We discuss the $({g}_{0}{\ensuremath{\varphi}}^{p}{)}_{d}$ self-interacting scalar field theory, in the strong-coupling regime. We assume the presence of macroscopic boundaries confining the field in a hypercube of side $L$. We also consider that the system is in thermal equilibrium at temperature ${\ensuremath{\beta}}^{\ensuremath{-}1}$. For spatially bounded free fields, the Bekenstein bound states that the specific entropy satisfies the inequality $\frac{S}{E}<2\ensuremath{\pi}R$, where $R$ stands for the radius of the smallest sphere that circumscribes the system. Employing the strong-coupling perturbative expansion, we obtain the renormalized mean energy $E$ and entropy $S$ for the system up to the order $({g}_{0}{)}^{\ensuremath{-}(2/p)}$, presenting an analytical proof that the specific entropy also satisfies in some situations a quantum bound. Defining ${\ensuremath{\epsilon}}_{d}^{(r)}$ as the renormalized zero-point energy for the free theory per unit length, the dimensionless quantity $\ensuremath{\xi}=\frac{\ensuremath{\beta}}{L}$ and ${h}_{1}(d)$ and ${h}_{2}(d)$ as positive analytic functions of $d$, for the case of high temperature, we get that the specific entropy satisfies $\frac{S}{E}<2\ensuremath{\pi}R\frac{{h}_{1}(d)}{{h}_{2}(d)}\ensuremath{\xi}$. When considering the low-temperature behavior of the specific entropy, we have $\frac{S}{E}<2\ensuremath{\pi}R\frac{{h}_{1}(d)}{{\ensuremath{\epsilon}}_{d}^{(r)}}{\ensuremath{\xi}}^{1\ensuremath{-}d}$. Therefore the sign of the renormalized zero-point energy can invalidate this quantum bound. If the renormalized zero-point energy is a positive quantity, at intermediate temperatures and in the low-temperature limit, there is a quantum bound.