Analyticity and the renormalization group

Based on the use of general principles of local quantum field theory we have proved the inequalities 0< or =psi(y, α)< or =α Vertical Bar #betta#(α/sub s/)/α/sub s/ Vertical Bar < or =1 (for the Peterman--Stueckelberg--Gell-Mann--Low function in quantum electrodynamics and in quantum chromodynamics), as well as the inequality #betta#0(α/sub s/)< or =1 (for the anomalous dimension of the gauge-invariant operator O(x) of quantum chromodynamics). Consequences of these inequalities are discussed. A comparison at present-day energies between the theoretical and experimental moments of the structure functions of deep inelastic lepton-hadron scattering is meaningful only for N< or =7 (the singlet case) and N< or =50 (the non-singlet case)