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.
Highlights
AND PRELIMINARIESThe field of mathematical inequalities and their applications has recorded an exponential and significant growth in the last three decades with considerable impact in various areas of Science such as Engineering [12], Economics [25], Mathematical Statistics [24], Qualitative Theory of Integral and Differential Equations [21], Information Theory and Coding [16], [18] etc
Adil Khan et al.: New Bound for the Jensen Gap With Applications in Information Theory of the Hölder inequality, Corollary 1 demonstrate another converse of the Hölder inequality while Corollary 2 presents a bound for the Hermite-Hadamard gap
The Jensen inequality has generalized the concept of classical convexity
Summary
The field of mathematical inequalities and their applications has recorded an exponential and significant growth in the last three decades with considerable impact in various areas of Science such as Engineering [12], Economics [25], Mathematical Statistics [24], Qualitative Theory of Integral and Differential Equations [21], Information Theory and Coding [16], [18] etc. An extensive literature exists regarding estimates for the Jensen gap and their applications in many branches of Science [1]–[8], [12], [15], [20], [24]–[26], [28] In this manuscript, we present a new bound as an estimate for the Jensen gap. We organize the remaining paper as: In Section II, we present a new main result following by a remark, two numerical experiments, a proposition, two corollaries and an another remark which completes the section. M. Adil Khan et al.: New Bound for the Jensen Gap With Applications in Information Theory of the Hölder inequality, Corollary 1 demonstrate another converse of the Hölder inequality while Corollary 2 presents a bound for the Hermite-Hadamard gap.
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