  We investigate the influence of the thermal properties of the boundaries in
turbulent Rayleigh-B\'enard convection on analytical upper bounds on convective
heat transport. We model imperfectly conducting bounding plates in two ways:
using idealized mixed thermal boundary conditions of constant Biot number
$\eta$, continuously interpolating between the previously studied fixed
temperature ($\eta = 0$) and fixed flux ($\eta = \infty$) cases; and by
explicitly coupling the evolution equations in the fluid in the Boussinesq
approximation through temperature and flux continuity to identical upper and
lower conducting plates. In both cases, we systematically formulate a bounding
principle and obtain explicit upper bounds on the Nusselt number $Nu$ in terms
of the usual Rayleigh number $Ra$ measuring the average temperature drop across
the fluid layer, using the ``background method'' developed by Doering and
Constantin. In the presence of plates, we find that the bounds depend on
$\sigma = d/\lambda$, where $d$ is the ratio of plate to fluid thickness and
$\lambda$ is the conductivity ratio, and that the bounding problem may be
mapped onto that for Biot number $\eta = \sigma$. In particular, for each
$\sigma > 0$, for sufficiently large $Ra$ (depending on $\sigma$) we show that
$Nu \leq c(\sigma) R^{1/3} \leq C Ra^{1/2}$, where $C$ is a
$\sigma$-independent constant, and where the control parameter $R$ is a
Rayleigh number defined in terms of the full temperature drop across the entire
plate-fluid-plate system. In the $Ra \to \infty$ limit, the usual fixed
temperature assumption is a singular limit of the general bounding problem,
while fixed flux conditions appear most relevant to the asymptotic $Nu$--$Ra$
scaling even for highly conducting plates.
