Properties of the power-mean and their applications

Jing-Feng Tian; Ming-Hu Ha; Hong-Jie Xing; Jing-Feng Tian; Ming-Hu Ha; Hong-Jie Xing

doi:10.3934/math.2020466

AIMS Mathematics

2020, Volume 5, Issue 6: 7285-7300. doi: 10.3934/math.2020466

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Research article

Properties of the power-mean and their applications

1.
School of Management, Hebei University, Wusi Road 180, Baoding 071002, P. R. China
2.
Department of Mathematics and Physics, North China Electric Power University, Yonghua Street 619, Baoding 071003, P. R. China
3.
School of Science, Hebei University of Engineering, Taiji Road 19, Handan 056038, P. R. China
4.
College of Mathematics and Information Science, Hebei University, Wusi Road 180, Baoding 071002, P. R. China

Received: 11 August 2020 Accepted: 15 September 2020 Published: 17 September 2020
MSC : 26E60, 26A51

Suppose $w, v>0$, $w\neq v$ and $A_{u}\left (w, v\right) $ is the $u$-order power mean (PM) of $w$ and $v$. In this paper, we completely describe the convexity of $u\mapsto A_{u}\left (w, v\right) $ on $\mathbb{R}$ and $% s\mapsto A_{u\left (s\right) }\left (w, v\right) $ with $u\left (s\right) = \left (\ln 2\right) /\ln \left (1/s\right) $ on $\left (0, \infty \right) $. These yield some new inequalities for PMs, and give an answer to an open problem.
- power mean,
- power-type mean,
- convexity,
- inequality
Citation: Jing-Feng Tian, Ming-Hu Ha, Hong-Jie Xing. Properties of the power-mean and their applications[J]. AIMS Mathematics, 2020, 5(6): 7285-7300. doi: 10.3934/math.2020466

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Abstract

Suppose $w, v>0$, $w\neq v$ and $A_{u}\left (w, v\right) $ is the $u$-order power mean (PM) of $w$ and $v$. In this paper, we completely describe the convexity of $u\mapsto A_{u}\left (w, v\right) $ on $\mathbb{R}$ and $% s\mapsto A_{u\left (s\right) }\left (w, v\right) $ with $u\left (s\right) = \left (\ln 2\right) /\ln \left (1/s\right) $ on $\left (0, \infty \right) $. These yield some new inequalities for PMs, and give an answer to an open problem.

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