Abstract
In the article, we prove that the double inequalities
hold for all \(a, b>0\) with \(a\ne b\) if and only if \(\alpha \le 2\log 2/(2+\log 2)=0.5147\cdots \), \(\beta \ge 2/3\), \(\lambda \le 2\log 2/(2-\log 2)=1.0607\cdots \) and \(\mu \ge 4/3\), where \(M_{p}(a,b)=[(a^{p}+b^{p})/2]^{1/p}\)\((p\ne 0)\), \(M_{0}(a,b)=G(a,b)=\sqrt{ab}\), \(Q(a,b)=\sqrt{(a^{2}+b^{2})/2}\), \(U(a,b)=(a-b)/[\sqrt{2\breve{}}\arctan ((a-b)/\sqrt{2ab})]\) and \(V(a,b)=(a-b)/[\sqrt{2}\sinh ^{-1}((a-b)/\sqrt{2ab})]\) are respectively the pth power, geometric, quadratic, first Yang and second Yang means, and \(\sinh ^{-1}(x)\) is the inverse hyperbolic sine function.
Similar content being viewed by others
References
Neuman, E., Sándor, J.: On the Schwab–Borchardt mean. Math. Pannon. 14(2), 253–266 (2003)
Neuman, E., Sándor, J.: On the Schwab–Borchardt mean II. Math. Pannon. 17(1), 49–59 (2006)
Sándor, J.: Two sharp inequalities for trigonometric and hyperbolic functions. Math. Inequal. Appl. 15(2), 409–413 (2012)
Yang, Z.-H., Chu, Y.-M.: Optimal evaluations for the Sándor–Yang mean by power mean. Math. Inequal. Appl. 19(3), 1031–1038 (2016)
Qian, W.-M., Chu, Y.-M.: Best possbie bounds for Yang mean using generalized logarithmic mean. Math. Probl. Eng. 2016, 7 (2016) (Article ID 8901258)
Qian, W.-M., Chu, Y.-M., Zhang, X.-H.: Sharp one-parameter mean bounds for Yang mean. Math. Probl. Eng. 2016, 5 (2016) (Article ID 1579468)
Zhou, S.-S., Qian, W.-M., Chu, Y.-M., Zhang, X.-H.: Sharp power-type Heronian mean bounds for the Sándor and Yang means. J. Inequal. Appl. 2015, 10 (2015) (Article ID 159)
Toader, Gh.: Some mean values related to the arithmetic-geometric mean. J. Math. Anal. Appl. 218(2), 358–368 (1998)
Xia, W.-F., Chu, Y.-M.: Optimal inequalities between Neuman–Sándor, centroidal and harmonic means. J. Math. Inequal. 7(4), 593–600 (2013)
He, Z.-Y., Qian, W.-M., Jiang, Y.-L., Song, Y.-Q., Chu, Y.-M.: Bounds for the combinations of Neuman–Sándor, arithmetic, and second Seiffert means in terms of contrahrmonic mean. Abstr. Appl. Anal. 2013, 5 (2013) (Article ID 903982)
Chu, Y.-M., Long, B.-Y.: Bounds of the Neuman–Sándor mean using power and identric means. Abstr. Appl. Anal. 2013, 6 (2013) (Article ID 832591)
Chu, Y.-M., Wang, M.-K., Wang, Z.-K.: A best-possible double inequality between Seiffert and harmonic means. J. Inequal. Appl. 2011, 7 (2011) (Article 94)
Chu, Y.-M., Wang, M.-K., Qiu, S.-L., Qiu, Y.-F.: Sharp generalized Seiffert mean bounds for Toader mean. Abstr. Appl. Anal. 2011, 8 (2011) (Article ID 605259)
Chu, Y.-M., Wang, M.-K., Gong, W.-M.: Two sharp double inequalities for Seiffert mean. J. Inequal. Appl. 2011, 7(2011) (Article 44)
Wang, M.-K., Qiu, Y.-F., Chu, Y.-M.: Sharp bounds for Seiffert means in terms of Lehmer means. J. Math. Inequal. 4(4), 581–586 (2010)
Chu, Y.-M., Wang, M.-K., Qiu, Y.-F.: An optimal double inequality between power-type Heron and Seiffert means. J. Inequal. Appl. 2010, 11 (2010) (Article ID 146945)
Qiu, Y.-F., Wang, M.-K., Chu, Y.-M., Wang, G.-D.: Two sharp inequalities for Lehmer mean, identric mean and logarithmic mean. J. Math. Inequal. 5(3), 301–306 (2011)
Xia, W.-F., Chu, Y.-M., Wang, G.-D.: The optimal upper and lower power mean bounds for a convex combination of the arithmetic and logarithmic means. Abstr. Appl. Anal. 2010, 9 (2010) (Article ID 604804)
