Refinements of Jensen’s Inequality via Majorization Results with Applications in the Information Theory

In this study, we present some new refinements of the Jensen inequality with the help of majorization results. We use the concept of convexity along with the theory of majorization and obtain refinements of the Jensen inequality. Moreover, as consequences of the refined Jensen inequality, we derive some bounds for power means and quasiarithmetic means. Furthermore, as applications of the refined Jensen inequality, we give some bounds for divergences, Shannon entropy, and various distances associated with probability distributions.

On the Jensen convex and Jensen concave envelopes of means

In recent papers, the convexity of quasiarithmetic means was characterized under twice differentiability assumptions. One of the main goals of this paper is to show that the convexity or concavity of a quasiarithmetic mean implies the twice continuous differentiability of its generator. As a consequence of this result, we can characterize those quasiarithmetic means which admit a lower convex and upper concave quasiarithmetic envelope.

Clustering Methods Based on Weighted Quasi-Arithmetic Means of T-Transitive Fuzzy Relations

In this paper we propose clustering methods based on weighted quasiarithmetic means of T-transitive fuzzy relations. We first generate a T-transitive closure RT from a proximity relation R based on a max-T composition and produce a T-transitive lower approximation or opening RT from the proximity relation R through the residuation operator. We then aggregate a new T-indistinguishability fuzzy relation by using a weighted quasiarithmetic mean of RT and RT. A clustering algorithm based on the proposed T-indistinguishability is thus created. We compare clustering results from three critical ti-indistinguishabilities: minimum (t3), product (t2), and Łukasiewicz (t1). A weighted quasiarithmetic mean of a t1-transitive closure [Formula: see text] and a t1-transitive lower approximation or opening [Formula: see text] with the weight [Formula: see text], demonstrates the superiority and usefulness of clustering begun by using a proximity relation R based on the proposed clustering algorithm. The algorithm is then applied to the practical evaluation of the performance of higher education in Taiwan.

Special Issue on Aggregation Operators and Clustering

This issue features decision making and other tools used in artificial intelligence applications. More specifically, the issue includes five papers focused on aggregation operators and clustering. The series starts with a paper by Yoshida on weighted quasiarithmetic means that focuses on their monotonicity viewed from utility and weighting functions. In the second paper, Nohmi, Honda and Okazaki focus on trust evaluation for networks, studying matrix operations based on t-norms and t-conorms. The authors also propose fuzzy graphs using adjacent matrices. These works are followed by three on fuzzy clustering. Kanzawa, Endo and Miyamoto present a variation of fuzzy c-means based on kernel functions in an approach developed for data with tolerance. Endo covers clustering using kernel functions. The paper is based on a fuzzy nonmetric model including pairwise constraints in the clustering process. The concluding paper also uses pairwise constraints, but within agglomerative hierarchical clustering. Hamasuna, Endo and Miyamoto include clusterwise tolerance in their mode. As the editors of this issue, we would like to thank the referees for their work in the reviews and journal editors-in-chief Profs. Toshio Fukuda and Kaoru Hirota and the journal staff for their support.

Stability of some functional equations defined by quasiarithmetic means

We obtain results on Hyers-Ulam stability for some functional equations defined by quasiarithmetic means.

The Inequalities for Quasiarithmetic Means

Overview and refinements of the results are given for discrete, integral, functional and operator variants of inequalities for quasiarithmetic means. The general results are applied to further refinements of the power means. Jensen's inequalities have been systematically presented, from the older variants, to the most recent ones for the operators without operator convexity.

Extension of Jensen's Inequality for Operators without Operator Convexity

We give an extension of Jensen's inequality for -tuples of self-adjoint operators, unital -tuples of positive linear mappings, and real-valued continuous convex functions with conditions on the operators' bounds. We also study operator quasiarithmetic means under the same conditions.