- Research
- Open access
- Published:
Schur quadratic concavity of the elliptic Neuman mean and its application
Journal of Inequalities and Applications volume 2014, Article number: 397 (2014)
Abstract
For and , we prove that the elliptic Neuman mean is strictly Schur quadratically concave on if and only if . As an application, the bounds for elliptic Neuman mean in terms of the quadratic mean are presented.
MSC:26B25, 26E60.
1 Introduction
Let and . Then the elliptic Neuman mean , see [1], is defined by
where and are the inverse functions of Jacobian elliptic functions cn and nc, see [2, 3], respectively, and . In particular, is the well-known complete elliptic integral of the first kind.
In [1] Neuman proved that is symmetric and homogeneous on , and strictly decreasing with respect to for fixed with . In this context let us note that if a mean is homogeneous, then the order of its homogeneity must be 1; see [4].
Let us recall the notion of Schur quadratic convexity (concavity) [5–7] for a real-valued function on .
A real-valued function is said to be strictly Schur quadratically convex on if for each pair of 2-tuples with and . f is said to be strictly Schur quadratically concave if −f is strictly Schur quadratically convex.
The main purpose of this paper is to present the range of k such that the elliptic Neuman mean is strictly Schur quadratically concave on . As an application, an inequality between the elliptic Neuman mean and the quadratic mean is also given.
2 Two lemmas
In order to prove our main results we need two lemmas, which we present in this section.
Lemma 2.1 (See [[5], Corollary 2.1], [[6], Corollary 1], [[7], Corollary 1])
Suppose that is a continuous symmetric function. If f is differentiable in , then f is strictly Schur quadratically convex on if and only if
for all with , and f is strictly Schur quadratically concave on if and only if inequality (2.1) is reversed.
Lemma 2.2 Let , , and
Then for all if and only if , and there exists such that for and for if .
Proof We distinguish for the proof two cases.
Case 1. . Then from (2.2) one has
for all . (Here and in the sequel, and denote, respectively, the left and right limit of f at t.)
From (2.3) and (2.4) we clearly see that for all .
Case 2. . Then (2.2) leads to
Let
Then simple computations lead to
We distinguish for the proof three subcases.
Subcase 2.1. . Then (2.7) leads to the conclusion that
for all .
Therefore, for all follows from (2.6) and (2.17).
Subcase 2.2. . Then (2.10) and (2.15) lead to
It follows from (2.14) that is strictly increasing on , then (2.16) and (2.19) lead to the conclusion that there exists such that is strictly decreasing on and strictly increasing on .
From (2.12) and (2.13) together with the piecewise monotonicity of we clearly see that there exists such that is strictly decreasing on and strictly increasing on . Then (2.9) and (2.18) lead to the conclusion that
for all .
It follows from (2.7) and (2.8) together with (2.20) that is strictly increasing on . Therefore, for all follows easily from (2.6) and the monotonicity of .
Subcase 2.3. . Then from (2.10) and (2.11) we know that is strictly increasing on and
It follows from (2.9) and (2.21) together with the monotonicity of that there exists such that for and for . Then (2.7) and (2.8) lead to the conclusion that is strictly increasing on and strictly decreasing on . Therefore, there exists such that for and for follows from (2.5) and (2.6) together with the piecewise monotonicity of . □
3 Main results
Theorem 3.1 The elliptic Neuman mean is strictly Schur quadratically concave on if and only if , and is not Schur quadratically convex on if .
Proof Since is symmetric and homogeneous of degree 1, without loss of generality, we assume that . Let , then
Note that
It follows from (1.1) and (3.2) together with (3.3) that
where is defined as in Lemma 2.2.
Therefore, Theorem 3.1 follows easily from Lemmas 2.1 and 2.2 together with (3.4). □
Theorem 3.2 The elliptic Neuman mean is strictly Schur quadratically concave (or convex, respectively) on if and only if the function is strictly increasing (or decreasing, respectively) in , where is the quadratic mean of x and y.
Proof Without loss of generality, we assume that . Let , then from (3.1) and (3.2) together with (3.4) we get
Therefore, Theorem 3.2 follows easily from Lemma 2.1 and (3.5). □
Theorem 3.3 The inequalities
and
hold for all with , and , and is the best possible lower elliptic Neuman mean bound for the quadratic mean .
Proof Without loss of generality, we assume that . Let and . Then
We distinguish for the proof two cases.
Case 1. . Then from Theorems 3.1 and 3.2 we clearly see that is strictly increasing on . Then (3.8)-(3.10) lead to the conclusion that
In particular, for we have
Therefore, inequalities (3.6) and (3.7) follow from (3.11) and (3.12).
Case 2. . Then (3.4) and (3.5) together with the Subcase 2.3 in Lemma 2.2 lead to the conclusion that there exists such that for and for , hence is strictly increasing on and strictly decreasing on . Therefore, for all with follows from (3.8) and (3.10) together with the monotonicity of on , and the optimality of inequality (3.6) follows.
Note that
From (3.8), (3.9), (3.13), and the piecewise monotonicity of we clearly see that
Therefore, inequality (3.7) follows from (3.14). □
References
Neuman E: On one-parameter family of bivariate means. Aequ. Math. 2012,83(1-2):191-197. 10.1007/s00010-011-0099-5
Carlson BC: Special Functions of Applied Mathematics. Academic Press, New York; 1977.
Olver FWJ, Lozier DW, Boisvert RF, Clark CW (Eds): NIST Handbook of Mathematical Functions. Cambridge University Press, Cambridge; 2010.
Matkowski J: Convex functions with respect to a mean and a characterization of quasi-arithmetic means. Real Anal. Exch. 2003/2004,29(1):229-246.
Yang Z-H: Schur power convexity of Gini means. Bull. Korean Math. Soc. 2013,50(2):485-498. 10.4134/BKMS.2013.50.2.485
Yang Z-H: Schur power convexity of the Daróczy means. Math. Inequal. Appl. 2013,16(3):751-762.
Yang Z-H: Schur power convexity of Stolarsky means. Publ. Math. (Debr.) 2012,80(1-2):43-66.
Acknowledgements
This research was supported by the Natural Science Foundation of China under Grants 11171307 and 61374086, and the Natural Science Foundation of Zhejiang Province under Grant LY13A010004.
Author information
Authors and Affiliations
Corresponding author
Additional information
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (https://creativecommons.org/licenses/by/4.0), which permits use, duplication, adaptation, distribution, and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
About this article
Cite this article
Chu, YM., Zhang, Y. & Qiu, SL. Schur quadratic concavity of the elliptic Neuman mean and its application. J Inequal Appl 2014, 397 (2014). https://doi.org/10.1186/1029-242X-2014-397
Received:
Accepted:
Published:
DOI: https://doi.org/10.1186/1029-242X-2014-397