Adil Khan, M., Zheer Ullah, S., Chu, Y.-M.: The concept of coordinate strongly convex functions and related inequalities. Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. RACSAM. https://doi.org/10.1007/s13398-018-0615-8
Wang, M.-K., Chu, Y.-M., Zhang, W.: The precise estimates for the solution of Ramanujan’s generalized modular equation. Ramanujan J. https://doi.org/10.1007/s11139-018-0130-8
Adil Khan, M., Wu, S.-H., Hidayat, U., Chu, Y.-M.: Discrete majorization type inequalities for convex functions on rectangles. J. Inequal. Appl. 2019, 18 (2019) (Article 16)
Yousaf, K., Adil Khan, M., Chu, Y.-M.: Conformable integral inequalities of the Hermite-Hadamard type in terms of GG- and GA-convexities. J. Funct. Sp. 2019, 8 (2019) (Article ID 6926107)
Yousaf, K., Adil Khan, M., Chu, Y.-M.: Hermite–Hadamard–Fejér inequalities for conformable fractional integrals via preinvex functions. J. Funct. Sp. 2019, 9 (2019) (Article ID 3146210)
Chu, Y.-M., Wang, M.-K., Qiu, S.-L.: Optimal combinations bounds of root-square and arithmetic means for Toader mean. Proc. Indian Acad. Sci. Math. Sci. 122(1), 41–51 (2012)
Chu, Y.-M., Wang, M.-K.: Optimal Lehmer mean bounds for the Toader mean. Results Math. 61(3–4), 223–229 (2012)
Qian, W.-M., Zhang, X.-H., Chu, Y.-M.: Sharp bounds for the Toader-Qi mean in terms of harmonic and geometric means. J. Math. Inequal. 11(1), 121–127 (2017)
Wang, M.-K., Chu, Y.-M.: Refinements of transformation inequalities for zero-balanced hypergeometric functions. Acta Math. Sci. 37B(3), 607–622 (2017)
Chu, Y.-M., Adil Khan, M., Ali, T., Dragomir, S.S.: Inequalities for \(\alpha \)-fractional differentiable functions. J. Inequal. Appl. 2017, 12 (2017) (Article 93)
Yang, Z.-H., Qian, W.-M., Chu, Y.-M., Zhang, W.: Monotonicity rule for the quotient of two functions and its applications. J. Inequal. Appl. 2017, 13 (2017) (Article 106)
Hu, H.-N., Tu, G.-Y., Chu, Y.-M.: Optimal bounds for Seiffert mean in terms of one-parameter means. J. Appl. Math. 2012, 7 (2012) (Article ID 917120)
Chu, Y.-M., Wang, M.-K., Wang, G.-D.: The optimal generalized logarithmic mean bounds for Seiffert’s mean. Acta Math. Sci. 32B(4), 1619–1626 (2012)
Yang, Z.-H., Chu, Y.-M.: A monotonicity property involving the generalized elliptic integral of the first kind. Math. Inequal. Appl. 20(3), 729–735 (2017)
Chu, Y.-M., Hou, S.-W.: Sharp bounds for Seiffert mean in terms of contraharmonic mean. Abstr. Appl. Anal. 2012, 6 (2012) (Article ID 425175)
Yang, Z.-H., Qian, W.-M., Chu, Y.-M., Zhang, W.: On rational bounds for the gamma function. J. Inequal. Appl. 2017, 17 (2017) (Article 210)
Yang, Z.-H., Zhang, W., Chu, Y.-M.: Sharp Gautschi inequality for parameter \(0<p<1\) with applications. Math. Inequal. Appl. 20(4), 1107–1120 (2017)
Chu, Y.-M., Wang, M.-K., Qiu, Y.-F., Ma, X.-Y.: Sharp two parameter bounds for the logarithmic mean and the arithmetic-geometric mean of Gauss. J. Math. Inequal. 7(3), 349–355 (2013)
Chu, Y.-M., Hou, S.-W., Xia, W.-F.: Optimal convex combinations bounds of centroidal and harmonic means for logarithmic and identric means. Bull. Iran. Math. Soc. 39(2), 259–269 (2013)
Qian, W.-M., Chu, Y.-M.: Sharp bounds for a special quasi-arithmetic mean in terms of arithmetic and geometric means with two-parameters. J. Inequal. Appl. 2017, 10 (2017) (Article 274)
