Characterization of equality in a generalized Dunkl--Williams inequality

We establish a generalization of the Dunkl–Williams inequality and its inverse in the framework of Hilbert $C^{\ast }$-modules and characterize the equality case. As applications, we get some new results and some known results due to Pe\v{c}ari'c and Raji'c [Linear Algebra Appl. 425 (2007), no. 1, 16–25].