Research article

Properties of the power-mean and their applications

  • 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.

    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

    Related Papers:

  • 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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