Abstract
In this paper, we prove that the double inequalities \t\t\tαNQA(a,b)+(1−α)G(a,b)<TD[A(a,b),G(a,b)]<βNQA(a,b)+(1−β)G(a,b),λNAQ(a,b)+(1−λ)G(a,b)<TD[A(a,b),G(a,b)]<μNAQ(a,b)+(1−μ)G(a,b)\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document} $$\\begin{aligned}& {\\alpha }N_{QA}(a,b)+({1-\\alpha })G(a,b)< TD \\bigl[A(a,b),G(a,b) \\bigr]< {\\beta }N_{QA}(a,b)+({1-\\beta })G(a,b), \\\\& {\\lambda }N_{AQ}(a,b)+({1-\\lambda })G(a,b)< TD \\bigl[A(a,b),G(a,b) \\bigr]< {\\mu }N_{AQ}(a,b)+({1-\\mu })G(a,b) \\end{aligned}$$ \\end{document} hold for all a,b>0 with aneq b if and only if alpha leq 3/8, beta geq 4/ [pi ( log (1+sqrt{2})+sqrt{2}) ]=0.5546 cdots , lambda leq 3/10 and mu geq 8/ [pi (pi +2) ]=0.4952 cdots , where TD(a,b), G(a,b), A(a,b) and N_{QA}(a,b), N_{AQ}(a,b) are the Toader, geometric, arithmetic and two Neuman means of a and b, respectively.
Highlights
1 Introduction For x, y, z ≥ with xy + xz + yz = and r ∈ (, ), the symmetric integrals RF (x, y, z) and RG(x, y, z) [ ] of the first and second kinds, and the complete elliptic integrals K(r) and E(r) of the first and second kinds are defined by RF (x, y, z) =
TD(a, b) > M / (a, b) for all a, b > with a = b. This conjecture was proved by Qiu and Shen [ ], and Barnard et al [ ], respectively, and Alzer and Qiu [ ] presented the best possible upper power mean bound for the Toader mean as follows: TD(a, b) < Mlog / log(π/ )(a, b) for all a, b > with a = b
4 Results and discussion In this paper, we provide the sharp bounds for the Toader-type mean in terms of the convex combination of geometric and Neuman means
Summary
Abstract In this paper, we prove that the double inequalities αNQA(a, b) + (1 – α)G(a, b) < TD A(a, b), G(a, b) < βNQA(a, b) + (1 – β)G(a, b), λNAQ(a, b) + (1 – λ)G(a, b) < TD A(a, b), G(a, b) < μNAQ(a, b) + (1 – μ)G(a, b) hold for all a, b > 0√with a√= b if and only if α ≤ 3/8, β ≥ 4/[π (log(1 + 2) + 2)] = 0.5546 · · · , λ ≤ 3/10 and μ ≥ 8/[π (π + 2)] = 0.4952 · · · , where TD(a, b), G(a, b), A(a, b) and NQA(a, b), NAQ(a, b) are the Toader, geometric, arithmetic and two Neuman means of a and b, respectively. The Toader mean TD(a, b) [ ] and the Schwab-Borchardt mean SB(a, b) [ – ] are respectively defined by TD(a, b) =
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