Means And The Chebyshev Functional

Although means is an ancient concept, their algebraic and geometric appealtogether with their numerous applications in areas such as probability, statistics,and engineering made means an indispensable tool in science and have kept thesubject alive and dynamically growing.From a general point of view, mean is a kind of multivariate function. Meanscan divide into two types: symmetric means and nonsymmetric means. Symmetricmeans have excellent properties and wide applications. In geometry and topology,many basic invariants are just defned in terms of the symmetric means, for in-stance, the rth mean curvature of a Riemannian submanifold et cetera. We mainlyfocus on the symmetric means in the following.In the concrete bivariate symmetric means, it is well known that arithmetic,geometric and harmonic means are special cases of power and the frst type gen-eralized logarithmic means. However, it still unknown that the power or the frsttype generalized logarithmic means bounds for the combination of arithmetic, geo-metric and harmonic means. So, we give the lower and upper power mean boundsfor the geometric combination of arithmetic, geometric and harmonic means, thelower and upper power mean bounds for the geometric mean of two power means,the frst type generalized logarithmic mean lower (or upper) bound for the convexcombination of arithmetic and geometric means. The frst, second Seifert andNeuman-Sándor means are three new means without parameters, and they are notspecial cases of any known means with parameters. So, it is meaningful to com-pare this three means with other means which have parameters. The second typegeneralized logarithmic means is an new means with one parameter which general-ized from classical logarithmic means. The research result about Neuman-Sa′ndorand the second type generalized logarithmic means is almost blank. We obtainthe second type generalized logarithmic mean lower bounds for the frst, secondSeifert and Neuman-Sándor means. Meanwhile, the conclusion are also obtainedthat it does not exists the second type generalized logarithmic mean upper boundsfor the frst, second Seifert and Neuman-Sándor means. All the bounds obtainedby us are sharp!Relative to concrete means, abstract means have more generality. So thereis important academic signifcance and value to investigate the abstract means. The weighted arithmetic integral means is the most important and basic abstractmean, and the generalized weighted quasi-arithmetic integral mean is the gener-alization of the weighted arithmetic integral mean. In recent years, a large num-ber of scholars have made many achievements in the weighted arithmetic integralmean, but the study on the generalized weighted quasi-arithmetic integral meanhas just begun. Using the convexity, Jensen inequality and Chebyshev inequality,we compare diferent generalized weighted quasi-arithmetic integral means. Somesufcient conditions are obtained. Furthermore, a type of integral means whichdefned by Toader and Sándor in [42] with a integral expression are investigatedin the same way.Abstract means have many applications; and Chebyshev functional is one ofthe most important applications. Due to the importance of its theory and wide ap-plicability in integral transforms, probability problems and the bounding of specialfunctions, the Chebyshev functional, which is defned by use of arithmetic integralmean, draws the attention of the mathematician for a long time. Particularly, toestimate the Chebyshev functional is the best important problem in Chebyshevfunctional theory. Due to predecessors’ work, it is toward true perfect to boundthe classical (weighted) Chebyshev functional. But the related results for bound-ing the generalized Chebyshev functional which defned over two diferent inter-vals are very rare. So, based on an identity for generalized Chebyshev functional,which established by us, we estimate the generalized Chebyshev functional. As theapplication of the Chebyshev functional, making use of some identities of Cheby-shev functional, Grüss inequality and constructing appropriate kernel, we get anOstrowski-Grüss type integral inequalities which generalized the Ostrowski-Grüssinequality. Meanwhile, making use of Chebyshev inequality and Grüss inequality,we provide several integral inequalities involving Taylor’s remainder.Schur convexity plays a important role in the research of the multivariate sym-metric function. Combination of Schur convexity with the majorization theory isan important method of conducing and producing new inequalities. To discuss theSchur convexity of some concrete or abstract multivariate functions is hot issue fora long time. The sufcient and necessary conditions for the weighted arithmeticintegral mean and the weighted Chebyshev functional are already established. Butthe criterion for Schur geometrical convexity and Schur harmonic convexity of theweighted arithmetic integral mean and the weighted Chebyshev functional are notbeen established. So, we explore the sufcient and necessary conditions for Schur geometrical convexity and Schur harmonic convexity of weighted arithmetic inte-gral mean. Furthermore, the ease-to-operate sufcient conditions for Schur con-vexity, Schur geometrical and harmonic convexities of weighted arithmetic integralmean are provided. By an identity for the weighted Chebyshev functional, we fndthat the weighted Chebyshev functional can be seen as the weighted arithmeticintegral mean of a function. Based on the conditions for Schur convexity of weight-ed arithmetic integral mean, similar conditions for weighted Chebyshev functionalare presented. At last, Schur convexity of several multivariate symmetric functionsare discussed and some new inequalities are established by combining the obtainedresults of Schur convexity with the majorization theory.