  The entanglement entropy of an arbitrary spacetime region $A$ in a
three-dimensional conformal field theory (CFT) contains a constant universal
coefficient, $F(A)$. For general theories, the value of $F(A)$ is minimized
when $A$ is a round disk, $F_0$, and in that case it coincides with the
Euclidean free energy on the sphere. We conjecture that, for general CFTs, the
quantity $F(A)/F_0$ is bounded above by the free scalar field result and below
by the Maxwell field one. We provide strong evidence in favor of this claim and
argue that an analogous conjecture in the four-dimensional case is equivalent
to the Hofman-Maldacena bounds. In three dimensions, our conjecture gives rise
to similar bounds on the quotients of various constants characterizing the CFT.
In particular, it implies that the quotient of the stress-tensor two-point
function coefficient and the sphere free energy satisfies $C_{
\scriptscriptstyle T} / F_0 \leq 3/ (4\pi^2 \log 2 - 6\zeta[3]) \simeq 0.14887$
for general CFTs. We verify the validity of this bound for free scalars and
fermions, general $O(N)$ and Gross-Neveu models, holographic theories,
$\mathcal{N}=2$ Wess-Zumino models and general ABJM theories.
