| 摘要: |
| 本文研究了算术-调和平均不等式的加细.首先利用经典分析的方法给出了关于标量情形的不等式，进而推广到算子的情形，得出了若0 < ν，τ < 1，a，b > 0且使（b-a）（τ-ν）> 0，则有a∇νb-a!νb/a∇τb-a!τb ≤ ν（1-ν）/τ（1-τ）及（a∇νb）2-（a!νb）2/（a∇τb）2-（a!τb）2 ≤ ν（1-ν）/τ（1-τ）.推广了W.Liao等人的结果. |
| 关键词: 算术-调和平均 算子不等式 Hilbert-Schmidt范数 |
| DOI： |
| 分类号:O177.1 |
| 基金项目:Supported by National Natural Science Foundation of China (11271112; 11771126; 11701154). |
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| IMPROVE INEQUALITIES OF ARITHMETIC-HARMONIC MEAN |
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YANG Chang-sen, REN Yong-hui, ZHANG Hai-xia
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College of Mathematics and Information Science, Henan Normal University, Xinxiang 453007, China
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| Abstract: |
| We study the refinement of arithmetic-harmonic mean inequalities. First, through the classical analysis method, the scalar inequalities are obtained, and then extended to the operator cases. Specifically, we have the following main results:for 0 < ν, τ < 1, a, b > 0 with (b-a)(τ-ν) >0, we have a∇νb-a!νb/a∇τb-a!τb ≤ ν(1-ν)/τ(1-τ) and (a∇νb)2-(a!νb)2/(a∇τb)2-(a!τb)2 ≤ ν(1-ν)/τ(1-τ), which are generalizations of the results of W. Liao et al. |
| Key words: arithmetic-harmonic mean operator inequality Hilbert-Schmidt norm |
