  Bounds on heat transfer have been the subject of previous studies concerning
convection in the Boussinesq approximation: in the Rayleigh-B\'enard
configuration, the first result obtained by \cite{howard63} states that $Nu <
(3/64 \ Ra)^{1/2}$ for large values of the Rayleigh number $Ra$, independently
of the Prandtl number $Pr$. This is still the best known upper bound, only with
the prefactor improved to $Nu < 1/6 \ Ra^{1/2}$ by \cite{DoeringConstantin96}.
In the present paper, this result is extended to compressible convection. An
upper bound is obtained for the anelastic liquid approximation, which is
similar to the anelastic model used in astrophysics based on a turbulent
diffusivity for entropy. The anelastic bound is still scaling as $Ra^{1/2}$,
independently of $Pr$, but depends on the dissipation number $\mathcal{D}$ and
on the equation of state. For monatomic gases and large Rayleigh numbers, the
bound is $Nu < 146\, Ra^{\frac{1}{2}} / (2-\mathcal{D} )^{\frac{5}{2}}$.