Chu, Y.-M., Wang, M.-K., Wang, Z.-K.: Best possible inequalities among harmonic, logarithmic and Seiffert means. Math. Inequal. Appl. 15(2), 415–422 (2012)
Yang, Z.-H., Chu, Y.-M., Zhang, W.: High accuracy asymptotic bounds for the complete elliptic integral of the second kind. Appl. Math. Comput. 348, 552–564 (2019)
Yang, Z.-H., Qian, W.-M., Chu, Y.-M.: Monotonicity properties and bounds involving the complete elliptic integrals of the first kind. Math. Inequal. Appl. 21(4), 1185–1199 (2018)
Zhao, T.-H., Wang, M.-K., Zhang, W., Chu, Y.-M.: Quadratic transformation inequalities for Gaussian hypergeometric function. J. Inequal. Appl. 2018, 15 (2018) (Article 251)
Wang, M.-K., Qiu, S.-L., Chu, Y.-M.: Infinite series formula for Hübner upper bound function with applications to Hersch-Pfluger distortion function. Math. Inequal. Appl. 21(3), 629–648 (2018)
Adil Khan, M., Iqbal, A., Suleman, M., Chu, Y.-M.: Hermite–Hadamard type inequalities for fractional integrals via Green’s function. J. Inequal. Appl. 2018, 15 (2018) (Article 161)
Song, Y.-Q., Adil Khan, M., Zaheer Ullah, S., Chu, Y.-M.: Integral inequalities involving strongly convex functions. J. Funct. Sp. 2018, 8 (2018) (Article ID 6595921)
Adil Khan, M., Chu, Y.-M., Kashuri, A., Liko, R., Ali, G.: Conformable fractional integrals versions of Hermite–Hadamard inequalities and their generalizations. J. Funct. Sp. 2018, 9 (2018) (Article ID 6928130)
Xu, H.-Z., Chu, Y.-M., Qian, W.-M.: Sharp bounds for the Sándor–Yang means in terms of arithmetic and contra-harmonic means. J. Inequal. Appl. 2018, 13 (2018) (Article 127)
Huang, T.-R., Han, B.-W., Ma, X.-Y., Chu, Y.-M.: Optimal bounds for the generalized Euler–Mascheroni constant. J. Inequal. Appl. 2018, 9 (2018) (Article 118)
Wang, M.-K., Li, Y.-M., Chu, Y.-M.: Inequalities and infinite product formula for Ramanujan generalized modular equation function. Ramanujan J. 46(1), 189–200 (2018)
Adil Khan, M., Begum, S., Khurshid, Y., Chu, Y.-M.: Ostrowski type inequalities involving conformable fractional integrals. J. Inequal. Appl. 2018, 14 (2018) (Article 70)
Wang, M.-K., Chu, Y.-M.: Landen inequalities for a class of hypergeometric functions with applications. Math. Inequal. Appl. 21(2), 521–537 (2018)
Yang, Z.-H., Qian, W.-M., Chu, Y.-M., Zhang, W.: On approximating the error function. Math. Inequal. Appl. 21(2), 469–479 (2018)
Yang, Z.-H., Qian, W.-M., Chu, Y.-M., Zhang, W.: On approximating the arithmetic-geometric mean and complete elliptic integral of the first kind. J. Math. Anal. Appl. 462(2), 1714–1726 (2018)
Adil Khan, M., Chu, Y.-M., Khan, T.U., Khan, J.: Some new inequalities of Hermite–Hadamard type for \(s\)-convex functions with applications. Open Math. 15, 1414–1430 (2017)
Wang, J.-L., Qian, W.-M., He, Z.-Y., Chu, Y.-M.: On approximating the Toader mean by other bivariate means. J. Funct. Sp. 2019, 6 (2019) (Article ID 6082413)
Qiu, S.-L., Ma, X.-Y., Chu, Y.-M.: Sharp Landen transformation inequalities for hypergeoemtric functions, with applications. J. Math. Anal. Appl. https://doi.org/10.1016/j.jmaa.2019.02.018
Chu, Y.-M., Xia, W.-F.: Two optimal double inequalities between power mean and logarithmic mean. Comput. Math. Appl. 60(1), 83–89 (2010)
