Rigorous upper bounds, for all bound-state systems, for the zeroth-order kinetic and exchange-energy functionals, viz., T₀[ρ] = 0.3(3π²)^2/3∫ρ^5/3(r) d³r ≤ 0.3(3π²)^2/3N^1/3⟨ρ⟩^2/3 and |K₀[ρ]| = 0.75(3/π)^1/3∫ρ^4/3(r) d³r ≤ 0.75(3/π)^1/3 N^2/3⟨ρ⟩^1/3, have been derived (where N is the number of electrons, ρ(r) is the electron density and ⟨ρ⟩ is the average electron density denoted by ⟨ρ⟩ = ∫ ρ²(r) d³r ). Numerical investigations on these bounds employing Hartree-Fock densities have been carried out for atomic systems. An empirical relationship between the zeroth-order kinetic energy functional and the average electron density for atomic systems has been proposed. Atomic units are used throughout.