期刊：
Journal of Computational Analysis and Applications,2020年28(4):646-653 ISSN：1521-1398
作者机构：
Friedman Brain Institute, Icahn School of Medicine at Mount Sinai, New York, NY 10029,USA;College of Science, Hunan City University, Yiyang 413000, Hunan, China;Department of Mathematics, Huzhou University, Huzhou 313000, Zhejiang, China
摘要：
In the article, we prove that a = 3, β = log4/(π/2 - log 4)= 7.51371. ··· ,γ = 1/4 and δ = 1 + log2 - π/2 = 0.122351 ··? are the best possible constants such that the double inequalities hold for all r ∈ (0,1), where r' = √1-r~2, and K(r) = ∫_0~(π/2) dθ/√1-r~2sin~2θ and ε(r) =∫_0~(π/2)√1-r~2sin~2θdθ are the complete elliptic integrals of the first and second kinds.
摘要：
In the article, we prove that the function x -> (1-x)K-P(root x) is logarithmically concave on (0, 1) if and only if p >= 7/32, the function x -> K(root x)/ log(1 + 4/root 1 - x) is convex on (0, 1) and the function x -> d(2)/dx(2) [K(root x) - log (1+ 4/root 1 - x)] is absolutely monotonic on (0,1), where K(x) = integral(pi/2)(0) (1 - x(2) sin(2) t)(-1/2) dt ( 0 < x < 1) is the complete elliptic integral of the first kind.
期刊：
Journal of Computational Analysis and Applications,2020年28(6):929-940 ISSN：1521-1398
作者机构：
College of Science, Hunan City University, Yiyang 413000. Hunan. China;College of Mathematics and Econometrics, Hunan University, Changsha 410082,Hunan, China;Department of Mathematics. Huzhou University, Huzhou 313000, Zhejiang, China
关键词：
Toader mean;Neuman mean;geometric mean;arithmetic mean;quadratic mean;contra-harmonic mean
摘要：
In the article, we prove that the double inequalities α_1Q(α, b) + (1 - a_1)N_(GA)(a, b) < T(a, b) < β_1Q(a, b) + (1 - β_1)N_(GA)(a, b), α_2Q(α, b) + (1 - a_2)N_(QA)(a, b) < T(α, b) < β_(2Q)(α,b) + (1 - β_2)N_(QA)(a, b), α_3C(α, b) + (1 - α_3)N_(GA)(α, b) < T(a, b) < β_3C(a, b) + (1 - β_3)N_(GA)(a, b), a_4C(a,b) + (1 - a_4)N_(QA)(a,b) < T(a, b) < β_4C(a,b) + (1 - β_4)N_(QA)(a,b) hold for all a,b>0 with a≠b if and only if α_1 ≤ 5/8, β_1 ≥ (16- π~2)/[(4√2-π)π] = 0,7758 …, α_2 ≤ 1/4, β_2≥ 1 - 2(√2π - 4)/[(√2 - log(1 + √2))π]= 0.4708…, α_3 ≤ 5/14 = 0.3571 …,β_3 ≥ (16 - π~2)/[(8 - π)π] = 0.4016…, α_4 ≤ 1/10 and β_4 ≥ 1-4(π-2)/[(4-+√2)) = 0.1472 …, where Q(a, b), C(α, b) and T(α, b) are respectively the quadratic, contra-harmonic and Toader means, and NGA(a, b) and NQA (a, b) are the Neuman means.
期刊：
Journal of Computational Analysis and Applications,2020年28(3):514-525 ISSN：1521-1398
作者机构：
Corresponding author), 4Department of Mathematics, Huzhou University, Huzhou 313000, Zhejiang, China;Friedman Brain Institute, Icahn School of Medicine at Mount Sinai, New York, NY 10029,USA;College of Science, Hunan City University, Yiyang 413000, Hunan, China
关键词：
Toader mean;geometric mean;contraharmonic mean;complete elliptic integral
摘要：
In this paper, we present the best possible parameters α_1, α_2, α_3 and β_1,β_2, β_3 such that the double inequalities hold for all a,b>0 with a ≠ b, where G(a, b) = √ab, C(a, b) = (a~2 + b~2) /(a + b) and T(a, b) = 2 f_0~(π/2) √a~2 cos~2(t) + b~2 sin~2(t)dt/π are the geometric, contraharmonic and Toader means of a and b, respectively.
期刊：
Journal of Computational Analysis and Applications,2020年28(5):814-823 ISSN：1521-1398
作者机构：
Department of Mathematics, Huzhou University, Huzhou 313000, China;College of Science, Hunan City University, Yiyang 413000,China;Department of Mathematics, Zhejiang Sci-Tech University, Hangzhou 310018, China
摘要：
In the article, we provide several sharp bounds for the the generalized Euler-Mascheroni constant, which are the generalizations of the previously results on the Euler-Mascheroni constant.
