Related papers: Jensen's Inequalities in Two and Three Dimensions
Convexity is a key concept in information theory, namely via the many implications of Jensen's inequality, such as the non-negativity of the Kullback-Leibler divergence (KLD). Jensen's inequality also underlies the concept of Jensen-Shannon…
We discuss a rather general condition under which the inequality of Jensen works for certain convex combinations of points not all in the domain of convexity of the function under attention. Based on this fact, an extension of the…
We prove a refinement of the inequality by Hoffmann-Jorgensen that is significant for three reasons. First, our result improves on the state-of-the-art even for real-valued random variables. Second, the result unifies several versions in…
Jensen's inequality, attributed to Johan Jensen -- a Danish mathematician and engineer noted for his contributions to the theory of functions -- is a ubiquitous result in convex analysis, providing a fundamental lower bound for the…
In this paper we improve results related to Normalized Jensen Functional for convex functions and Uniformly Convex Functions.
Divergence functions play a key role as to measure the discrepancy between two points in the field of machine learning, statistics and signal processing. Well-known divergences are the Bregman divergences, the Jensen divergences and the…
Comparative convexity is a generalization of convexity relying on abstract notions of means. We define the Jensen divergence and the Jensen diversity from the viewpoint of comparative convexity, and show how to obtain the generalized…
We study the Mercer inequality and its operator extension for superquadratic functions. In particular, we give a more general form of the Mercer inequality by replacing some constants by positive operators. As some consequences, our results…
Famous Redheffer's inequality is generalized to a class of anti-periodic functions. We apply the novel inequality to the generalized trigonometric functions and establish several Redheffer-type inequalities for these functions.
Jensen inequalities for positive linear maps of Choi and Hansen-Pedersen type are established for a large class of operator/matrix means. These results are also extensions of the Minkowski determinantal inequality. To this end we develop…
We study some properties convex functions fulfill. Among the conclusions we obtain from such result, we are able to prove some nontrivial inequalities among real numbers, and we give an improvement of the reverse triangle inequality in the…
Problems pointwise estimates from above functions or its averages often arise in the function theory under known integral restrictions on the growth of this function. We offer an approach to such problems based on the integral Jensen's…
Let $I$ and $J$ be two intervals, and let $f, g: I \rightarrow \mathbb{R}$. If for any points $a$ and $b$ in $I$ and any positive numbers $p$ and $q$ such that $p + q = 1$, we have \begin{align} \nonumber p f(a) + q f(b) + g(pa + qb) \in J,…
We give a Jensen operator inequality for strongly convex functions. As a corollary, we improve operator Holder-McCarthy inequality under suitable conditions.
In this paper we prove results on the difference between a normalized Jensen functional and the sum of other normalized Jensen functionals for convex function.
We extend Dragomir's refinement of Jensen's inequality from the dicrete to the general case, identifying the equality conditions.
Jensen's operator inequality for convexifiable functions is obtained. This result contains classical Jensen's operator inequality as a particular case. As a consequence, a new refinement and a reverse of Young's inequality is given.
We give improvements and generalizations of both the classical Hardy inequality and the geometric Hardy inequality based on the divergence theorem. Especially, our improved Hardy type inequality derives both two Hardy type inequalities with…
We discuss the extension of inequality R_A >= c/a * r_b + b/a * r_c to the plane of triangle ABC. Based on the obtained extension, in regard to all three vertices of the triangle, we get the extension of Erdos-Mordell inequality, and some…
We prove that a symmetric nonnegative function of two variables on a Lebesgue space that satisfies the triangle inequality for almost all triples of points is equivalent to some semimetric. Some other properties of metric triples (spaces…