Sharp inequalities for Toader mean in terms of other bivariate means

Year 2023, , 841 - 849, 15.08.2023

Weidong Jıang

https://doi.org/10.15672/hujms.1106426

Abstract

In the paper, the author discovers the best constants $\alpha_1$, $\alpha_2$, $\alpha_3$, $\beta_1$, $\beta_2$ and $\beta_3$ for the double inequalities
\[
\alpha_1 A\left(\frac{a-b}{a+b}\right)^{2n+2} < T(a,b)-\frac{1}{4}C-\frac{3}{4}A-A\sum_{k=1}^{n-1}\frac{(\frac{1}{2},k)^2}{4((k+1)!)^2}\left(\frac{a-b}{a+b}\right)^{2k+2}<\beta_1 A\left(\frac{a-b}{a+b}\right)^{2n+2}
\]
\[
\alpha_2 A\left(\frac{a-b}{a+b}\right)^{2n+2} < T(a,b)-\frac{3}{4}\overline{C}-\frac{1}{4}A-A\sum_{k=1}^{n-1}\frac{(\frac{1}{2},k)^2}{4((k+1)!)^2}\left(\frac{a-b}{a+b}\right)^{2k+2}<\beta_2 A\left(\frac{a-b}{a+b}\right)^{2n+2}
\]
and
\[
\alpha_3 A\left(\frac{a-b}{a+b}\right)^{2n+2} < \frac{4}{5}T(a,b)+\frac{1}{5}H-A-A\sum_{k=1}^{n-1}\frac{(\frac{1}{2},k)^2}{5((k+1)!)^2}\left(\frac{a-b}{a+b}\right)^{2k+2}<\beta_3 A\left(\frac{a-b}{a+b}\right)^{2n+2}
\]
to be valid for all $a,b>0$ with $a\ne b$ and $n=1,2,\cdots$, where
\[
C\equiv C(a,b)=\frac{a^2+b^2}{a+b},\,\overline{C}\equiv\overline{C}(a,b)=\frac{2(a^2+ab+b^2)}{3(a+b)},\, A\equiv A(a,b)=\frac{a+b}{2},
\]
\[
H\equiv H(a,b) =\frac{2ab}{a+b},\quad T(a,b)=\frac2{\pi}\int_0^{\pi/2}\sqrt{a^2\cos^2\theta+b^2\sin^2\theta}\,{\rm d}\theta
\]
are respectively the contraharmonic, centroidal, arithmetic, harmonic and Toader means of two positive numbers $a$ and $b$, $ (a,n)=a(a+1)(a+2)(a+3)\cdots (a+n-1)$ denotes the shifted factorial function. As an application of the above inequalities, the author also find a new bounds for the complete elliptic integral of the second kind.

Keywords

Toader mean, complete elliptic integrals, arithmetic mean, centroidal mean, contraharmonic mean

