The Ramanujan Journal

, Volume 48, Issue 1, pp 91–116 | Cite as

Sharp inequalities for the generalized elliptic integrals of the first kind

  • Zhen-Hang Yang
  • Jingfeng TianEmail author


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 \(\mathcal {K}_{a}\left( r\right) \).


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

Mathematics Subject Classification

33C05 33E05 


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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

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