  Using the Euclidean path integral approach with functional methods, we
discuss the $(g_{0} \phi^{p})_{d}$ 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 $\beta^{-1}$. For spatially
bounded free fields, the Bekenstein bound states that the specific entropy
satisfies the inequality $\frac{S}{E} < 2 \pi R$, 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 $(g_{0})^{-\frac{2}{p}}$,
presenting an analytical proof that the specific entropy also satisfies in some
situations a quantum bound. Defining $\epsilon_d^{(r)}$ as the renormalized
zero-point energy for the free theory per unit length, the dimensionless
quantity $\xi=\frac{\beta}{L}$ 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 $\frac{S}{E} < 2\pi R \frac{h_1(d)}{h_2(d)} \xi$. When
considering the low temperature behavior of the specific entropy, we have
$\frac{S}{E} <2\pi R \frac{h_1(d)}{\epsilon_d^{(r)}}\xi^{1-d}$. 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.
