Abstract: 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. I will present a new conjecture stating 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. I will 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_T/F_0 ≤3/(4\pi \log2−6\zeta[3]) \simeq 0.14887 for general CFTs. I will show that the bound is satisfied by free scalars and fermions, general O(N) and Gross-Neveu models, holographic theories, N=2 Wess-Zumino models and general ABJM theories.