Inequalities between Power Means and Convex Combinations of the Harmonic and Logarithmic Means
نویسندگان
چکیده
منابع مشابه
Inequalities between Power Means and Convex Combinations of the Harmonic and Logarithmic Means
We prove that αH a, b 1 − α L a, b > M 1−4α /3 a, b for α ∈ 0, 1 and all a, b > 0 with a/ b if and only if α ∈ 1/4, 1 and αH a, b 1 − α L a, b < M 1−4α /3 a, b if and only if α ∈ 0, 3√345/80 − 11/16 , and the parameter 1 − 4α /3 is the best possible in either case. Here, H a, b 2ab/ a b , L a, b a − b / loga − log b , and Mp a, b a b /2 1/p p / 0 and M0 a, b √ ab are the harmonic, logarithmic, ...
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For all $a,b>0$, the following two optimal inequalities are presented: $H^{alpha}(a,b)L^{1-alpha}(a,b)geq M_{frac{1-4alpha}{3}}(a,b)$ for $alphain[frac{1}{4},1)$, and $ H^{alpha}(a,b)L^{1-alpha}(a,b)leq M_{frac{1-4alpha}{3}}(a,b)$ for $alphain(0,frac{3sqrt{5}-5}{40}]$. Here, $H(a,b)$, $L(a,b)$, and $M_p(a,b)$ denote the harmonic, logarithmic, and power means of order $p$ of two positive numbers...
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for all $a,b>0$, the following two optimal inequalities are presented: $h^{alpha}(a,b)l^{1-alpha}(a,b)geq m_{frac{1-4alpha}{3}}(a,b)$ for $alphain[frac{1}{4},1)$, and $ h^{alpha}(a,b)l^{1-alpha}(a,b)leq m_{frac{1-4alpha}{3}}(a,b)$ for $alphain(0,frac{3sqrt{5}-5}{40}]$. here, $h(a,b)$, $l(a,b)$, and $m_p(a,b)$ denote the harmonic, logarithmic, and power means of order $p$ of two positive numbers...
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ژورنال
عنوان ژورنال: Journal of Applied Mathematics
سال: 2012
ISSN: 1110-757X,1687-0042
DOI: 10.1155/2012/471096