Startseite Mathematik Sharp bounds for the Toader mean of order 3 in terms of arithmetic, quadratic and contraharmonic means
Artikel
Lizenziert
Nicht lizenziert Erfordert eine Authentifizierung

Sharp bounds for the Toader mean of order 3 in terms of arithmetic, quadratic and contraharmonic means

  • Hong-Hu Chu , Tie-Hong Zhao und Yu-Ming Chu EMAIL logo
Veröffentlicht/Copyright: 27. September 2020
Veröffentlichen auch Sie bei De Gruyter Brill
Informationen für Autor*innen Erkunden Sie dieses Fachgebiet
Mathematica Slovaca
Aus der Zeitschrift Mathematica Slovaca Band 70 Heft 5

Abstract

In the article, we present the best possible parameters α1, β1, α2, β2 ∈ ℝ and α3, β3 ∈ [1/2, 1] such that the double inequalities

α1C(a,b)+(1−α1)A(a,b)<T3(a,b)<β1C(a,b)+(1−β1)A(a,b),α2C(a,b)+(1−α2)Q(a,b)<T3(a,b)<β2C(a,b)+(1−β2)Q(a,b),C(α3;a,b)<T3(a,b)<C(β3;a,b)

hold for a, b > 0 with a ≠ b, and provide new bounds for the complete elliptic integral of the second kind, where A(a, b) = (a + b)/2 is the arithmetic mean, Q(a,b)=a2+b2/2 is the quadratic mean, C(a, b) = (a2 + b2)/(a + b) is the contra-harmonic mean, C(p; a, b) = C[pa + (1 – p)b, pb + (1 – p)a] is the one-parameter contra-harmonic mean and T3(a,b)=(2π∫0π/2a3cos2⁡θ+b3sin2⁡θdθ)2/3 is the Toader mean of order 3.

Keywords: Toader mean; complete elliptic integral; arithmetic mean; quadratic mean; contraharmonic mean
MSC 2010: Primary 26E60; Secondary 33E05

This research was supported by the Natural Science Foundation of China (Grant Nos. 11971142, 61673169, 11871202, 11701176, 11626101, 11601485) and the Natural Science Foundation of Zhejiang Province (Grant No. LY19A010012).

  1. (Communicated by Tomasz Natkaniec)

References

[1] Abbas Baloch, I.—Chu, Y.-M.: Petrović-type inequalities for harmonic h-convex Functions, J. Funct. Spaces 2020 (2020), Article ID 3075390, 7 pp.10.1155/2020/3075390Suche in Google Scholar

[2] Alzer, H.—Qiu, S.-L.: Monotonicity theorems and inequalities for the complete elliptic integrals, J. Comput. Appl. Math. 172 (2004), 289–312.10.1016/j.cam.2004.02.009Suche in Google Scholar

[3] Anderson, G. D.—Vamanamurthy, M. K.— Vuorinen, M.: Conformal Invariants, Inequalities, and Quasiconformal Maps, John Wiley & Sons, New York, 1997.Suche in Google Scholar

[4] 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 (2000), 693–699.10.1137/S0036141098341575Suche in Google Scholar

[5] Carlson, B. C.—Gustafson, J. L.: Asymptotic expansion of the first elliptic integral, SIAM J. Math. Anal. 16 (1985), 1072–1092.10.1137/0516080Suche in Google Scholar

[6] Chu, H.-H.—Qian, W.-M.— Chu, Y.-M.—Song, Y.-Q.: Optimal bounds for a Toader-type mean in terms of one-parameter quadratic and contraharmonic means, J. Nonlinear Sci. Appl. 9 (2016), 3424–3432.10.22436/jnsa.009.05.126Suche in Google Scholar

[7] Chu, Y.-M.—Qiu, Y.-F.—Wang, M.-K.: Hölder mean inequalities for the complete elliptic integrals, Integral Transforms Spec. Funct. 23 (2012), 521–527.10.1080/10652469.2011.609482Suche in Google Scholar

[8] Chu, Y.-M.—Qiu, S.-L.—Wang, M.-K.: Sharp inequalities involving the power mean and complete elliptic integral of the first kind, Rocky Mountain J. Math. 43 (2013), 1489–1496.10.1216/RMJ-2013-43-5-1489Suche in Google Scholar

