International Journal of Mathematics and Mathematical Sciences

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 Schur-concave function on if and only if is Schur-convex.

Definition 2.2 (see [15]). Let and . (i) is called a geometrically convex set if for all , and , such that . (ii) Let . The function : is said to be Schur-geometrically convex function on if on implies . The function is said to be a Schur-geometrically concave on if and only if is Schur-geometrically convex.

Definition 2.3 (see [4, 20]). (i) The set is called symmetric set, if implies for every permutation matrix .
(ii) The function is called symmetric if, for every permutation matrix , for all .

Lemma 2.4 (see [15]). Let be a symmetric and geometrically convex set with a nonempty interior . Let be continuous on and differentiable in . If is symmetric on and holds for any , then is a Schur-geometrically convex (Schur-geometrically concave) function.

Lemma 2.5. For one has

Proof. It is easy to see that the left-hand inequality in (2.2) is equivalent to , and the right-hand inequality in (2.2) is equivalent to that is, Indeed, from the left-hand inequality in (2.2) we have so the right-hand inequality in (2.2) holds.
The inequality in (2.3) is equivalent to Since so it is sufficient prove that that is, and, from the left-hand inequalities in (2.2), we have so the inequality in (2.3) holds.
Notice that the functions in the inequalities (2.4) are homogeneous. So, without loss of generality, we may assume , and set . Then and (2.4) reduces to Squaring every side in the above inequalities yields Reducing to common denominator and rearranging, the right-hand inequality in (2.14) reduces to and the left-hand inequality in (2.14) reduces to so two inequalities in (2.4) hold.

Lemma 2.6 (see [16]). Let . If or , then

3. Proof of Main Result

Proof of Theorem I. Let .
(1) For we have whence From (2.3) we have which implies and, by Lemma 2.4, it follows that is Schur-geometrically convex in .
(2) For
To prove that the function is Schur-geometrically convex in it is enough to notice that .
(3) For we have and then
From (2.2) we have , so by Lemma 2.4, it follows that is Schur-geometrically convex in .
(4) For we have and then By (2.2) we infer that so . By Lemma 2.4, we get that is Schur-geometrically convex in .
(5) For we have and then From (2.4) we have so ; it follows that is Schur-geometrically convex in .
(6) For we have and then By (2.4) we infer that , which proves that is Schur-geometrically convex in .
(7) For we have and then From (2.4) we have , and, consequently, by Lemma 2.4, we obtain that is Schur-geometrically convex in .
(8) In order to prove that the function is Schur-geometrically convex in it is enough to notice that
(9) For we have and then From (2.2) and (2.4) we obtain that so , which proves that the function is Schur-geometrically convex in .
(10) One can easily check that and, consequently, the function is Schur-geometrically convex in .
(11) To prove that the function is Schur-geometrically convex in it is enough to notice that
(12) For we have and then
By the inequality (2.2) we get that , which proves that is Schur-geometrically convex in .
(13) It is easy to check that which means that the function is Schur-geometrically convex in .
(14) To prove that the function is Schur-geometrically convex in it is enough to notice that
The proof of Theorem I is complete.