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# Sharp inequalities for the generalized elliptic integrals of the first kind

• Zhen-Hang Yang
• Jingfeng Tian
Article
• 52 Downloads

## Abstract

Elliptic integrals are of cardinal importance in mathematical analysis and in the field of applied mathematics. Since they cannot be represented by the elementary transcendental functions, there is a need for sharp computable bounds for the family of integrals. In this paper, by studying the monotonicity of the functions
\begin{aligned} \mathcal {G}_{p}\left( r\right) =\frac{\left( p+r^{2}\right) \mathcal {K} _{a}\left( r\right) }{\ln \left( e^{R\left( a\right) /2}/r^{\prime }\right) } \quad \text {and }\quad \mathcal {I}_{p}\left( r\right) =\frac{\left( p+r^{2}\right) \mathcal {K}_{a}\left( r\right) -p\pi /2}{\ln \left( 1/r^{\prime }\right) } \end{aligned}
on $$\left( 0,1\right)$$, we establish some new sharp lower and upper bounds for the generalized elliptic integrals of the first kind $$\mathcal {K} _{a}\left( r\right)$$, where $$R\left( x\right) \equiv R\left( x,1-x\right)$$ is the Ramanujan constant function defined on (0, 1 / 2], $$r^{\prime }=\sqrt{ 1-r^{2}}$$, $$p\in \mathbb {R}$$ is a parameter. These results not only improve some known bounds in the literature, but also yield some new inequalities for $$\mathcal {K}_{a}\left( r\right)$$.

## Keywords

Gaussian hypergeometric function Generalized elliptic integral of the first kind Monotonicity Inequality

