New Extensions of Popoviciu’s Inequality

Popoviciu’s inequality is extended to the framework of h-convexity and also to convexity with respect to a pair of quasi-arithmetic means. Several applications are included.

     

The converse of the Hermite–Hadamard inequality on simplices

We prove that the Hermite–Hadamard inequality on simplices characterizes convex functions under some assumptions on the measure.     

On the Jensen functional and superquadraticity

In this note we give a recipe which describes upper and lower bounds for the Jensen functional under superquadraticity conditions. Some results involve the Chebychev functional. We give a more general definition of these functionals and establish analogous results.     

Structural results on convexity relative to cost functions

Mass transportation problems appear in various areas of mathematics, their solutions involving cost convex potentials. Fenchel duality also represents an important concept for a wide variety of optimization problems, both from the theoretical and the computational viewpoints. We drew a parallel to the classical theory of convex functions by investigating the cost convexity and its connections with the usual convexity. We give a generalization of Jensen’s inequality for c-convex functions.     

Calculation of the quantum efficiency for the absorption on confinement levels in quantum dots

The quantum efficiency of the absorption on quantum confinement levels is investigated. This is achieved by modeling the electron confinement in a spherical quantum dot (QD). The confinement levels are calculated using both infinite and finite rectangular quantum wells. The spectral internal quantum efficiency is evaluated within both the models, by computing Einstein’s coefficients for the transitions between confinement levels. The size of QDs (1–3 nm radius) leads to negligible many body effects. The nature of the QD material and of the matrix embedding is taken into account in the finite rectangular quantum well approximation and introduces only a small correction. The temperature dependence of the efficiency is also taken into account. A numerical application is performed for a silicon QD of 2.5 nm radius, embedded in amorphous silica. It is proved that the absorption threshold shifts toward the far infrared limit and that the spectral internal quantum efficiency reaches 4–5% at the threshold.     

Ferroelectricity in mixtures of mesogenic and nonmesogenic compounds

It is shown that by adding potassium nitrate (1 % by weight) to compensated cholesteric mixtures (cholesteryl chloride-cholesteryl myristate 63.63:36.37% by weight and cholesteryl laurate-cholesteryl chloride 35.65% by weight) or to smectic sitosteryl undecilenate, the mixtures exhibit ferroelectric behaviour. Values of P_s = 10-⁹ - 10-⁸ C/cm² were obtained in the high-temperature range. In compensated cholesteric mixtures, the spontaneous polarisation reached a minimum at a temperature corresponding to the cholesteric-nematic transition. The results are explained by assuming coupling between the dipoles of the antiferroelectric KNO₃ and the dipoles of the mesogenic compound.     

Mathematical inequalities for some divergences

Some natural phenomena are deviating from standard statistical behavior and their study has increased interest in obtaining new definitions of information measures. But the steps for deriving the best definition of the entropy of a given dynamical system remain unknown. In this paper, we introduce some parametric extended divergences combining Jeffreys divergence and Tsallis entropy defined by generalized logarithmic functions, which lead to new inequalities. In addition, we give lower bounds for one-parameter extended Fermi–Dirac and Bose–Einstein divergences. Finally, we establish some inequalities for the Tsallis entropy, the Tsallis relative entropy and some divergences by the use of the Young’s inequality.