Convex inequalities in Hilbert $C^{\ast }$-modules

The H$\ddot{\mathrm{o}}$lder-McCarty inequalities are originally derived in the Hilbert space case and have been generalized via a convex inequality. The main purpose of this paper is to extend this convex inequality to the Hilbert $C^{\ast }$-module case, and meanwhile to make some investigations on the H$\ddot{\mathrm{o}}$lder-McCarty inequalities in the Hilbert $C^{\ast }$-module case.