Let U be a domain, convex in $x$ and symmetric about the $y$-axis, which is contained in a centered and oriented rectangle $S$. It is proved that $H_t{(U^+)}/H_t{(U)}\,{\leq}\, H_t{(S^+)}/H_t{(S)}$ where $H_t$ stands for heat content, that is, the remaining heat in the domain at time $t$ if it initially has uniform temperature 1, with Dirichlet boundary conditions, where $A^+\,{=}\,A\,{\cap}\, \{(x,y)\,{:}\,x\,{>}\,0\}$. It is also shown that the analog of this inequality holds for some other Schrödinger operators.