  We obtain an analytical bound on the mean vertical convective heat flux
$\langle w T \rangle$ between two parallel boundaries driven by uniform
internal heating. We consider two configurations, one with both boundaries held
at the same constant temperature, and the other one with a top boundary held at
constant temperature and a perfectly insulating bottom boundary. For the first
configuration, Arslan et al. (J. Fluid Mech. 919:A15, 2021) recently provided
numerical evidence that Rayleigh-number-dependent corrections to the only known
rigorous bound $\langle w T \rangle \leq 1/2$ may be provable if the classical
background method is augmented with a minimum principle stating that the
fluid's temperature is no smaller than that of the top boundary. Here, we
confirm this fact rigorously for both configurations by proving bounds on
$\langle wT \rangle$ that approach $1/2$ exponentially from below as the
Rayleigh number is increased. The key to obtaining these bounds are inner
boundary layers in the background fields with a particular inverse-power
scaling, which can be controlled in the spectral constraint using Hardy and
Rellich inequalities. These allow for qualitative improvements in the analysis
not available to standard constructions.
