Related papers: Refinement of Inequalities among Means
Spherical means are well-known useful tool in the theory of partial differential equations with applications to solving hyperbolic and ultrahyperbolic equations and problems of integral geometry, tomography and Radon transforms. We…
In this paper we first introduce the Heron and Heinz means of two convex functionals. Afterwards, some inequalities involving these functional means are investigated. The operator versions of our theoretical functional results are…
It is well-known that if a real valued function acting on a convex set satisfies the $n$-variable Jensen inequality, for some natural number $n\geq 2$, then, for all $k\in\{1,\dots, n\}$, it fulfills the $k$-variable Jensen inequality as…
In this work, we present continued fractions for the arithmetic, geometric, harmonic and cotangent means of $[a_0,a_1,\dots,a_k]$ and $[a_0,a_1,\dots,a_k,a_{k+1}]$, and some of their applications.
We obtain simple proofs of certain inequalites for bivariate means.
This paper provides an overview of the Pythagorean centrality measures, which are the arithmetic, geometric, and harmonic means. Both the evolution of their meaning through history and their geometrical interpretation are outlined. Relevant…
In this paper, we give the explicit formulas for the Neuman means $N_{AH}$, $N_{HA}$, $N_{AC}$ and $N_{CA}$, and present the best possible upper and lower bounds for theses means in terms of the combinations of harmonic mean $H$, arithmetic…
The notion of an equational shell is studied to involve the objects and their environment. Appropriate methods are studied as valid embeddings of refined objects. The refinement process determines the linkages between the variety of…
In this paper, we present some extensions of interpolation between the arithmetic-geometric means inequality. Among other inequalities, it is shown that if $A, B, X$ are $n\times n$ matrices, then \begin{align*}…
In this paper we have considered two one parametric generalizations. These two generalizations have in articular the well known measures such as: J-divergence, Jensen-Shannon divergence and Arithmetic-Geometric mean divergence. These three…
In this paper we have considered a difference of Jensen's inequality for convex functions and proved some of its properties. In particular, we have obtained results for Csisz\'{a}r \cite{csi1} $f-$divergence. A result is established that…
We start to investigate how small changes on the definition of ordinary means affect their properties. Especially the property of being a mean. In that direction we are looking for weakenings of the basic defining property of means. Hence…
The aim of this paper is to characterize in broad classes of means the so-called Hardy means, i.e., those means $M\colon\bigcup_{n=1}^\infty \mathbb{R}_+^n\to\mathbb{R}_+$ that satisfy the inequality $$ \sum_{n=1}^\infty M(x_1,\dots,x_n)…
We define so-called residual means, which have a Taylor expansion of the form $M(x)=\bar x +\tfrac12 \xi_M(\bar x) \text{Var}(x)+o(\|x-\bar x\|^\alpha)$ for some $\alpha>2$ and a single-variable function $\xi_M$ ($\bar x$ stands for the…
The goal of this note is to provide a geometric setting in which generalized arithmetic means are best predictors in an appropriate metric. This characterization provides a geometric interpretation to the concept of certainty equivalent.…
Using known entropic and information inequalities new inequalities for some classical polynomials are obtained. Examples of Jacobi and Legendre polynomials are considered.
New measures for the quantization of systems with constraints are discussed and applied to several examples, in particular, examples of alternative but equivalent formulations of given first-class constraints, as well as a comparison of…
In this paper we present a new approach to proving some exponential inequalities involving the sinc function. Power series expansions are used to generate new polynomial inequalities that are sufficient to prove the given exponential…
Binomial coefficients have been used for centuries in a variety of fields and have accumulated numerous definitions. In this paper, we introduce a new way of defining binomial coefficients as repeated sums of ones. A multitude of binomial…
We present correspondences induced by some classical mappings between measures on an interval and measures on the unit circle. More precisely, we link their sequences of orthogonal polynomial and their recursion coefficients. We also deduce…