Optimal sublinear inequalities involving geometric and power means

Jiajin Wen; Sui-Sun Cheng; Chaobang Gao

Mathematica Bohemica (2009)

  • Volume: 134, Issue: 2, page 133-149
  • ISSN: 0862-7959

Abstract

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There are many relations involving the geometric means G n ( x ) and power means [ A n ( x γ ) ] 1 / γ for positive n -vectors x . Some of them assume the form of inequalities involving parameters. There then is the question of sharpness, which is quite difficult in general. In this paper we are concerned with inequalities of the form ( 1 - λ ) G n γ ( x ) + λ A n γ ( x ) ≥ A n ( x γ ) and ( 1 - λ ) G n γ ( x ) + λ A n γ ( x ) ≤ A n ( x γ ) with parameters λ ∈ ℝ and γ ∈ ( 0 , 1 ) . We obtain a necessary and sufficient condition for the former inequality, and a sharp condition for the latter. Several applications of our results are also demonstrated.

How to cite

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Wen, Jiajin, Cheng, Sui-Sun, and Gao, Chaobang. "Optimal sublinear inequalities involving geometric and power means." Mathematica Bohemica 134.2 (2009): 133-149. <http://eudml.org/doc/38081>.

@article{Wen2009,
abstract = {There are many relations involving the geometric means $G_n(x)$ and power means $[A_n(x^\{\gamma \})]^\{1/\gamma \}$ for positive $n$-vectors $x$. Some of them assume the form of inequalities involving parameters. There then is the question of sharpness, which is quite difficult in general. In this paper we are concerned with inequalities of the form $(1-\lambda )G_n^\{\gamma \}(x)+\lambda A_n^\{\gamma \}(x)\ge A_n(x^\{\gamma \})$ and $(1-\lambda )G_n^\{\gamma \}(x)+\lambda A_n^\{\gamma \}(x)\le A_n(x^\{\gamma \})$ with parameters $\lambda \in \mathbb \{R\}$ and $\gamma \in (0,1).$ We obtain a necessary and sufficient condition for the former inequality, and a sharp condition for the latter. Several applications of our results are also demonstrated.},
author = {Wen, Jiajin, Cheng, Sui-Sun, Gao, Chaobang},
journal = {Mathematica Bohemica},
keywords = {geometric mean; power mean; Hermitian matrix; permanent of a complex; simplex; arithmetic-geometric inequality; geometric mean; power mean; Hermitian matrix; permanent of a complex; simplex; arithmetic-geometric inequality},
language = {eng},
number = {2},
pages = {133-149},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Optimal sublinear inequalities involving geometric and power means},
url = {http://eudml.org/doc/38081},
volume = {134},
year = {2009},
}

TY - JOUR
AU - Wen, Jiajin
AU - Cheng, Sui-Sun
AU - Gao, Chaobang
TI - Optimal sublinear inequalities involving geometric and power means
JO - Mathematica Bohemica
PY - 2009
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 134
IS - 2
SP - 133
EP - 149
AB - There are many relations involving the geometric means $G_n(x)$ and power means $[A_n(x^{\gamma })]^{1/\gamma }$ for positive $n$-vectors $x$. Some of them assume the form of inequalities involving parameters. There then is the question of sharpness, which is quite difficult in general. In this paper we are concerned with inequalities of the form $(1-\lambda )G_n^{\gamma }(x)+\lambda A_n^{\gamma }(x)\ge A_n(x^{\gamma })$ and $(1-\lambda )G_n^{\gamma }(x)+\lambda A_n^{\gamma }(x)\le A_n(x^{\gamma })$ with parameters $\lambda \in \mathbb {R}$ and $\gamma \in (0,1).$ We obtain a necessary and sufficient condition for the former inequality, and a sharp condition for the latter. Several applications of our results are also demonstrated.
LA - eng
KW - geometric mean; power mean; Hermitian matrix; permanent of a complex; simplex; arithmetic-geometric inequality; geometric mean; power mean; Hermitian matrix; permanent of a complex; simplex; arithmetic-geometric inequality
UR - http://eudml.org/doc/38081
ER -

References

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  3. Pečarić, J. E., Wen, J. J., Wang, W. L., Tao, L., A generalization of Maclaurin's inequalities and its applications, Math. Inequal. Appl. 8 (2005), 583-598. (2005) MR2174887
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  8. Wen, J. J., Wang, W. L., Chebyshev type inequalities involving permanents and their application, Linear Alg. Appl. (2007), 422 295-303. (2007) MR2299014
  9. Wen, J. J., Shi, H. N., Optimizing sharpening for Maclaurin inequality, Journal of Chengdu University (Natural Science Edition) 19 (2000), 1-8 Chinese. (2000) 
  10. Wen, J. J., Wang, W. L., 10.1016/j.camwa.2008.01.032, Computer and Mathematics with Applications 56 (2008), 1045-1058 [Online: http://dx.doi.org/10.1016/j.camwa.2008.01.032]. (2008) MR2435283DOI10.1016/j.camwa.2008.01.032
  11. Wen, J. J., Gao, C. B., Geometric inequalities involving the central distance of the centered 2-surround system, Acta. Math. Sin. 51 (2008), 815-832 Chinese. (2008) Zbl1174.26015MR2454021

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