Sharp weighted Hölder mean bounds for the second kind generalized elliptic integral

Zixuan Wang; Chuanlong Sun; Tiren Huang; Zixuan Wang; Chuanlong Sun; Tiren Huang

doi:10.3934/math.2025511

AIMS Mathematics

2025, Volume 10, Issue 5: 11271-11289. doi: 10.3934/math.2025511

Previous Article Next Article

Research article Special Issues

Sharp weighted Hölder mean bounds for the second kind generalized elliptic integral

Department of Mathematics, Zhejiang Sci-Tech University, Hangzhou 310018, China

Received: 28 December 2024 Revised: 05 May 2025 Accepted: 13 May 2025 Published: 19 May 2025
MSC : 33C05, 33E05, 26E60

This paper deals with the second kind of generalized elliptic integral $\mathcal{E}_a$, for $a$ in the interval $\left[\frac{1}{2}, 1\right)$, approximated by the weighted Hölder mean. It establishes sharp bounds of the weighted Hölder mean of $\mathcal{E}_a$ in terms of weight, accordingly extending the existing results for the complete case when $ a = \frac{1}{2} $ and establishing new inequality relationships.
- generalized elliptic integral,
- weighted Hölder mean,
- hypergeometric function,
- inequality
Citation: Zixuan Wang, Chuanlong Sun, Tiren Huang. Sharp weighted Hölder mean bounds for the second kind generalized elliptic integral[J]. AIMS Mathematics, 2025, 10(5): 11271-11289. doi: 10.3934/math.2025511

Related Papers:

Abstract

This paper deals with the second kind of generalized elliptic integral $\mathcal{E}_a$, for $a$ in the interval $\left[\frac{1}{2}, 1\right)$, approximated by the weighted Hölder mean. It establishes sharp bounds of the weighted Hölder mean of $\mathcal{E}_a$ in terms of weight, accordingly extending the existing results for the complete case when $ a = \frac{1}{2} $ and establishing new inequality relationships.

References

[1]	M. Abramowitz, I. A. Stegun, D. Miller, Handbook of mathematical functions with formulas, graphs, and mathematical tables (National Bureau of Standards Applied Mathematics Series No. 55), J. Appl. Mech., 32 (1965), 239. https://doi.org/10.1115/1.3625776
[2]	J. M. Borwein, P. B. Borwein, Pi and the AGM, John Wiley & Sons, 1987.
[3]	G. D. Anderson, M. K. Vamanamurthy, M. Vuorinen, Conformal invariants, inequalities, and quasiconformal maps, John Wiley & Sons, 1997.
[4]	S. L. Qiu, M. K. Vamanamurthy, M. Vuorinen, Some inequalities for the Hersch-Pfluger distortion function, J. Inequal. Appl., 4 (1999), 115–139. https://doi.org/10.1155/S1025583499000326
[5]	G. D. Anderson, S. L. Qiu, M. K. Vamanamurthy, M. Vuorinen, Generalized elliptic integrals and modular equations, Pacific J. Math., 192 (2000), 1–37. https://doi.org/10.2140/pjm.2000.192.1
[6]	H. Alzer, K. Richards, On the modulus of the Grötzsch ring, J. Math. Anal. Appl., 432 (2015), 134–141. https://doi.org/10.1016/j.jmaa.2015.06.057 doi: 10.1016/j.jmaa.2015.06.057
[7]	W. M. Qian, Z. Y. He, Y. M. Chu, Approximation for the complete elliptic integral of the first kind, RACSAM, 114 (2020), 57. https://doi.org/10.1007/s13398-020-00784-9 doi: 10.1007/s13398-020-00784-9
[8]	J. F. Tian, Z. H. Yang, M. H. Ha, H. J. Xing, A family of high order approximations of Ramanujan type for perimeter of an ellipse, RACSAM, 115 (2021), 85. https://doi.org/10.1007/s13398-021-01021-7 doi: 10.1007/s13398-021-01021-7
[9]	T. H. Zhao, M. K. Wang, Discrete approximation of complete $p$-elliptic integral of the second kind and its application, RACSAM, 118 (2024), 37. https://doi.org/10.1007/s13398-023-01537-0 doi: 10.1007/s13398-023-01537-0
[10]	M. K. Wang, H. H. Chu, Y. M. Chu, Precise bounds for the weighted Hölder mean of the complete $p$-elliptic integrals, J. Math. Anal. Appl., 480 (2019), 123388. https://doi.org/10.1016/j.jmaa.2019.123388 doi: 10.1016/j.jmaa.2019.123388
[11]	Z. H. Yang, J. F. Tian, Sharp bounds for the Toader mean in terms of arithmetic and geometric means, RACSAM, 115 (2021), 99. https://doi.org/10.1007/s13398-021-01040-4 doi: 10.1007/s13398-021-01040-4
[12]	R. W. Barnard, K. Pearce, K. C. Richards, An inequality involving the generalized hypergeometric function and the arc length of an ellipse, SIAM J. Math. Anal., 31 (2000), 693–699. https://doi.org/10.1137/S0036141098341575
[13]	H. Alzer, S. L. Qiu, Monotonicity theorems and inequalities for the complete elliptic integrals, J. Comput. Appl. Math., 172 (2004), 289–312. https://doi.org/10.1016/j.cam.2004.02.009
[14]	P. S. Bullen, Handbook of means and their inequalities, Springer Science & Business Media, 2003.
[15]	M. K. Wang, Z. Y. He, T. H. Zhao, Q. Bao, Sharp weighted Hölder mean bounds for the complete elliptic integral of the second kind, Integr. Transf. Spec. Funct., 34 (2023), 537–551. https://doi.org/10.1080/10652469.2022.2155819
[16]	V. Heikkala, M. K. Vamanamurthy, M. Vuorinen, Generalized elliptic integrals, Comput. Methods Funct. Theory, 9 (2009), 75–109. https://doi.org/10.1007/BF03321716 doi: 10.1007/BF03321716
[17]	Q. I. Rahman, On the monotonicity of certain functionals in the theory of analytic functions, Can. Math. Bull., 10 (1967), 723–729. https://doi.org/10.4153/CMB-1967-074-9 doi: 10.4153/CMB-1967-074-9
[18]	Z. H. Yang, Recurrence relations of coefficients involving hypergeometric function with an application, arXiv, 2022. https://doi.org/10.48550/arXiv.2204.04709

Reader Comments

Your name:*

Email:*
© 2025 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)