Rigorous lower bounds for the first-gradient corrections to the kinetic and exchange-energy functionals in atoms are derived using Benson's inequality. They are formulated in terms of the expectation values and the momentum expectation value in a simple manner namelyT₂[ρ] = (1/72)∫(|ρ(r)|²/ρ(r)) dr ≥ (n²/72) ⟨r^{n − 3}⟩²⟨r^{2n − 4}⟩⁻¹and|K₂[ρ]| = β ∫(|ρ(r)|²/ρ^4/3(r)) dr ≥ 0.05105 ⟨r⁻¹⟩²⟨p⟩⁻¹,where n, the natural number = 1, 2, 3,..., ⟨r^{n − 3}⟩ and ⟨r^{2n − 4}⟩ are the expectation values, ⟨p⟩ is the momentum value, ρ(r) is the electron density with the normalization condition ∫ ρ(r) dr = N, the number of electrons. The bounds derived in this work are tested for the atoms Z = 3 to 36. A comparison is also made with the previously derived lower bound estimates for T₂[ρ] and K₂[ρ]. The bounds presented are sharper than the previous ones given by Pathak and Gadre (atomic units are used throughout the paper).