Some New Karamata Type Inequalities and Their Applications to Some Entropies

Abstract

Some new inequalities of Karamata type are established with a convex function in this paper. The methods of our proof allow us to obtain an extended version of the reverse of Jensen inequality given by Pečarić and Mićić.Applying the obtained results, we give reverses for information inequality (Shannon inequality) in different types, namely ratio type and difference type, under some conditions. Also, we provide interesting inequalities for von Neumann entropy and quantum Tsallis entropy which is a parametric extension of von Neumann entropy. The inequality for von Neumann entropy recovers the nonnegativity and gives a refinement for the weaker version of Fannes's inequality for only special cases.Finally, we estimate bounds for the Tsallis relative operator entropy.

Symmetry-resolved entanglement: general considerations, calculation from correlation functions, and bounds for symmetry-protected topological phases

“…Table taken from [33] and extended by the probabilities pn which give a lower bound on the number entropy S N . pn and B ∆ n are defined in equation (37). The shaded areas in the 'cut' column indicate the considered subsystem after taking the partial trace.…”

“…If we can show the above inequality, then it will also be true for S N [ρ Y ] and thus for the maximum of the number entropies of the two subsystems. Since f (p) is concave, we can use Karamata's inequality [36][37][38]. Let σ(i) and α(i) be permutations of (0, 1, .…”

Estimations of the Jensen Gap and Their Applications Based on 6-Convexity

“…Obviously, the function Φ(δ) = exp(δ) is convex and 6-convex because both Φ (δ) = exp(δ) and the function Φ (δ) = exp(δ) are positive. Therefore, by applying (8) for ς = ln b ς , δ ς = a ς , and Φ(δ) = exp (δ), we deduce (28) .…”

“…In recent years, the extensive applications of convex functions and their generalizations have been observed in the field of mathematical inequalities [24][25][26][27]. There are many inequalities, which it would not be possible to establish without convex functions [28][29][30]. Some of the interesting inequalities that have been acquired by convex functions are the majorization inequality [21], Slater's inequality [31], Hermite-Hadamard's inequality [7], Jensen-Mercer's [32] inequality, and many more [33][34][35].…”

A Note on Kantorovich and Ando Inequalities

“…In [8,Lemma 2.1] it is proved that if f : J → R is a convex function and A ∈ B (H) is a self-adjoint operator with spectrum in the interval J, then for any unital positive linear map…”