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 ⁎ -modules and characterize the equality case. As applications, we get some new results and some known results due to J.E. Pečarić and R. Rajić [The Dunkl–Williams equality in pre-Hilbert C ⁎ -modules, Linear Algebra Appl. 425 (1) (2007) 16–25].