Refinements of Inequalities among Difference of Means

In this paper, for the difference of famous means discussed by Taneja in 2005, we study the Schur-geometric convexity in (0, ∞) × (0, ∞) of the difference between them. Moreover some inequalities related to the difference of those means are obtained. 1. Introduction In 2005, Taneja [1] proved the following chain of inequalities for the binary means for : where The means , , , , ？ and are called, respectively, the arithmetic mean, the geometric mean, the harmonic mean, the root-square mean, the square-root mean, and Heron’s mean. The one can be found in Taneja [2, 3]. Furthermore Taneja considered the following difference of means: and established the following. Theorem A. The difference of means given by (1.4) is nonnegative and convex in . Further, using Theorem A, Taneja proved several chains of inequalities; they are refinements of inequalities in (1.1). Theorem B. The following inequalities among the mean differences hold: For the difference of means given by (1.4), we study the Schur-geometric convexity of difference between these differences in order to further improve the inequalities in (1.1). The main result of this paper reads as follows. Theorem I. The following differences are Schur-geometrically convex in : The proof of this theorem will be given in Section 3. Applying this result, in Section 4, we prove some inequalities related to the considered differences of means. Obtained inequalities are refinements of inequalities (1.5)–(1.9). 2. Definitions and Auxiliary Lemmas The Schur-convex function was introduced by Schur in 1923, and it has many important applications in analytic inequalities, linear regression, graphs and matrices, combinatorial optimization, information-theoretic topics, Gamma functions, stochastic orderings, reliability, and other related fields (cf. [4–14]). In 2003, Zhang first proposed concepts of “Schur-geometrically convex function” which is extension of “Schur-convex function” and established corresponding decision theorem [15]. Since then, Schur-geometric convexity has evoked the interest of many researchers and numerous applications and extensions have appeared in the literature (cf. [16–19]). In order to prove the main result of this paper we need the following definitions and auxiliary lemmas. Definition 2.1 (see [4, 20]). Let and . (i) is said to be majorized by (in symbols ) if for and , where and are rearrangements of and in a descending order.(ii) is called a convex set if for every and , where and with . (iii) Let . The function : is said to be a Schur-convex function on if on implies is said to be a