Optimal inequalities for bounding Toader mean by arithmetic and quadratic means

详细信息    查看全文

• 作者：Tie-Hong Zhao ; Yu-Ming Chu ; Wen Zhang
• 关键词：arithmetic mean ; Toader mean ; quadratic mean ; complete elliptic integral
• 刊名：Journal of Inequalities and Applications
• 出版年：2017
• 出版时间：December 2017
• 年：2017
• 卷：2017
• 期：1
• 全文大小：1345KB
• 刊物主题：Analysis; Applications of Mathematics; Mathematics, general;
• 出版者：Springer International Publishing
• ISSN：1029-242X
• 卷排序：2017

文摘

In this paper, we present the best possible parameters \(\alpha(r)\) and \(\beta(r)\) such that the double inequality $$\begin{aligned} \bigl[\alpha(r)A^{r}(a,b)+ \bigl(1-\alpha(r) \bigr)Q^{r}(a,b) \bigr]^{1/r} < & TD \bigl[A(a,b), Q(a,b) \bigr] \\ < & \bigl[\beta(r)A^{r}(a,b)+ \bigl(1-\beta(r) \bigr)Q^{r}(a,b) \bigr]^{1/r} \end{aligned}$$ holds for all \(r\leq 1\) and \(a, b>0\) with \(a\neq b\), and we provide new bounds for the complete elliptic integral \(\mathcal{E}(r)=\int_{0}^{\pi/2}(1-r^{2}\sin^{2}\theta)^{1/2}\,d\theta\)\((r\in (0, \sqrt{2}/2))\) of the second kind, where \(TD(a,b)=\frac{2}{\pi}\int_{0}^{\pi/2}\sqrt{a^{2}\cos^{2}\theta+b^{2}\sin^{2}\theta}\,d\theta\), \(A(a,b)=(a+b)/2\) and \(Q(a,b)=\sqrt{(a^{2}+b^{2})/2}\) are the Toader, arithmetic, and quadratic means of a and b, respectively.Keywordsarithmetic meanToader meanquadratic meancomplete elliptic integralMSC26E601 IntroductionFor \(p\in [0, 1]\), \(q\in \mathbb{R}\) and \(a,b>0\) with \(a\neq b\), 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 \(TD(a,b)\) [1], centroidal mean \(\overline{C}(a,b)\), contraharmonic mean \(C(a,b)\) are, respectively, defined by $$\begin{aligned}& S_{p}(a,b)= \textstyle\begin{cases} \displaystyle\frac{p(a-b)}{\arctan [2p(a-b)/(a+b) ]}, &0< p\leq 1,\\ (a+b)/2, &p=0, \end{cases}\displaystyle \\& G_{q}(a,b)= \textstyle\begin{cases} [(a^{q-1}+b^{q-1})/(a+b) ]^{1/(q-2)}, &q\neq 2,\\ (a^{a}b^{b} )^{1/(a+b)}, &q=2, \end{cases}\displaystyle \\& M_{q}(a,b)= \textstyle\begin{cases} [(a^{q}+b^{q})/2 ]^{1/q}, &q\neq 0,\\ \sqrt{ab}, &q=0, \end{cases}\displaystyle \\& L_{q}(a,b)=\frac{a^{q+1}+b^{q+1}}{a^{q}+b^{q}}, \qquad H(a,b)=\frac{2ab}{a+b},\qquad G(a,b)= \sqrt{ab}, \\& A(a,b)=\frac{a+b}{2},\qquad Q(a,b)=\sqrt{\frac{a^{2}+b^{2}}{2}}, \\& TD(a,b)=\frac{2}{\pi} \int_{0}^{{\pi}/{2}}\sqrt{a^{2} \cos^{2}{\theta}+b^{2}\sin^{2}{\theta}}\,d\theta, \\& \overline{C}(a,b)=\frac{2(a^{2}+ab+b^{2})}{3(a+b)}, \qquad C(a,b)=\frac{a^{2}+b^{2}}{a+b}. \end{aligned}$$ (1.1)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 \(p\in [0, 1]\) and \(q\in \mathbb{R}\) for fixed \(a, b>0\) with \(a\neq b\), and the inequalities $$\begin{aligned} H(a,b)&=M_{-1}(a,b)=L_{-1}(a,b)< G(a,b)=M_{0}(a,b)=L_{-1/2}(a,b) \\ & < A(a,b)=M_{1}(a,b)=L_{0}(a,b)< TD(a,b)< \overline{C}(a,b) \\ &< Q(a,b)=M_{2}(a,b)< C(a,b)=L_{1}(a,b) \end{aligned}$$ hold for all \(a, b>0\) with \(a\neq b\).The Toader mean \(TD(a,b)\) has been well known in the mathematical literature for many years, it satisfies $$ TD(a,b)=R_{E} \bigl(a^{2}, b^{2} \bigr), $$ where $$ R_{E}(a,b)=\frac{1}{\pi} \int_{0}^{\infty}\frac{[a(t+b)+b(t+a)]t}{(t+a)^{3/2}(t+b)^{3/2}}\,dt $$ stands for the symmetric complete elliptic integral of the second kind (see [2–4]), therefore it cannot be expressed in terms of the elementary transcendental functions.Let \(r\in (0, 1)\), \(\mathcal{K}(r)=\int _{0}^{\pi/2}(1-r^{2}\sin^{2}\theta)^{-1/2}\,d\theta\) and \(\mathcal{E}(r)=\int _{0}^{\pi/2}(1-r^{2}\sin^{2}\theta)^{1/2}\,d\theta\) be, respectively, the complete elliptic integrals of the first and second kind. Then \(\mathcal{K}(0^{+})=\mathcal{E}(0^{+})=\pi/2\), \(\mathcal{K}(r)\), and \(\mathcal{E}(r)\) satisfy the derivatives formulas (see [5], Appendix E, p.474-475) $$\begin{aligned}& \frac{d\mathcal{K}(r)}{dr}=\frac{\mathcal{E}(r)-(1-r^{2})\mathcal{K}(r)}{r(1-r^{2})}, \qquad \frac{d\mathcal{E}(r)}{dr}=\frac{\mathcal{E}(r)-\mathcal{K}(r)}{r}, \\& \frac{d[\mathcal{K}(r)-\mathcal{E}(r)]}{dr}=\frac{r\mathcal{E}(r)}{1-r^{2}}, \end{aligned}$$ the values \(\mathcal{K}(\sqrt{2}/2)\) and \(\mathcal{E}(\sqrt{2}/2)\) can be expressed as (see [6], Theorem 1.7) $$ \mathcal{K} \biggl(\frac{\sqrt{2}}{2} \biggr)=\frac{\Gamma^{2} ({1}/{4} )}{4\sqrt{\pi}}=1.854\ldots, \qquad \mathcal{E} \biggl(\frac{\sqrt{2}}{2} \biggr)=\frac{4\Gamma^{2} ({3}/{4} )+\Gamma^{2} ({1}/{4} )}{8\sqrt{\pi}}=1.350 \ldots, $$ where \(\Gamma(x)=\int_{0}^{\infty}t^{x-1}e^{-t}\,dt(\operatorname{Re}{x}>0)\) is the Euler gamma function, and the Toader mean \(TD(a,b)\) can be rewritten as $$ TD(a,b)= \textstyle\begin{cases} {2a}\mathcal{E} (\sqrt{1- ({b}/{a} )^{2}} )/\pi, \quad a\geq b,\\ {2b}\mathcal{E} (\sqrt{1- ({a}/{b} )^{2}} )/\pi, \quad a< b. \end{cases} $$ (1.2)Recently, the Toader mean \(TD(a,b)\) has been the subject of intensive research. Vuorinen [7] conjectured that the inequality $$ TD(a,b)>M_{3/2}(a,b) $$ holds for all \(a, b>0\) with \(a\neq b\). This conjecture was proved by Qiu and Shen [8], and Barnard, Pearce and Richards [9], respectively.Alzer and Qiu [10] presented a best possible upper power mean bound for the Toader mean as follows: $$ TD(a,b)< M_{\log 2/(\log\pi-\log 2)}(a,b) $$ for all \(a, b>0\) with \(a\neq b\).Neuman [2], and Kazi and Neuman [3] proved that the inequalities $$\begin{aligned}& \frac{(a+b)\sqrt{ab}-ab}{AGM(a,b)}< TD(a,b)< \frac{4(a+b)\sqrt{ab}+(a-b)^{2}}{8AGM(a,b)}, \\& TD(a,b)< \frac{1}{4} \bigl(\sqrt{(2+\sqrt{2})a^{2}+(2- \sqrt{2})b^{2}}+\sqrt{(2+\sqrt{2})b^{2}+(2- \sqrt{2})a^{2}} \bigr) \end{aligned}$$ hold for all \(a, b>0\) with \(a\neq b\), where \(AGM(a,b)\) is the arithmetic-geometric mean of a and b.In [11–13], the authors presented the best possible parameters \(\lambda_{1}, \mu_{1}\in [0, 1]\) and \(\lambda_{2}, \mu_{2}, \lambda_{3}, \mu_{3}\in \mathbb{R}\) such that the double inequalities \(S_{\lambda_{1}}(a,b)< TD(a,b)< S_{\mu_{1}}(a,b)\), \(G_{\lambda_{2}}(a,b)< TD(a,b)< G_{\mu_{2}}(a,b)\) and \(L_{\lambda_{3}}(a,b)< TD(a,b)< L_{\mu_{3}}(a,b)\) hold for all \(a, b>0\) with \(a\neq b\).