We investigate some issues concerning the zero-momentum four-point renormalized coupling constant g in the symmetric phase of O(N) models, and the corresponding Callan-Symanzik β-function. In the framework of the 1/N expansion we show that the Callan-Symanzik β-function is non-analytic at its zero, i.e. at the fixed-point value g ^∗ of g. This fact calls for a check of the actual accuracy of the determination of g ^∗ from the resummation of the d = 3 perturbative g-expansion, which is usually performed assuming the analyticity of the β-function. Two alternative approaches are exploited. We extend the ϵ-expansion of g ^∗ to O(ϵ⁴). Quite accurate estimates of g ^∗ are obtained by an analysis that exploits the analytic behavior of g ^∗ as a function of d and the known values of g ^∗ for lower-dimensional O(N) models, i.e. for d = 2, 1, 0. Accurate estimates of g ^∗ are also obtained by a reanalysis of the strong-coupling expansion of the lattice N-vector model allowing for the leading confluent singularity. The agreement among the g-, ϵ-, and strong-coupling expansion results is good for all values of N. However, at N = 0, 1, ϵ- and strong-coupling expansion favor values of g ^∗ which are slightly lower than those obtained by the resummation of the g-expansion assuming the analyticity of the Callan-Symanzik β-function.