A New Bound for the Jensen Gap With Applications in Information Theory

Abstract

In this manuscript, we adopt a novel approach to present a new bound for the Jensen gap for functions whose double derivatives in absolute function, are convex. We demonstrate two numerical experiments to verify the main result and to discuss the tightness of the bound. Then we utilize the bound for deriving two new converses of the Hölder inequality and a bound for the Hermite-Hadamard gap. Finally, we demonstrate applications of the main result for various divergences in information theory. Also, we present a numerical example to verify the bound for Shannon entropy.

Keywords

• Jensen inequality
• Hölder inequality
• Hermite-Hadamard inequality
• convex function
• Green function
• Csiszár divergence