Generalizations of Bohr's inequality in Hilbert $C^{\ast }$-modules

We present a new operator equality in the framework of Hilbert $C^{\ast }$-modules. As a consequence, we get an extension of the Euler–Lagrange type identity in the setting of Hilbert bundles as well as several generalized operator Bohr’s inequalities due to O. Hirzallah, W.-S. Cheung-J.E. Pecaric and F. Zhang.