delta$ are given for the
three cases $s=\frac{1}{2}$, $s=1$ and $s=\frac{3}{2}$, while those for
$\rho_{s}$ and $\chi$ are given for the two cases $s=\frac{1}{2}$ and $s=1$. On
the basis of all these results we find that the spin-$\frac{1}{2}$ and spin-1
models both have an intermediate paramagnetic phase, with no discernible
magnetic long-range order, between the two AFM phases in their $T=0$ phase
diagrams, while for $s > 1$ there is a direct transition between them. Accurate
values are found for all of the associated quantum critical points. While the
results also provide strong evidence for the intermediate phase being gapped
for the case $s=\frac{1}{2}$, they are less conclusive for the case $s=1$. On
balance however, at least the transition in the latter case at the striped
phase boundary seems to be to a gapped intermediate state.