Wang, M.-K., Chu, Y.-M., Qiu, Y.-F., Qiu, S.-L.: An optimal power mean inequality for the complete elliptic integrals. Appl. Math. Lett. 24(6), 887–890 (2011)
Wang, G.-D., Zhang, X.-H., Chu, Y.-M.: A powr mean inequality for the Grötzsch ring function. Math. Inequal. Appl. 14(4), 833–837 (2011)
Wang, G.-D., Zhang, Z.-H., Chu, Y.-M.: A power mean inequality involving the complete elliptic integrals. Rocky Mt. J. Math. 44(5), 1661–1667 (2014)
Radó, T.: On convex functions. Trans. Am. Math. Soc. 37(2), 266–285 (1935)
Lin, T.P.: The power mean and the logarithmic mean. Am. Math. Mon. 81(2), 879–883 (1974)
Stolarsky, K.B.: The power and generalized logarithmic means. Am. Math. Mon. 87(7), 545–548 (1980)
Pittenger, A.O.: Inequalities between arithmetic and logarithmic means. Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. Fiz. 678(715), 15–18 (1980)
Jagers, A.A.: Solution of problem 887. Nieuw Arch. Wisk. 12(4), 230–231 (1994)
Hästö, P.A.: A monotonicity property of ratios of symmetric homogeneous means. JIPAM. J. Inequal. Pure Appl. Math. 3(5), 23 (2002) (Article 71)
Hästö, P.A.: Optimal inequalities between Seiffert mean and power means. Math. Inequal. Appl. 7(1), 47–53 (2004)
Costin, I., Toader, G.: Optimal evaluations of some Seiffert-type means by power means. Appl. Math. Comput. 219(9), 4745–4754 (2013)
Li, Y.-M., Wang, M.-K., Chu, Y.-M.: Sharp power mean bounds for Seiffert mean. Appl. Math. J. Chin. Univ. 29B(1), 101–107 (2014)
Yang, Z.-H.: Estimates for Neuman–Sándor mean by power means and their relative errors. J. Math. Inequal. 7(4), 711–726 (2013)
Chu, Y.-M., Yang, Z.-H., Wu, L.-M.: Sharp power mean bounds for Sándor mean. Abstr. Appl. Anal. 2015, 5(2015) (Article ID 172867)
Yang, Z.-H., Wu, L.-M., Chu, Y.-M.: Optimal power mean bounds for Yang mean. J. Inequal. Appl. 2014, 10 (2014) (Article 401)
Li, J.-F., Yang, Z.-H., Chu, Y.-M.: Optimal power mean bounds for the second Yang mean. J. Inequal. Appl. 2016, 9 (2016) (Article 31)
Barnard, R.W., Pearce, K., Richards, K.C.: An inequality involving the generalized hypergeometric function and the arc length of an ellipse. SIAM. J. Math. Anal. 31(3), 693–699 (2000)
Alzer, H., Qiu, S.-L.: Monotonicity theorems and inequalities for the complete elliptic integrals. J. Comput. Appl. Math. 172(2), 289–312 (2004)
Yang, Z.-H.: Three families of two-parameter means constructed by trigonometric functions. J. Inequal. Appl. 2013, 27 (2013) (Article 541)
Zhao, T.-H., Qian, W.-M., Song, Y.-Q.: Optimal bounds for two Sándor-type means in terms of power means. J. Inequal. Appl. 2016, 10 (2016) (Article 64)
Acknowledgements
Funding was provided by National Natural Science Foundation of China (Grant no. 61673169) and Natural Science Foundation of Huzhou City (Grant no. 2018YZ07).
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
This work was supported by the Natural Science Foundation of China (Grant no. 61673169) and the Natural Science Foundation of Huzhou City (Grant no. 2018YZ07).
Rights and permissions
About this article
Cite this article
He, XH., Qian, WM., Xu, HZ. et al. Sharp power mean bounds for two Sándor–Yang means. RACSAM 113, 2627–2638 (2019). https://doi.org/10.1007/s13398-019-00643-2
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s13398-019-00643-2