[9] Chu, Y.-M.—Wang, M.-K.: Inequalities between arithmetic-geometric, Gini, and Toader means, Abstr. Appl. Anal. 2012 (2012), Article ID 830585, 11 pp.10.1155/2012/830585Suche in Google Scholar

[10] Chu, Y.-M.—Wang, M.-K.: Optimal Lehmer mean bounds for the Toader mean, Results Math. 61 (2012), 223–229.10.1007/s00025-010-0090-9Suche in Google Scholar

[11] Chu, Y.-M.—Wang, M.-K.—Qiu, Y.-F.: On Alzer and Qiu’s conjecture for complete elliptic integral and inverse hyperbolic tangent function, Abstr. Appl. Anal. 2011 (2011), Article ID 697547, 7 pp.10.1155/2011/697547Suche in Google Scholar

[12] 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 (2012), 41–51.10.1007/s12044-012-0062-ySuche in Google Scholar

[13] Chu, Y.-M.—Wang, M.-K.—Qiu, S.-L.—Qiu, Y.-F.: Sharp generalized Seiffert mean bounds for Toader mean, Abstr. Appl. Anal. 2011 (2011), Article ID 605259, 8 pp.10.1155/2011/605259Suche in Google Scholar

[14] Duan, L.—Fang, X.-W.—Huang, C.-X.: Global exponential convergence in a delayed almost periodic Nicholson’s blowflies model with discontinuous harvesting, Math. Methods Appl. Sci. 41 (2018), 1954–1965.10.1002/mma.4722Suche in Google Scholar

[15] Huang, T.-R.—Han, B.-W.—Ma, X.-Y.—Chu, Y.-M.: Optimal bounds for the generalized Euler-Mascheroni constant, J. Inequal. Appl. 2018 (2018), Article 118, 9 pp.10.1186/s13660-018-1711-1Suche in Google Scholar PubMed PubMed Central

[16] Huang, C.-X.—Liu, L.-Z.: Boundedness of multilinear singular integral operator with a non-smooth kernel and mean oscillation, Quaest. Math. 40 (2017), 295–312.10.2989/16073606.2017.1287136Suche in Google Scholar

[17] Huang, C.-X.—Qiao, Y.-C.—Huang, L.-H.—Agarwal, R. P.: Dynamical behaviors of a food-chain model with stage structure and time delays, Adv. Difference Equ. 2018 (2018), Article 186, 26 pp.10.1186/s13662-018-1589-8Suche in Google Scholar

[18] Huang, T.-R.—Tan, S.-Y.—Ma, X.-Y.—Chu, Y.-M.: Monotonicity properties and bounds for the complete p-elliptic integrals, J. Inequal. Appl. 2018 (2018), Article 239, 11 pp.10.1186/s13660-018-1828-2Suche in Google Scholar

[19] Huang, C.-X.—Yang, Z.-C.—Yi, T.-S.—Zou, X.-F.: On the basins of attraction for a class of delay differential equations with non-monotone bistable nonlinearities, J. Differential Equations 256 (2014), 2101–2114.10.1016/j.jde.2013.12.015Suche in Google Scholar

[20] Huang, C.-X.—Zhang, H.—Cao, J.-D.—Hu, H.-J.: Stability and Hopf bifurcation of a delayed prey-predator model with disease in the predator, Internat. J. Bifur. Chaos Appl. Sci. Engrg. 29 (2019), Article ID 1950091, 23 pp.10.1142/S0218127419500913Suche in Google Scholar

[21] Huang, C.-X.—Zhang, H.—Huang, L.-H.:: Almost periodicity analysis for a delayed Nicholson’s blowflies model with nonlinear density-dependent mortality term, Commun. Pure Appl. Anal. 18 (2019), 3337–3349.10.3934/cpaa.2019150Suche in Google Scholar

[22] Kazi, H.—Neuman, E.: Inequalities and bounds for elliptic integrals, J. Approx. Theory 146(2) (2007), 212–226.10.1016/j.jat.2006.12.004Suche in Google Scholar

[23] Latif, M. A.—Rashid, S.—Dragomir, S. S.—Chu, Y.-M.: Hermite-Hadamard type inequalities for co-ordinated convex and quasi-convex functions and their applications, J. Inequal. Appl. 2019 (2019), Article 317, 33 pp.10.1186/s13660-019-2272-7Suche in Google Scholar

