Determination of Bounds for the Jensen Gap and Its Applications

The Jensen inequality has been reported as one of the most consequential inequalities that has a lot of applications in diverse fields of science. For this reason, the Jensen inequality has become one of the most discussed developmental inequalities in the current literature on mathematical inequalities. The main intention of this article is to find some novel bounds for the Jensen difference while using some classes of twice differentiable convex functions. We obtain the proposed bounds by utilizing the power mean and Höilder inequalities, the notion of convexity and the prominent Jensen inequality for concave function. We deduce several inequalities for power and quasi-arithmetic means as a consequence of main results. Furthermore, we also establish different improvements for Hölder inequality with the help of obtained results. Moreover, we present some applications of the main results in information theory.

Some remarks on the stability of the multi-Jensen equation

First a stability result of Prager-Schwaiger [Prager W., Schwaiger J., Stability of the multi-Jensen equation, Bull. Korean Math. Soc., 2008, 45(1), 133–142] is generalized by admitting more general domains of the involved function and by allowing the bound to be not constant. Next a result by Cieplinski [Cieplinski K., On multi-Jensen functions and Jensen difference, Bull. Korean Math. Soc., 2008, 45(4), 729–737] is discussed. Finally a characterization of the completeness of a normed space in terms of stability requirements for multi-Jensen functions is presented.

Divergence Function, Duality, and Convex Analysis

From a smooth, strictly convex function Φ: Rn → R, a parametric family of divergence function DΦ(α) may be introduced: [Formula: see text] for x, y, ε int dom(Φ) and for α ε R, with DΦ(±1 defined through taking the limit of α. Each member is shown to induce an α-independent Riemannian metric, as well as a pair of dual α-connections, which are generally nonflat, except for α = ±1. In the latter case, D(±1)Φ reduces to the (nonparametric) Bregman divergence, which is representable using and its convex conjugate Φ * and becomes the canonical divergence for dually flat spaces (Amari, 1982, 1985; Amari & Nagaoka, 2000). This formulation based on convex analysis naturally extends the information-geometric interpretation of divergence functions (Eguchi, 1983) to allow the distinction between two different kinds of duality: referential duality (α -α) and representational duality (Φ  Φ *). When applied to (not necessarily normalized) probability densities, the concept of conjugated representations of densities is introduced, so that ± α-connections defined on probability densities embody both referential and representational duality and are hence themselves bidual. When restricted to a finite-dimensional affine submanifold, the natural parameters of a certain representation of densities and the expectation parameters under its conjugate representation form biorthogonal coordinates. The alpha representation (indexed by β now, β ε [−1, 1]) is shown to be the only measure-invariant representation. The resulting two-parameter family of divergence functionals D(α, β), (α, β) ε [−1, 1] × [-1, 1] induces identical Fisher information but bidual alpha-connection pairs; it reduces in form to Amari's alpha-divergence family when α =±1 or when β = 1, but to the family of Jensen difference (Rao, 1987) when β = 1.

GENERALIZED MEAN OF ORDER $t$ VIA BOX AND COX'S TRANSFORMATION

Mean of order $t$ and Box and Cox's transformation function are very famous in the literature of mathematics and statistics respectively. In this paper, we have derived some standard inequalities from the mean of order $t$ and studied some interesting properties of Box and Cox's transformation function. A compos- iterelation of these two measures, calling generalized mean of order $t$ or unified $(t,s)$-mean is considered. The unified $(t,s)$-mean leads us to very important gen- eralized information theoretic measures. These measures include generalizations of Shannon's entropy, Kullback-leibler's relative information, Kerridge's inaccu- racy, J-divergence, Jensen difference dive~gence measure, etc. Properties of unified $(t,s)$-mean are also studied.