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Huan-Nan Shi, Da-Mao Li, Jian Zhang, "Refinements of Inequalities among Difference of Means", International Journal of Mathematics and Mathematical Sciences, vol. 2012, Article ID 315697, 15 pages, 2012. https://doi.org/10.1155/2012/315697
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.
In 2005, Taneja  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 . 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 ). 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.
Lemma 2.4 (see ). 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
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 ). 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 .
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.
Theorem II. Let . or , and . Then
The authors are grateful to the referees for their helpful comments and suggestions. The first author was supported in part by the Scientific Research Common Program of Beijing Municipal Commission of Education (KM201011417013).
- I. J. Taneja, “Refinement of inequalities among means,” Journal of Combinatorics, Information & System Sciences, vol. 31, no. 1–4, pp. 343–364, 2006.
- I. J. Taneja, “On a Difference of Jensen Inequality and its Applications to Mean Divergence Measures,” RGMIA Research Report Collection, vol. 7, article 16, no. 4, 2004, http://rgmia.vu.edu.au/.
- I. J. Taneja, “On symmetric and nonsymmetric divergence measures and their generalizations,” Advances in Imaging and Electron Physics, vol. 138, pp. 177–250, 2005.
- A. W. Marshall and I. Olkin, Inequalities: Theory of Majorization and Its Applications, vol. 143 of Mathematics in Science and Engineering, Academic Press, New York, NY, USA, 1979.
- X. Zhang and Y. Chu, “The Schur geometrical convexity of integral arithmetic mean,” International Journal of Pure and Applied Mathematics, vol. 41, no. 7, pp. 919–925, 2007.
- K. Guan, “Schur-convexity of the complete symmetric function,” Mathematical Inequalities & Applications, vol. 9, no. 4, pp. 567–576, 2006.
- K. Guan, “Some properties of a class of symmetric functions,” Journal of Mathematical Analysis and Applications, vol. 336, no. 1, pp. 70–80, 2007.
- C. Stepniak, “An effective characterization of Schur-convex functions with applications,” Journal of Convex Analysis, vol. 14, no. 1, pp. 103–108, 2007.
- H.-N. Shi, “Schur-convex functions related to Hadamard-type inequalities,” Journal of Mathematical Inequalities, vol. 1, no. 1, pp. 127–136, 2007.
- H.-N. Shi, D.-M. Li, and C. Gu, “The Schur-convexity of the mean of a convex function,” Applied Mathematics Letters, vol. 22, no. 6, pp. 932–937, 2009.
- Y. Chu and X. Zhang, “Necessary and sufficient conditions such that extended mean values are Schur-convex or Schur-concave,” Journal of Mathematics of Kyoto University, vol. 48, no. 1, pp. 229–238, 2008.
- N. Elezović and J. Pečarić, “A note on Schur-convex functions,” The Rocky Mountain Journal of Mathematics, vol. 30, no. 3, pp. 853–856, 2000.
- J. Sándor, “The Schur-convexity of Stolarsky and Gini means,” Banach Journal of Mathematical Analysis, vol. 1, no. 2, pp. 212–215, 2007.
- H.-N. Shi, S.-H. Wu, and F. Qi, “An alternative note on the Schur-convexity of the extended mean values,” Mathematical Inequalities & Applications, vol. 9, no. 2, pp. 219–224, 2006.
- X. M. Zhang, Geometrically Convex Functions, An'hui University Press, Hefei, China, 2004.
- H.-N. Shi, Y.-M. Jiang, and W.-D. Jiang, “Schur-convexity and Schur-geometrically concavity of Gini means,” Computers & Mathematics with Applications, vol. 57, no. 2, pp. 266–274, 2009.
- Y. Chu, X. Zhang, and G. Wang, “The Schur geometrical convexity of the extended mean values,” Journal of Convex Analysis, vol. 15, no. 4, pp. 707–718, 2008.
- K. Guan, “A class of symmetric functions for multiplicatively convex function,” Mathematical Inequalities & Applications, vol. 10, no. 4, pp. 745–753, 2007.
- H.-N. Shi, M. Bencze, S.-H. Wu, and D.-M. Li, “Schur convexity of generalized Heronian means involving two parameters,” Journal of Inequalities and Applications, vol. 2008, Article ID 879273, 9 pages, 2008.
- B. Y. Wang, Foundations of Majorization Inequalities, Beijing Normal University Press, Beijing, China, 1990.
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