The key result of this paper is a strengthened version for universal Teichmüller space of the fundamental Gardiner--Royden theorem on coincidence of the Kobayashi and Teichmüller metrics for Teichmüller spaces. Using the Grunsky coefficient inequalities for univalent functions, we show that the Teichmüller metric is logarithmically plurisubharmonic and has constant holomorphic sectional curvature $\kappa_{\mathcal K}(\psi,v)=-4$. This result has various important applications in geometric function theory and geometry. Some applications to complex geometry of Teichmüller spaces are given.