Best Possible Inequalities between Generalized Logarithmic Mean and Classical Means
نویسندگان
چکیده
منابع مشابه
Best Possible Inequalities between Generalized Logarithmic Mean and Classical Means
and Applied Analysis 3 Theorem B. For all positive real numbers a and b with a/ b, we have √ G a, b A a, b < √ L a, b I a, b
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In this paper, we find the greatest value α and the least values β , p , q and r in (0,1/2) such that the inequalities L(αa+ (1−α)b,αb+ (1−α)a) < P(a,b) < L(βa + (1− β)b,βb + (1− β)a) , H(pa + (1− p)b, pb + (1− p)a) > G(a,b) , H(qa+ (1− q)b,qb +(1− q)a) > L(a,b) , and G(ra+(1− r)b,rb+(1− r)a) > L(a,b) hold for all a,b > 0 with a = b . Here, H(a,b) , G(a,b) , L(a,b) and P(a,b) denote the harmoni...
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For p ∈ R, the generalized logarithmic mean Lp of two positive numbers a and b is defined as Lp a, b a, for a b, LP a, b b 1 − a 1 / p 1 b − a 1/p , for a/ b, p / − 1, p / 0, LP a, b b − a / log b − loga , for a/ b, p −1, and LP a, b 1/e b/a 1/ b−a , for a/ b, p 0. In this paper, we prove that G a, b H a, b 2L−7/2 a, b , A a, b H a, b 2L−2 a, b , and L−5 a, b H a, b for all a, b > 0, and the co...
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Copyright q 2010 B.-Y. Long and Y.-M. Chu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. For p ∈ R, the generalized logarithmic mean L p a, b, arithmetic mean Aa, b, and geometric mean Ga, b of two positive numbers a and b are d...
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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: Abstract and Applied Analysis
سال: 2010
ISSN: 1085-3375,1687-0409
DOI: 10.1155/2010/303286