[24] Meng, M.-L.: Inequalities for a Class of New Arithmetic Means, Thesis (B.S.), Huzhou University, 2017 (in Chinese).Suche in Google Scholar

[25] Neuman, E.: Bounds for symmetric elliptic integrals, J. Approx. Theory 122 (2003), 249–259.10.1016/S0021-9045(03)00077-7Suche in Google Scholar

[26] 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 (2017), Article 274, 10 pp.10.1186/s13660-017-1550-5Suche in Google Scholar PubMed PubMed Central

[27] Qian, W.-M.—He, Z.-Y.—Chu, Y.-M.: Approximation for the complete elliptic integral of the first kind, Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. 114 (2020), https://doi.org/10.1007/s13398-020-00784-9, 12 pp.Suche in Google Scholar

[28] Qian, W.-M.—He, Z.-Y.—Zhang, H.-W.—Chu, Y.-M.: Sharp bounds for Neuman means in terms of two-parameter contraharmonic and arithmetic mean, J. Inequal. Appl. 2019 (2019), Article 168, 13 pp.10.1186/s13660-019-2124-5Suche in Google Scholar

[29] Qian, W.-M.—Yang, Y.-Y.—Zhang, H.-W.—Chu, Y.-M.: Optimal two-parameter geometric and arithmetic mean bounds for the Sándor-Yang mean, J. Inequal. Appl. 2019 (2019), Article 287, 12 pp.10.1186/s13660-019-2245-xSuche in Google Scholar

[30] Qian, W.-M.—Zhang, W.—Chu, Y.-M.: Bounding the convex combination of arithmetic and integral means in terms of one-parameter harmonic and geometric means, Miskolc Math. Notes 20 (2019), 1157–1166.10.18514/MMN.2019.2334Suche in Google Scholar

[31] Qiu, S.-L.—Ma, X.-Y.—Chu, Y.-M.: Sharp Landen transformation inequalities for hypergeometric functions, with applications, J. Math. Anal. Appl. 474(2) (2019), 1306–1337.10.1016/j.jmaa.2019.02.018Suche in Google Scholar

[32] Rafeeq, S.—Kalsoom, H.—Hussain, S.—Rashid, S.—Chu, Y.-M.: Delay dynamic double integral inequalities on time scales with applications, Adv. Difference Equ. 2020 (2020), Article 40, 32 pp.10.1186/s13662-020-2516-3Suche in Google Scholar

[33] Tan, Y.-X.—Huang, C.-X.—Sun, B.—Wang, T.: Dynamics of a class of delayed reaction-diffusion systems with Neumann boundary condition, J. Math. Anal. Appl. 458 (2018), 1115–1130.10.1016/j.jmaa.2017.09.045Suche in Google Scholar

[34] Toader, Gh.: Some mean values related to the arithmetic-geometric mean, J. Math. Anal. Appl. 218 (1998), 358–368.10.1006/jmaa.1997.5766Suche in Google Scholar

[35] Toader, Gh.: The monotonicity of a family of means, Bull. Appl. Comp. Math. 85-A (1998), 189–198.Suche in Google Scholar

[36] Vuorinen, M.: Hypergeometric functions in geometric function theory. In: Special Functions and Differential Equations (Madras, 1997), Allied Publ., New Delhi, 1998, pp. 119–126.Suche in Google Scholar

[37] Wang, J.-F.—Chen, X.-Y.—Huang, L.-H.: The number and stability of limit cycles for planar piecewise linear systems of node-saddle type, J. Math. Anal. Appl. 469 (2019), 405–427.10.1016/j.jmaa.2018.09.024Suche in Google Scholar

[38] Wang, M.-K.—Chu, H.-H.—Chu, Y.-M.: Precise bounds for the weighted Hölder mean of the complete p-elliptic integrals, J. Math. Anal. Appl. 480(2) (2019), 9 pp.10.1016/j.jmaa.2019.123388Suche in Google Scholar

[39] Wang, M.-K.—Chu, Y.-M.—Qiu, S.-L.—Jiang, Y.-P.: Convexity of the complete elliptic integrals of the first kind with respect to Hölder means, J. Math. Anal. Appl. 388 (2012), 1141–1146.10.1016/j.jmaa.2011.10.063Suche in Google Scholar

