Toader平均的二次与调和平均界

数学物理学报 ›› 2015, Vol. 35 ›› Issue (1): 36-42.

Toader平均的二次与调和平均界

孙惠, 褚玉明

湖南城市学院数学与计算科学学院湖南益阳 413000

收稿日期:2013-04-08 修回日期:2014-02-15 出版日期:2015-02-25 发布日期:2015-02-25
基金资助:
国家自然科学基金(61374086, 11171307)和湖南省教育厅自然科学基金(13C127)资助

Bounds for Toader Mean by Quadratic and Harmonic Means

SUN Hui, CHU Yu-Meng

School of Mathematics and Computation Sciences, Hunan City University, Hunan |Yiyang 413000

Received:2013-04-08 Revised:2014-02-15 Online:2015-02-25 Published:2015-02-25

摘要/Abstract

摘要：

文证明了双向不等式 $\100dpi \inline $\alphaQ(a,b)+(1-\alpha)H(a,b)<T(a,b)<\beta Q(a,b)+(1-\beta)H(a,b)$$ 和
$\100dpi \inline $\lambda/H(a,b)+(1-\lambda)/Q(a,b)<1/T(a,b)<\mu/H(a,b)+(1-\mu)/Q(a,b)$$ 对所有a, b>0 且 $\100dpi \inline $a\neq b$$ 成立的充分和必要条件是
$\100dpi \inline $\alpha\leq 5/6$, $\beta\geq 2\sqrt{2}/\pi$, $\lambda\leq 0$$ 和 $\100dpi \inline $\mu\geq 1/6$$ . 其中 $\100dpi \inline $Q(a,b)=\sqrt{(a^{2}+b^{2})/2}$, $H(a,b)=2ab/(a+b)$$ 和

$\100dpi \inline $T(a,b)=\frac{2}{\pi}\int_{0}^{{\pi}/{2}}\sqrt{a^2{\cos^2{\theta}}+b^2{\sin^2{\theta}}}{\rm d}\theta$$

分别表示正数 $\100dpi \inline $a$$ 和 $\100dpi \inline $b$$ 的二次平均, 调和平均和 Toader 平均.

关键词: 二次平均 , 调和平均 , Toader 平均

Abstract:

We prove that the double inequalities $\100dpi \inline $\alphaQ(a,b)+(1-\alpha)H(a,b)<T(a,b)<\beta Q(a,b)+(1-\beta)H(a,b)$$ and
$\100dpi \inline $\lambda/H(a,b)+(1-\lambda)/Q(a,b)<1/T(a,b)<\mu/H(a,b)+(1-\mu)/Q(a,b)$$ hold for all a,b>0 with $\100dpi \inline $a\neq b$$ if and only if $\100dpi \inline $\alpha\leq5/6$, $\beta\geq 2\sqrt{2}/\pi$, $\lambda\leq 0$$ and $\100dpi \inline $\mu\geq 1/6$$ .Here, $\100dpi \inline $Q(a,b)=\sqrt{(a^{2}+b^{2})/2}$, $H(a,b)=2ab/(a+b)$$ and
$\100dpi \inline $T(a,b)=\frac{2}{\pi}\int_{0}^{{\pi}/{2}}\sqrt{a^2{\cos^2{\theta}}+b^2{\sin^2{\theta}}}{\rm d}\theta$$ denote the quadratic, harmonic and Toader means of two positive
numbers $\100dpi \inline $a$$ and $\100dpi \inline $b$$ , respectively.

Key words: Quadratic mean , Harmonic mean , Toader mean

中图分类号:

26E60

孙惠, 褚玉明. Toader平均的二次与调和平均界[J]. 数学物理学报, 2015, 35(1): 36-42.

SUN Hui, CHU Yu-Meng. Bounds for Toader Mean by Quadratic and Harmonic Means[J]. Acta mathematica scientia,Series A, 2015, 35(1): 36-42.

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