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Monotonicity, Convexity and Inequalities Involving the Generalized Elliptic Integrals

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Abstract

We establish the monotonicity and convexity properties for several special functions involving the generalized elliptic integrals, and present some new analytic inequalities.

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References

  1. Abramowitz M, Stegun I A. Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. Washington: US Government Printing Office, 1964

    MATH  Google Scholar 

  2. Wang M K, Chu Y M. Refinements of transformation inequalities for zero-balanced hypergeometric functions. Acta Math Sci, 2017, 37B(3): 607622

    MathSciNet  Google Scholar 

  3. Qiu S L, Ma X Y, Chu Y M. Sharp Landen transformation inequalities for hypergeometric functions, with applications. J Math Anal Appl, 2019, 474(2): 13061337

    Article  MathSciNet  MATH  Google Scholar 

  4. Wang M K, Chu Y M, Zhang W. Monotonicity and inequalties involving zero-balanced hypergeometric function. Math Inequal Appl, 2019, 22(2): 601617

    Google Scholar 

  5. Anderson G D, Vamanamurthy M K, Vuorinen M. Conformal Invariants, Inequalities, and Quasiconformal Maps. New York: John Wiley & Sons, 1997

    MATH  Google Scholar 

  6. Anderson G D, Qiu S L, Vamanamurthy M K, et al. Generalized elliptic integrals and modular equations. Pacific J Math, 2000, 192(1): 137

    Article  MathSciNet  Google Scholar 

  7. Almkvist G, Berndt B. Gauss, Landen, Ramanujan, the arithmetic-geometric mean, ellipse, π, and the Ladies diary. Amer Math Monthly, 1988, 95(7): 585608

    MathSciNet  MATH  Google Scholar 

  8. Borwein J M, Borwein P B. Pi and the AGM. New York: John Wiley & Sons, 1987

    MATH  Google Scholar 

  9. Borwein J M, Borwein P B. Inequalities for compound mean iterations with logarithmic asymptotes. J Math Anal Appl, 1993, 177(2): 572582

    Article  MathSciNet  MATH  Google Scholar 

  10. Barnard R W, Pearce K, Richards K C. A monotonicity property involving 3F2 and comparisons of the classical approximations of elliptical arc length. SIAM J Math Anal, 2000, 32(2): 403419

    Article  MATH  Google Scholar 

  11. Alzer H. Sharp inequalities for the complete elliptic integral of the first kind. Math Proc Cambridge Philos Soc, 1998, 124(2): 309314

    Article  MathSciNet  Google Scholar 

  12. Wang M K, Chu Y M. Asymptotical bounds for complete elliptic integrals of the second kind. J Math Anal Appl, 2013, 402(1): 119126

    Article  MathSciNet  Google Scholar 

  13. Anderson G D, Qiu S L, Vamanamurthy M K. Elliptic integral inequalities, with applications. Constr Approx, 1998, 14(2): 195207

    Article  MathSciNet  Google Scholar 

  14. Chu Y M, Wang M K, Qiu S L. Optimal combinations bounds of root-square and arithmetic means for Toadr mean. Proc Indian Acad Sci Math Sci, 2012, 122(1): 4151

    Article  Google Scholar 

  15. Chu Y M, Wang M K. Optimal Lehmer mean bounds for the Toader mean. Results Math, 2012, 61(3/4): 223229

    MathSciNet  MATH  Google Scholar 

  16. Chu Y M, Qiu Y F, Wang M K. Hölder mean inequalities for the complete elliptic integrals. Intetral Transfroms Spec Funct, 2012, 23(7): 521527

    Google Scholar 

  17. Qiu S L, Qiu Y F, Wang M K, Chu Y M. Hölder mean inequalities for the generalized Grötzsch ring and Hersch-Pfluger distortion functions. Math Inequal Appl, 2012, 15(1): 237245

    MATH  Google Scholar 

  18. 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, 2018, 21(3): 629648

    MATH  Google Scholar 

  19. Chu Y M, Cheng J F, Wang G D. Remarks on John disks. Acta Math Sci, 2009, 29B(1): 160168

    MathSciNet  MATH  Google Scholar 

  20. Chu Y M, Sun T C. The Schur harmonic convexity for a class of symmetric functions. Acta Math Sci, 2010, 30B(5): 15011506

    MathSciNet  Google Scholar 

  21. 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, 2014, 256(7): 21012114

    Article  MathSciNet  Google Scholar 

  22. Huang C X, Guo S, Liu L Z. Boundedness on Morrey space for Toeplitz type operator associated to singular integral operator with variable Calderón-Zygmund kernel. J Math Inequal, 2014, 8(3): 453464

    MATH  Google Scholar 

  23. Huang C X, Liu L Z. Boundedness of multilinear singular integral operator with a non-smooth kernel and mean oscillation. Quaest Math, 2017, 40(3): 295312

    Article  MathSciNet  Google Scholar 

  24. 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, 2018, 21(4): 11851199

    MathSciNet  Google Scholar 

  25. 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, 2018, 41(5): 19541965

    Article  MathSciNet  MATH  Google Scholar 

  26. Yang Z H, Chu Y M, Zhang W. High accuracy asymptotic bounds for the complete elliptic integral of the second kind. Appl Math Comput, 2019, 348: 552564

    Article  MathSciNet  Google Scholar 

  27. 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, 2019, 469(1): 405427

    MathSciNet  Google Scholar 

  28. Wang M K, Li Y M, Chu Y M. Inequalities and infinite product formula for Ramanujan generalized modular equation function. Ramanujan J, 2018, 46(1): 189200

    Article  MathSciNet  Google Scholar 

  29. Alzer H, Richards K C. A note on a function involving complete elliptic integrals: monotonicity, convexity, inequalities. Anal Math, 2015, 41(3): 133139

    Article  MathSciNet  MATH  Google Scholar 

  30. Yang Z H, Chu Y M, Wang M K. Monotonicity criterion for the quotient of power series with applications. J Math Anal Appl, 2015, 428(1): 587604

    MathSciNet  Google Scholar 

  31. Pinelis I. On L’Hospital-type rules for monotonicity. J Inequal Pure Appl Math, 2006, 7(2): Article 40

    Google Scholar 

  32. Huang T R, Tan S Y, Zhang X H. Monotonicity, convexity, inequalities for the generalized elliptic integrals. J Inequal Appl, 2017, 2017: Article 278

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Correspondence to Yuming Chu.

Additional information

This research was supported by the Natural Science Foundation of China (11701176, 61673169, 11301127, 11626101, 11601485), and the Science and Technology Research Program of Zhejiang Educational Committee (Y201635325).

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Wang, M., Zhang, W. & Chu, Y. Monotonicity, Convexity and Inequalities Involving the Generalized Elliptic Integrals. Acta Math Sci 39, 1440–1450 (2019). https://doi.org/10.1007/s10473-019-0520-z

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  • DOI: https://doi.org/10.1007/s10473-019-0520-z

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