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The classical AM-GM inequality has been generalized in a number of ways. Generalizations which incorporate variance appear to be the most useful in economics and finance, as well as mathematically natural. Previous work leaves unanswered…
This paper is dedicated to the analysis and detailed study of a procedure to generate both the weighted arithmetic and harmonic means of $n$ positive real numbers. Together with this interpretation, we prove some relevant properties that…
The problem of classification of cubic homogeneous Finslerian 3D metrics with respect to their isometries is considered. It is shown, that there are 6 different general affine types of such metrics. Algebras of isometries are presented in…
We offer new proofs, refinements as well as new results related to classical means of two variables, including the identric and logarithmic means.
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.…
Many classical geometric inequalities on functionals of convex bodies depend on the dimension of the ambient space. We show that this dimension dependence may often be replaced (totally or partially) by different symmetry measures of the…
We complement a recent result of S. Furuichi, by showing that the differences $\sum_{i=1}^n \alpha_i x_i - \prod_{i=1}^n x_i^{\alpha_i}$ associated to distinct sequences of weights are comparable, with constants that depend on the smallest…
We introduce some symmetric homogeneous means, and then show unitarily invariant norm inequalities for them, applying the method established by Hiai and Kosaki. Our new inequalities give the tighter bounds of the logarithmic mean than the…
We give a simpler proof of a result of Holland concerning a mixed arithmetic-geometric mean inequality. We also prove a result of mixed mean inequality involving the symmetric means.
Some new Hermite-Hadamard's inequalities for h-convex functions are proved, generalizing and unifying a number of known results. Some new applications for special Means of real numbers are also derived.
The concept of Loewner (partial) order for general complex matrices is introduced. After giving the definition of arithmetic, geometric, and harmonic mean for accretive-dissipative matrices, we study their basic properties. An AM-GM-HM…
Analogical proportions compare pairs of items (a, b) and (c, d) in terms of their differences and similarities. They play a key role in the formalization of analogical inference. The paper first discusses how to improve analogical inference…
This paper deals with four symmetrizations of a convex set $C$: the intersection, the harmonic and the arithmetic mean, and the convex hull of $C$ and $-C$. A well-known result of Firey shows that those means build up a subset-chain in the…
Motivated by the refinements and reverses of arithmetic-geometric mean and arithmetic-harmonic mean inequalities for scalars and matrices, in this article, we generalize the scalar and matrix inequalities for the difference between…
The main goal of this paper is to discuss the recent advancements of operator means for accretive matrices in a more general setting. In particular, we present the general form governing the well established definition of geometric mean,…
Let m(a,b) and M(a,b,c) be symmetric means. We say that M is type 1 invariant with respect to m if M(m(a,c),m(a,b),m(b,c)) = M(a,b,c) for all a, b, c > 0. If m is strict and isotone, then we show that there exists a unique M which is type 1…
In this note we prove an inequality for t-geometric means that immediately implies the recent results of Audenaert [2] and Hayajneh-Kittaneh [6].
In this article, we present the best possible upper and lower bounds for the Neuman-S\'andor mean in terms of the geometric combinations of harmonic and quadratic means, geometric and quadratic means, harmonic and contra-harmonic means, and…
Lie derivatives of various geometrical and physical quantities define symmetries and conformal symmetries in general relativity. Thus we obtain motions, collineations, conformal motions and conformal collineations. These symmetries are used…
We are going to classify sets by a given mean in two ways. Firstly we study small and big sets regarding a given mean. Secondly we study sets that have the same weight according to a mean. We also generalize the notion of roundness and get…