Normal structure and the generalized James and Zbăganu constants

We present several sufficient conditions for normal structure of a Banach space in terms of the generalized James, von Neumann–Jordan, Zbăganu constants, and the coefficient of weak orthogonality of Sims. Our results significantly improve many known results in the literatures. A negative answer is given to the question of Llorens–Fuster [E. Llorens-Fuster, Zbganu constant and normal structure, Fixed Point Theory 9 (2008) 159–172].