[40] 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 (2011), 887–890.10.1016/j.aml.2010.12.044Suche in Google Scholar

[41] Wang, M.-K.—Chu, Y.-M.—Zhang, W.: Monotonicity and inequalities involving zero-balanced hypergeometric function, Math. Inequal. Appl. 22 (2019), 601–617.10.7153/mia-2019-22-42Suche in Google Scholar

[42] Wang, M.-K.—Chu, Y.-M.—Zhang, W.: Precise estimates for the solution of Ramanujan’s generalized modular equation, Ramanujan J. 49 (2019), 653–668.10.1007/s11139-018-0130-8Suche in Google Scholar

[43] Wang, M.-K.—He, Z.-Y.—Chu, Y.-M.: Sharp power mean inequalities for the generalized elliptic integral of the first kind, Comput. Methods Funct. Theory 20 (2020), 111–124.10.1007/s40315-020-00298-wSuche in Google Scholar

[44] Wang, M. K.—Hong, M. Y.—Xu, Y.-F.—Shen, Z.-H.—Chu, Y.-M.: Inequalities for generalized trigonometric and hyperbolic functions with one parameter, J. Math. Inequal. 14 (2020), 1–21.10.7153/jmi-2020-14-01Suche in Google Scholar

[45] Wang, J.-F.—Huang, C.-X.—Huang, L.-H.: Discontinuity-induced limit cycles in a general planar piecewise linear system of saddle-focus type, Nonlinear Anal. Hybrid Syst. 33 (2019), 162–178.10.1016/j.nahs.2019.03.004Suche in Google Scholar

[46] Wang, B.—Luo, C.-L.—Li, S.-H.—Chu, Y.-M.: Sharp one-parameter geometric and quadratic means bounds for the Sándor-Yang means, Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. 114 (2020), 10 pages.10.1007/s13398-019-00734-0Suche in Google Scholar

[47] Wang, J.-L.—Qian, W.-M.—He, Z.-Y.—Chu, Y.-M.: On approximating the Toader mean by other bivariate means, J. Funct. Spaces 2019 (2019), Article ID 6082413, 7 pp.10.1155/2019/6082413Suche in Google Scholar

[48] Wang, G.-D.—Zhang, X.-H.—Chu, Y.-M.: A power mean inequality involving the complete elliptic integrals, Rocky Mountain J. Math. 44 (2014), 1661–1667.10.1216/RMJ-2014-44-5-1661Suche in Google Scholar

[49] Wang, M.-K.—Zhang, W.—Chu, Y.-M.: Monotonicity, convexity and inequalities involving the generalized elliptic integrals, Acta Math. Sci. 39B (2019), 1440–1450.10.1007/s10473-019-0520-zSuche in Google Scholar

[50] Yang, Z.-Y.—Chu, Y.-M.—Zhang, W.: High accuracy asymptotic bounds for the complete elliptic integral of the second kind, Appl. Math. Comput. 348 (2019), 552–564.10.1016/j.amc.2018.12.025Suche in Google Scholar

[51] 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 (2018), 1185–1199.10.7153/mia-2018-21-82Suche in Google Scholar

[52] Yang, Z.-H.—Qian, W.-M.—Chu, Y.-M.—Zhang, W.: Monotonicity rule for the quotient of two functions and its application, J. Inequal. Appl. 2017 (2017), Article 106, 13 pp.10.1186/s13660-017-1383-2Suche in Google Scholar PubMed PubMed Central

[53] Yang, Z.-H.—Qian, W.-M.—Chu, Y.-M.—Zhang, W.: On rational bounds for the gamma function, J. Inequal. Appl. 2017 (2017), Article 210, 17 pp.10.1186/s13660-017-1484-ySuche in Google Scholar PubMed PubMed Central

[54] 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 (2018), 1714–1726.10.1016/j.jmaa.2018.03.005Suche in Google Scholar

[55] Zaheer Ullah, S.—Adil Khan, M.—Chu, Y.-M.: A note on generalized convex functions, J. Inequal. Appl. 2019 (2019), Article 291, 10 pp.10.1186/s13660-019-2242-0Suche in Google Scholar

