Bounds for the zeroth-order exchange-energy functional for atomic systems are derived using the generalized Carlson’s inequality. They are formulated entirely in terms of the average electron density [rho], the expectation value [r(-1)], and the number of electrons N, viz., (3/4) x (3/pi)N-1/3(2/3)[rho](1/3) greater than or equal to K-0[rho] = (3/4)(3/pi)(1/3)integral rho(4/3)(r)dr greater than or equal to (1/18 pi(5))(1/3)[r(-1)](2)/[rho](1/3). quality of the bounds presented are numerically analyzed and discussed. Atomic units are used throughout.