The inverse mean problem of geometric mean and contraharmonic means
نویسندگان
چکیده
منابع مشابه
Sharp Bounds for Seiffert Mean in Terms of Contraharmonic Mean
and Applied Analysis 3 2. Proof of Theorem 1.1 Proof of Theorem 1.1. Let λ 1 √ 4/π − 1 /2 and μ 3 √3 /6. We first proof that the inequalities T a, b > C λa 1 − λ b, λb 1 − λ a , 2.1 T a, b < C ( μa ( 1 − μb, μb 1 − μa 2.2 hold for all a, b > 0 with a/ b. From 1.1 and 1.2 we clearly see that both T a, b and C a, b are symmetric and homogenous of degree 1. Without loss of generality, we assume th...
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and Applied Analysis 3 Theorem 1.1. The double inequality α1H a, b 1 − α1 Q a, b < M a, b < β1H a, b ( 1 − β1 ) Q a, b 1.7 holds for all a, b > 0with a/ b if and only if α1 ≥ 2/9 0.2222 . . . and β1 ≤ 1−1/ √ 2 log 1 √ 2 0.1977 . . . . Theorem 1.2. The double inequality α2G a, b 1 − α2 Q a, b < M a, b < β2G a, b ( 1 − β2 ) Q a, b 1.8 holds for all a, b > 0with a/ b if and only if α2 ≥ 1/3 0.3333...
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SB (a, b) = { √ b2−a2 cos−1(a/b) , a < b , √ a2−b2 cosh−1(a/b) , a > b . In this paper, we find the greatest values α1, α2, α3 and α4, and the least values β1, β2, β3 and β4 such that the double inequalities α1SAH(a, b) + (1 − α1)C(a, b) < A(a, b) < β1SAH(a, b) + (1 − β1)C(a, b), α2SHA(a, b) + (1 − α2)C(a, b) < A(a, b) < β2SHA(a, b) + (1 − β2)C(a, b), α3SCA(a, b) + (1 − α3)H(a, b) < A(a, b) < β...
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We find the greatest value λ and the least value μ such that the double inequality C(λa + (1 − λ)b, λb+ (1 − λ)a) < αA(a, b) + (1 − α)T (a, b) < C(μa + (1 − μ)b, μb+ (1− μ)a) holds for all α ∈ (0, 1) and a, b > 0 with a 6= b, where C(a, b), A(a, b), and T (a, b) denote respectively the contraharmonic, arithmetic, and Toader means of two positive numbers a and b.
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and Applied Analysis 3 Lemma 1. The double inequality
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ژورنال
عنوان ژورنال: Linear Algebra and its Applications
سال: 2005
ISSN: 0024-3795
DOI: 10.1016/j.laa.2005.06.013