[56] Zhao, T.-H.—Chu, Y.-M.—Wang, H.: Logarithmically complete monotonicity properties relating to the gamma function, Abstr. Appl. Anal. 2011 (2011), Article ID 896483, 13 pp.10.1155/2011/896483Suche in Google Scholar

[57] Zhao, T.-H.—Wang, M.-K.—Zhang, W.—Chu, Y.-M.: Quadratic transformation inequalities for Gaussian hypergeometric function, J. Inequal. Appl. 2018 (2018), Article 251, 15 pp.10.1186/s13660-018-1848-ySuche in Google Scholar PubMed PubMed Central

[58] Zhao, T.-H.—Zhou, B. C.—Wang, M.-K.—Chu, Y.-M.: On approximating the quasi-arithmetic mean, J. Inequal. Appl. 2019 (2019), Article 42, 12 pp.10.1186/s13660-019-1991-0Suche in Google Scholar

Received: 2019-10-07
Accepted: 2020-01-14
Published Online: 2020-09-27
Published in Print: 2020-10-27

© 2020 Mathematical Institute Slovak Academy of Sciences

Artikel in diesem Heft

  1. Regular papers
  2. Monadic pseudo BE-algebras
  3. Quadruple construction of decomposable double MS-algebras
  4. Fibonacci numbers in generalized Pell sequences
  5. Solutions of a generalized markoff equation in Fibonacci numbers
  6. Root separation for polynomials with reducible derivative
  7. Quadratic refinements of Young type inequalities
  8. Sharp bounds for the Toader mean of order 3 in terms of arithmetic, quadratic and contraharmonic means
  9. Integration with respect to deficient topological measures on locally compact spaces
  10. Differential subordinations and Pythagorean means
  11. Analogs of Hayman’s Theorem and of logarithmic criterion for analytic vector-valued functions in the unit ball having bounded L-index in joint variables
  12. Unbounded oscillation of fourth order functional differential equations
  13. Oscillation criteria for a class of nonlinear discrete fractional order equations with damping term
  14. Entropy as an integral operator: Erratum and modification
  15. Unital topology on a unital l-group
  16. On the topological complexity of Grassmann manifolds
  17. Properties and methods of estimation for a bivariate exponentiated Fréchet distribution
  18. Existence and Hyers-Ulam stability results for a class of fractional order delay differential equations with non-instantaneous impulses
  19. Two semigroup rings associated to a finite set of meromorphic functions
https://doi.org/10.1515/ms-2017-0417
Schlagwörter für diesen Artikel
Toader mean; complete elliptic integral; arithmetic mean; quadratic mean; contraharmonic mean

Artikel in diesem Heft

  1. Regular papers
  2. Monadic pseudo BE-algebras
  3. Quadruple construction of decomposable double MS-algebras
  4. Fibonacci numbers in generalized Pell sequences
  5. Solutions of a generalized markoff equation in Fibonacci numbers
  6. Root separation for polynomials with reducible derivative
  7. Quadratic refinements of Young type inequalities
  8. Sharp bounds for the Toader mean of order 3 in terms of arithmetic, quadratic and contraharmonic means
  9. Integration with respect to deficient topological measures on locally compact spaces
  10. Differential subordinations and Pythagorean means
  11. Analogs of Hayman’s Theorem and of logarithmic criterion for analytic vector-valued functions in the unit ball having bounded L-index in joint variables
  12. Unbounded oscillation of fourth order functional differential equations
  13. Oscillation criteria for a class of nonlinear discrete fractional order equations with damping term
  14. Entropy as an integral operator: Erratum and modification
  15. Unital topology on a unital l-group
  16. On the topological complexity of Grassmann manifolds
  17. Properties and methods of estimation for a bivariate exponentiated Fréchet distribution
  18. Existence and Hyers-Ulam stability results for a class of fractional order delay differential equations with non-instantaneous impulses
  19. Two semigroup rings associated to a finite set of meromorphic functions
Jetzt anmelden und 20% sparen
Institutioneller Zugang
Wie funktioniert Authentifizierung?
Haben Sie eine Idee, wie wir unsere Website verbessern können?
Bitte schreiben Sie uns.
© 2025 De Gruyter Brill
Heruntergeladen am 15.12.2025 von https://www.degruyterbrill.com/document/doi/10.1515/ms-2017-0417/pdf?lang=de
Button zum nach oben scrollen