References

• [1] H. Alzer, S.-L. Qiu, Monotonicity theorems and inequalities for the complete elliptic integrals, J. Comput. Appl. Math. 172, 289–312, 2004.
• [2] G.D. Anderson, M.K. Vamanamurthy, M. Vuorinen, Conformal Invariants, Inequalities, and Quasiconformal Maps, John Wiley & Sons, New York, 1997.
• [3] R.W. Barnard, K. Pearce, and K.C. Richards, An inequality involving the generalized hypergeometric function and the arc length of an ellipse, SIAM J. Math. Anal. 31(3), 693–699, 2000.
• [4] F. Bowman, Introduction to Elliptic Functions with Applications, Dover Publications, New York, 1961.
• [5] P.F. Byrd and M.D. Friedman, Handbook of Elliptic Integrals for Engineers and Scientists, Springer-Verlag, New York, 1971.
• [6] Y.-M. Chu, M.-K.Wang, Inequalities between arithmetic-geometric, Gini, and Toader means, Abstr. Appl. Anal. 2012, Art. ID 830585, 2012.
• [7] Y.-M. Chu, M.-K. Wang, X.-Y. Ma, Sharp bounds for Toader mean in terms of contraharmonic mean with applications, J. Math. Inequal. 7 (2), 161–166, 2013.
• [8] Y.-M. Chu, M.-K. Wang, S.-L. Qiu, Optimal combinations bounds of root-square and arithmetic means for Toader mean, Proc. Indian Acad. Sci. Math. Sci. 122, 41–51, 2012.
• [9] Y.-M. Chu, M.-K. Wang, S.-L. Qiu, Optimal Lehmer mean bounds for the Toader mean. Results Math. 61, 223–229, 2012.
• [10] Y.-M. Chu, M.-K. Wang, S.-L. Qiu, Y.-P. Jiang, Bounds for complete elliptic integrals of the second kind with applications, Comput. Math. Appl. 63, 1177–1184, 2012.
• [11] Y. Hua, Bounds For The Arithmetic Mean In Terms Of The Toader Mean And Other Bivariate Means, Miskolc Math. Notes 18 (1) , 203–210, 2017.
• [12] Y. Hua, F. Qi, The Best Bounds for Toader Mean in Terms of the Centroidal and Arithmetic Means, Filomat 28 (4), 775–780, 2014.
• [13] Y. Hua, F. Qi, A double inequality for bounding Toader mean by the centroidal mean, Proc. Indian Acad. Sci. (Math. Sci.) 124 (4), 527–531, 2014.
• [14] W.-D. Jiang, F. Qi, A double inequality for the combination of Toader mean and the arithmetic mean in terms of the contraharmonic mean, Publ. Inst. Math. 99 (113), 237–242, 2016.
• [15] W.-H. Li, M.-M. Zheng, Some inequalities for bounding Toader mean, J. Func. Spaces Appl. 2013, Art. ID 394194, 5 pages, 2013.
• [16] W.-M. Qian, H.-H. Chu, M.-K. Wang, Y.-M. Chu, Sharp inequalities for the Toader mean of order -1 in terms of other bivariate means, J. Math. Inequal 16 (1), 127–141, 2022.
• [17] S.-L. Qiu, J.-M. Shen, On two problems concerning means, J. Hangzhou Insitute Electronic Engineering (In Chinese) 17 (3), 1–7, 1997.
• [18] Y.-Q. Song, W.-D. Jiang, Y.-M. Chu, D.-D. Yan, Optimal bounds for Toader mean in terms of arithmetic and contraharmonic means, J. Math. Inequal 7 (4), 751–757, 2013.
• [19] Gh. Toader, Some mean values related to the arithmetic-geometric mean, J. Math. Anal. Appl. 218, 358–368, 1998.
• [20] M. Vuorinen, Hypergeometric functions in geometric function theory, in: Special Functions and Differential Equations, Proceedings of a Workshop held at The Institute of Mathematical Sciences, Madras, India, January 13-24, 1997, Allied Publ., New Delhi, 119–126, 1998.
• [21] M.-K. Wang, Y.-M. Chu, Y.-M. Li, W. Zhang, Asymptotic expansion and bounds for complete elliptic integrals, Math. Inequal. Appl. 23 (3), 821–841, 2020.
• [22] M.-K. Wang, Y.-M. Chu, S.-L. Qiu, Y.-P. Jiang, Bounds for the perimeter of an ellipse, J. Approx. Theory. 164, 928–937, 2012.
• [23] Z.-H. Yang, Sharp approximations for the complete elliptic integrals of the second kind by one-parameter means, J. Math. Anal. Appl. 467, 446–461, 2018.
• [24] Z.-H. Yang, Y.-M. Chu, W.Zhang, High accuracy asymptotic bounds for the complete elliptic integral of the second kind, Appl. Math. Comput. 348, 552–564, 2019.
• [25] Z.-H. Yang, Y.-M. Chu, W. Zhang, Monotonicity of the ratio for the complete elliptic integral and Stolarsky mean, J. Inequal. Appl. 2016, 176, 2016.
• [26] Z.-H. Yang„ Y.-M. Chu, W. Zhang, Accurate approximations for the complete elliptic integral of the second kind, J. Math. Anal. Appl. 438, 875–888, 2016.
• [27] Z.-H. Yang, Y.-M. Chu, X.-H. Zhang, Sharp Stolarsky mean bounds for the complete elliptic integral of the second kind, J. Nonlinear Sci. Appl. 10, 929–936, 2017.
• [28] Z.-H. Yang, W.-M. Qian, Y.-M. Chu, W.Zhang, Monotonicity rule for the quotient of two functions and its application, J. Inequal. Appl. 2017, 106, 2017.
• [29] Z.-H. Yang, J.-F. Tian, Sharp bounds for the Toader mean in terms of arithmetic and geometric means, RACSAM 115, 99, 2021.
• [30] F. Zhang, W.-M. Qian, H.-Z. Xu, Optimal bounds for Seiffert-like elliptic integral mean by harmonic, geometric, and arithmetic means, J. Inequal. Appl 2022, 33, 2022.
• [31] T.-H. Zhao, H.-H. Chu , Y.-M. Chu, Optimal Lehmer mean bounds for the nth powertype Toader means of $n=-1,1,3$, J. Math. Inequal 16 (1), 157–168, 2022.
• [32] T.-H. Zhao, M.-K. Wang, Y.-Q. Dai and Y.-M. Chu, On the generalized power-type Toader mean, J. Math. Inequal. 16(1), 247–264, 2022.