Optimal inequalities for bounding Toader mean by arithmetic and quadratic means

In this paper, we present the best possible parameters α(r)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\alpha(r)$\end{document} and β(r)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\beta(r)$\end{document} such that the double inequality [α(r)Ar(a,b)+(1−α(r))Qr(a,b)]1/r<TD[A(a,b),Q(a,b)]<[β(r)Ar(a,b)+(1−β(r))Qr(a,b)]1/r\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\begin{aligned} \bigl[\alpha(r)A^{r}(a,b)+ \bigl(1-\alpha(r) \bigr)Q^{r}(a,b) \bigr]^{1/r} < & TD \bigl[A(a,b), Q(a,b) \bigr] \\ < & \bigl[\beta(r)A^{r}(a,b)+ \bigl(1-\beta(r) \bigr)Q^{r}(a,b) \bigr]^{1/r} \end{aligned}$$ \end{document} holds for all r≤1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$r\leq 1$\end{document} and a,b>0\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} with a≠b\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$a\neq b$\end{document}, and we provide new bounds for the complete elliptic integral E(r)=∫0π/2(1−r2sin2θ)1/2dθ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathcal{E}(r)=\int_{0}^{\pi/2}(1-r^{2}\sin^{2}\theta)^{1/2}\,d\theta$\end{document}(r∈(0,2/2))\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$(r\in (0, \sqrt{2}/2))$\end{document} of the second kind, where TD(a,b)=2π∫0π/2a2cos2θ+b2sin2θdθ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$TD(a,b)=\frac{2}{\pi}\int_{0}^{\pi/2}\sqrt{a^{2}\cos^{2}\theta+b^{2}\sin^{2}\theta}\,d\theta$\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}$A(a,b)=(a+b)/2$\end{document} and Q(a,b)=(a2+b2)/2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$Q(a,b)=\sqrt{(a^{2}+b^{2})/2}$\end{document} are the Toader, arithmetic, and quadratic means of a and b, respectively.


Introduction
For p ∈ [, ], q ∈ R and a, b >  with a = b, the pth generalized Seiffert mean S p (a, b), qth Gini mean G q (a, b), qth power mean M q (a, b), qth Lehmer mean L q (a, b), harmonic mean H(a, b), geometric mean G(a, b), arithmetic mean A(a, b), quadratic mean Q(a, b), Toader mean TD(a, b) [], centroidal mean C(a, b), contraharmonic mean C(a, b) are, respectively, defined by ,  < p ≤ , It is well known that S p (a, b), G q (a, b), M q (a, b), and L q (a, b) are continuous and strictly increasing with respect to p ∈ [, ] and q ∈ R for fixed a, b >  with a = b, and the inequal- The Toader mean TD(a, b) has been well known in the mathematical literature for many years, it satisfies stands for the symmetric complete elliptic integral of the second kind (see [-]), therefore it cannot be expressed in terms of the elementary transcendental functions.
be, respectively, the complete elliptic integrals of the first and second kind. Then K( + ) = E( + ) = π/, K(r), and E(r) satisfy the derivatives formulas (see [], Appendix E, p.-) the values K( √ /) and E( √ /) can be expressed as (see [], Theorem .) is the Euler gamma function, and the Toader mean TD(a, b) can be rewritten as Recently, the Toader mean TD(a, b) has been the subject of intensive research. Vuorinen [] conjectured that the inequality holds for all a, b >  with a = b. This conjecture was proved by Qiu Neuman [], and Kazi and Neuman [] proved that the inequalities , is the arithmetic-geometric mean of a and b. In [-], the authors presented the best possible parameters λ  , μ  ∈ [, ] and

). Then Chu, Wang and Ma [], and Hua and Qi [] proved that the double inequalities
, the authors proved that the double inequalities The main purpose of this paper is to present the best possible parameters α(r) and β(r) such that the double inequality holds for all r ≤  and a, b >  with a = b.

Lemmas
In order to prove our main result we need two lemmas, which we present in this section.
Proof It follows from (.) that for all t ∈ (, √ /). It follows from (.) that f  (t) is strictly decreasing on (, √ /). We divide the proof into three cases.
Then from (.)-(.) one has where inequalities (.) and (.) hold due to c  > c  and the function t → log t/(t r -) is strictly decreasing on (, ∞). Note that λ(r) ∈ (, ) and the function x → V (r, x) is strictly increasing on (, √ ). Then (.)-(.) lead to the conclusion that there exists x  ∈ (, √ ) such that the function x → ∂ log U(r; a, b)/∂r is strictly decreasing on (, x  ) and strictly increasing on (x  , √ ). It follows from (.) and (.) together with the piecewise monotonicity of the function x → ∂ log U(r; a, b)/∂r on the interval (, for all a, b >  with  < b/a < √ . Therefore, Lemma . follows from (.). where f (t) is defined as in Lemma ..