33C05 33E05

## References

1. 1.
Abramowitz, M., Stegun, I.A. (eds.): Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables. Dover, New York (1965)
2. 2.
Alzer, H.: Sharp inequalities for the complete elliptic integral of the first kind. Math. Proc. Camb. Philos. Soc. 124(2), 309–314 (1998)
3. 3.
Anderson, G.D., Vamanamurthy, M.K., Vuorinen, M.: Functional inequalities for complete elliptic integrals and their ratios. SIAM J. Math. Anal. 21, 536–549 (1990)
4. 4.
Anderson, G.D., Vamanamurthy, M.K., Vuorinen, M.: Functional inequalities for hypergeometric functions and complete elliptic integrals. SIAM J. Math. Anal. 23, 512–524 (1992)
5. 5.
Anderson, G.D., Barnard, R.W., Richards, K.C., Vamanamurthy, M.K., Vuorinen, M.: Inequalities for zero-balanced hypergeometric functions. Trans. Am. Math. Soc. 347(5), 1713–1723 (1995)
6. 6.
Anderson, G.D., Qiu, S.-L., Vamanamurthy, M.K., Vuorinen, M.: Generalized elliptic integrals and modular equations. Pac. J. Math. 192(1), 1–37 (2000)
7. 7.
Anderson, G.D., Vamanamurthy, M., Vuorinen, M.: Monotonicity rules in calculus. Am. Math. Mon. 113(9), 805–816 (2006)
8. 8.
Askey, R.: Ramanujan and hypergeometric and basic hypergeometric series. Russ. Math. Surv. 451, 37–86 (1990)
9. 9.
Baricz, Á.: Turán type inequalities for generalized complete elliptic integrals. Math. Z. 256(4), 895–911 (2007)
10. 10.
Baricz, Á.: Landen inequalities for special functions. Proc. Am. Math. Soc. 142(9), 3059–3066 (2014)
11. 11.
Berndt, B.C.: Ramanujan’s Notebooks, Part I. Springer, New York (1985)
12. 12.
Berndt, B.C.: Ramanujan’s Notebooks, Part II. Springer, New York (1989)
13. 13.
Berndt, B.C.: Ramanujan’s Notebooks, Part III. Springer, New York (1991)
14. 14.
Biernacki, M., Krzyz, J.: On the monotonity of certain functionals in the theory of analytic functions. Ann. Univ. Mariae Curie-Sklodowska 9, 135–147 (1995)
15. 15.
Byrd, P.F., Friedman, M.D.: Handbook of Elliptic Integrals for Engineers and Scientists. Springer, New York (1971)
16. 16.
Carlson, B.C., Gustafson, J.L.: Asymptotic expansion of the first elliptic integral. SIAM J. Math. Anal. 16(5), 1072–1092 (1985)
17. 17.
Chu, H.-H., Yang, Zh-H, Zhang, W., Chu, Y.-M.: Improvements of the bounds for Ramanujan constant function. J. Inequal. Appl. 2016, 196 (2016).
18. 18.
Estrada, R., Pavlovic, M.: L’Hopital’s monotone rule, Gromov’s theorem, and operations that preserve the monotonicity of quotients. Publ. Inst. Math. (Beograd) (N.S.) 101(115), 11–24 (2017).
19. 19.
Heikkala, V., Lindén, H., Vamanamurthy, M.K., Vuorinen, M.: Generalized elliptic integrals and the Legendre M-function. J. Math. Anal. Appl. 338(2), 223–243 (2008)
20. 20.
Heikkala, V., Vamanamurthy, M.K., Vuorinen, M.: Generalized elliptic integrals. Comput. Methods Funct. Theory 9(1), 75–109 (2009)
21. 21.
Huang, T.-R., Tan, S., Zhang, X.-H.: Monotonicity, convexity, and inequalities for the generalized elliptic integrals. J. Inequal. Appl. 2017, 278 (2017).
22. 22.
Kühnau, R.: Eine Methode, die Positivität einer Funktion zu prüfen. Z. Angew. Math. Mech. 74, 140–143 (1994). (in German)
23. 23.
Luo, T.-Q., Lv, H.-L., Yang, Zh-H, Zheng, Sh-Zh: New sharp approximations involving incomplete Gamma functions. Results Math. 72, 1007–1020 (2017).
24. 24.
Lv, H.-L., Yang, Zh-H, Luo, T.-Q., Zheng, Sh-Zh: Sharp inequalities for tangent function with applications. J. Inequal. Appl. 2017, 94 (2017).
25. 25.
Neuman, E.: Inequalities and bounds for generalized complete elliptic integrals. J. Math. Anal. Appl. 373(1), 203–213 (2011)
26. 26.
Pinelis, I.: L’Hospital type rules for monotonicity, with applications. J. Inequal. Pure Appl. Math. 3(1), (2002). Art. 5Google Scholar
27. 27.
Ponnusamy, S., Vuorinen, M.: Asymptotic expansions and inequalities for hypergeometric functions. Mathematika 44, 278–301 (1997)
28. 28.
Qiu, S.-L.: The proof of a conjecture on the first elliptic integrals. J. Hangzhou Inst. Electric. Eng. 3, 29–36 (1993)Google Scholar
29. 29.
Qiu, S.-L.: Grötzsch ring and Ramanujan’s modular equations. Acta Math. Sin. 43, 283–290 (2000). (in Chinese)
30. 30.
Qiu, S.-L., Vamanamurthy, M.K.: Sharp estimates for complete elliptic integrals. SIAM J. Math. Anal. 27(3), 823–834 (1996)
31. 31.
Qiu, S.-L., Vuorinen, M.: Landen inequalities for hypergeometric functions. Nagoya Math. J. 154, 31–56 (1999)
32. 32.
Rainville, E.D.: Special Functions. MacMillan, New York (1960)
33. 33.
Varadarajan, V.S.: Linear meromorphic differential equations: a modern point of view. Bull. Am. Math. Soc. 33, 1–42 (1996)
34. 34.
Wang, M.-K., Chu, Y.-M., Qiu, S.-L.: Some monotonicity properties of generalized ellipitic integrals with applications. Math. Inequal. Appl. 16(3), 671–677 (2013)
35. 35.
Wang, M.-K., Chu, Y.-M., Qiu, S.-L.: Sharp bounds for generalized elliptic integrals of the first kind. J. Math. Anal. Appl. 429, 744–757 (2015)
36. 36.
Wang, M.-K., Chu, Y.-M., Song, Y.-Q.: Asymptotical formulas for Gaussian and generalized hypergeometric functions. Appl. Math. Comput. 276, 44–60 (2016)
37. 37.
Whittaker, E.T., Watson, G.N.: A Course of Modern Analysis, 4th edn. Cambridge University Press, London (1958)
38. 38.
Yang, Zh.-H.: A new way to prove L’Hospital Monotone Rules with applications. arXiv:1409.6408 [math.CA]
39. 39.
Yang, Zh-H, Chu, Y.-M.: On approximating the modified Bessel function of the first kind and Toader-Qi mean. J. Inequal. Appl. 2016, 40 (2016).
40. 40.
Yang, Zh-H, Chu, Y.-M.: A monotonicity property involving the generalized elliptic integrals of the first kind. Math. Inequal. Appl. 20(3), 729–735 (2017).
41. 41.
Yang, Zh-H, Tian, J.: Optimal inequalities involving power-exponential mean, arithmetic mean and geometric mean. J. Math. Inequal. 11(4), 1169–1183 (2017).
42. 42.
Yang, Zh-H, Tian, J.: Monotonicity and sharp inequalities related to gamma function. J. Math. Inequal. 12(1), 1–22 (2018)
43. 43.
Yang, Zh-H, Zheng, Sh-Zh: Sharp bounds for the ratio of modified Bessel functions. Mediterr. J. Math. 14, 169 (2017).
44. 44.
Yang, Zh-H, Chu, Y.-M., Wang, M.-K.: Monotonicity criterion for the quotient of power series with applications. J. Math. Anal. Appl. 428(1), 587–604 (2015)
45. 45.
Yin, L., Huang, L.-G., Wang, Y.-L., Lin, X.-L.: An inequality for generalized complete elliptic integral. J. Inequal. Appl. 2017, 303 (2017).
46. 46.
Zhang, X.-H., Wang, G.-D., Chu, Y.-M.: Remark on generalized elliptic integrals. Proc. R. Soc. Edinburgh A 139(2), 417–426 (2009)

## Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

## Authors and Affiliations

1. 1.College of Science and TechnologyNorth China Electric Power UniversityBaodingPeople’s Republic of China
2. 2.Department of Science and TechnologyState Grid Zhejiang Electric Power Company Research InstituteHangzhouChina