Sharp bounds for a special quasi-arithmetic mean in terms of arithmetic and geometric means with two parameters

In the article, we present the best possible parameters \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\lambda=\lambda (p)$\end{document}λ=λ(p) and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mu=\mu(p)$\end{document}μ=μ(p) on the interval \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$[0, 1/2]$\end{document}[0,1/2] such that the double inequality \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\begin{aligned}& G^{p}\bigl[\lambda a+(1-\lambda)b, \lambda b+(1-\lambda)a \bigr]A^{1-p}(a,b) \\& \quad< E(a,b) < G^{p}\bigl[\mu a+(1-\mu)b, \mu b+(1-\mu)a \bigr]A^{1-p}(a,b) \end{aligned}$$ \end{document}Gp[λa+(1−λ)b,λb+(1−λ)a]A1−p(a,b)<E(a,b)<Gp[μa+(1−μ)b,μb+(1−μ)a]A1−p(a,b) holds for any \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$p\in[1, \infty)$\end{document}p∈[1,∞) and all \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$a, b>0$\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, where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$A(a, b)=(a+b)/2$\end{document}A(a,b)=(a+b)/2, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$G(a,b)=\sqrt{ab}$\end{document}G(a,b)=ab and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$E(a,b)=[2\int_{0}^{\pi /2}\sqrt{a\cos^{2}\theta+b\sin^{2}\theta}\,d\theta/\pi]^{2}$\end{document}E(a,b)=[2∫0π/2acos2θ+bsin2θdθ/π]2 are the arithmetic, geometric and special quasi-arithmetic means of a and b, respectively.


Introduction
Let r ∈ (, ). Then the Legendre complete elliptic integrals K(r) and E(r) [, ] of the first and second kinds are defined as respectively. It is well known that the function r → K(r) is strictly increasing from (, ) onto (π/, ∞) and the function r → E(r) is strictly decreasing from (, ) onto (, π/), and they satisfy the formulas (see [, Appendix E, pp. ,]) The complete elliptic integrals K(r) and E(r) are the particular cases of the Gaussian hypergeometric function [-] where (a)  =  for a = , (a) n = a(a + )(a + ) · · · (a + n -) = (a + n)/ (a) is the shifted factorial function and (x) = ∞  t x- e -t dt (x > ) is the gamma function [-]. Indeed, Recently, the bounds for the complete elliptic integrals have attracted the attention of many researchers. In particular, many remarkable inequalities and properties for K(r), E(r) and F(a, b; c; x) can be found in the literature [-].
In , a class of quasi-arithmetic mean was introduced by Toader [] which is defined by where r n (θ ) = (a n cos  θ +b n sin  θ ) /n for n = , r  (θ ) = a cos  θ b sin  θ , and p is a strictly monotonic function. It is well known that many important means are the special cases of the quasi-arithmetic mean. For example, is the arithmetic-geometric mean of Gauss [-], is the Toader mean [-], and is the Toader-Qi mean [-]. Let p = √ x and n = . Then M p,n (a, b) reduces to a special quasi-arithmetic mean be the arithmetic, geometric and pth power means of a and b, respectively. Then it is well known that the inequality holds for all a, b >  with a = b, and the double inequality Then it is not difficult to verify that the function Motivated by inequalities (.) and the monotonicity of the function x → f (x; p; a, b) on the interval [, /], in the article, we shall find the best possible parameters λ = λ(p), μ = μ(p) on the interval [, /] such that the double inequality holds for any p ∈ [, ∞) and all a, b >  with a = b.
is strictly monotone, then the monotonicity in the conclusion is also strict.
Then simple computations lead to Lemma . The following statements are true: It follows from part () and (.) that for all r ∈ (, ). Therefore, part () follows from (.) and (.).
Proof Let g  (r) = r  K(r) and g  (r) = ( + r  )[K(r) -E(r)]. Then we clearly see that where f  (r) and g(r) are defined by (.) and Lemma ., respectively. From Lemma .() and Lemma . together with (.) we clearly see that the function r → h  (p; r) is strictly increasing on (, ) and From Lemma . we know that  -( √ /π) /p > /(p). Therefore, we only need to divide the proof into three cases as follows.
Case  u ≤ /(p). Then Lemma .(), (.), (.) and the monotonicity of the function r → h  (p; r) on the interval (, ) lead to the conclusion that the function r → h(u, p; r) is strictly increasing on (, ). Therefore, h(u, p; r) >  for all r ∈ (, ) follows from (.) and the monotonicity of the function r → h(u, p; r).

Results and discussion
In this paper, we provide the sharp bounds for the special quasi-arithmetic mean E(a, b) in terms of the arithmetic mean A(a, b) and geometric mean G(a, b) with two parameters. As consequences, we present the best possible one-parameter harmonic and geometric means bounds for E(a, b) and find new bounds for the complete elliptic integral of the second kind.

Conclusion
In the article, we derive a new bivariate mean E(a, b) from the quasi-arithmetic mean and provide its sharp upper and lower bounds in terms of the concave combination of arithmetic and geometric means.