Optimal convex combination bounds of geometric and Neuman means for Toader-type mean

In this paper, we prove that the double inequalities \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \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}αNQA(a,b)+(1−α)G(a,b)0$\end{document}a,b>0 with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$a\neq b$\end{document}a≠b if and only if \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\alpha \leq 3/8$\end{document}α≤3/8, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\beta \geq 4/ [\pi ( \log (1+\sqrt{2})+\sqrt{2}) ]=0.5546 \cdots $\end{document}β≥4/[π(log(1+2)+2)]=0.5546⋯ , \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\lambda \leq 3/10$\end{document}λ≤3/10 and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mu \geq 8/ [\pi (\pi +2) ]=0.4952 \cdots $\end{document}μ≥8/[π(π+2)]=0.4952⋯ , where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$TD(a,b)$\end{document}TD(a,b), \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$G(a,b)$\end{document}G(a,b), \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$A(a,b)$\end{document}A(a,b) and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$N_{QA}(a,b)$\end{document}NQA(a,b), \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$N_{AQ}(a,b)$\end{document}NAQ(a,b) are the Toader, geometric, arithmetic and two Neuman means of a and b, respectively.


Introduction
For x, y, z ≥  with xy + xz + yz =  and r ∈ (, ), the symmetric integrals R F (x, y, z) and R G (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 and where cos - (x) and cosh - (x) = log(x + √ x  -) are the inverse cosine and inverse hyperbolic cosine functions, respectively.
Very recently, Neuman [] introduced the Neuman mean N(a, b) of the second kind as follows: .
It is well known that the Toader mean TD(a, b), the Schwab-Borchardt mean SB(a, b) and the Neuman mean of the second kind N(a, b) satisfy the identities (see [, ]) Let p ∈ R and a, b > . Then the pth power mean M p (a, b) is defined by We clearly see that M p (a, b) is symmetric and homogeneous of degree one with respect to a and b, strictly increasing with respect to p ∈ R for fixed a, b >  with a = b, and the inequalities and proved that the inequalities Recently, the Toader mean has been the subject of intensive research. In particular, many remarkable inequalities for Toader mean and other related means can be found in the literature [-].
In [], Vuorinen conjectured that for all a, b >  with a = b. This conjecture was proved by Qiu Li, Qian and Chu [] proved that the inequality From inequalities (.) and (.) we clearly see that The main purpose of this paper is to find the greatest values α, λ and the least values β, μ such that the double inequalities hold for all a, b >  with a = b. As applications, we get two new bounds for the complete elliptic integral of the second kind in terms of elementary functions.

Lemmas
In order to prove our main results, we need several lemmas, which we present in this section.
For r ∈ (, ), we clearly see that and K(r) and E(r) satisfy the formulas (see [], Appendix E, pp.-) .
is strictly monotone, then the monotonicity in the conclusion is also strict.

Lemma . The function r
Proof It is easy to verify that for r ∈ (, ). Therefore, Lemma . follows easily from (.) and (.).
Proof It is not difficult to verify that From (.) and Lemma .() together with the monotonicity of E(r) on (, ) we clearly see that for r ∈ (, ). Therefore, Lemma . follows from (.) and (.).

Theorem . The double inequality
Proof Since G(a, b), TD(a, b) and N QA (a, b) are symmetric and homogenous of degree , without loss of generality, we assume that a > b >  and let r = (ab)/(a + b) ∈ (, ). Then (.)-(.) lead to   TD A(a, b), G(a, b)
From Theorems .-. we get the following Corollary . immediately.

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. As applications, we find new bounds for the complete elliptic integral of the second kind.

Conclusion
In the article, we present the optimal convex combination bounds of the geometric and Neuman means for the Toader-type mean, and give several new upper and lower bounds for the complete elliptic integral of the second kind. The given results are the improvements of some previously known results.