In order to understand and deal with the basic theoretical problems of the electron spin transport in mesoscopic systems, nonequilibrium spin transport properties of typical three site one-dimensional(1D) quantum chain system with electronic interactions are analyzed and the corresponding analytic formulas are deduced. Firstly, the initial charge on each site of the chain is calculated from the exactly derived ground state under the half-filling assumption. Then, based on Keldysh formalism, the basic theoretical methods are described in detail, and the main analytic formulas of the nonequilibrium spin transport properties(differential spin conductance, spin transport current and electronic charge distribution, etc.) of the 1D chain systems are derived considering Hartree-Fock approximation and Coulomb electronic interaction. Both the cases, namely, the scattering process considering the Coulomb repulsion force between the spin-up and spin-down electrons and the scattering process considering the spin-spin interaction due to spin degree of freedom, are considered. Finally, under the conditions of typical parameters, the initial charges and calculating formulas are used to calculate the nonequilibrium spin transport properties of the chain system self-consistently, and the corresponding physical interpretations of the numerical calculation data are provided, which verifies the correctness and validity of the formulas.