Approximations related to the complete p-elliptic integrals

Abstract

In this paper, the authors present some monotonicity properties for certain functions involving the complete p-elliptic integrals of the first and second kinds, by showing the monotonicity and concavity-convexity properties of certain combinations defined in terms of Kp{{\mathscr{K}}}_{p}, Ep{{\mathscr{E}}}_{p} and the inverse hyperbolic tangent arthp{{\rm{arth}}}_{p}, which is of importance in the computation of the generalized pi and in the elementary proof of Ramanujan’s cubic transformation. By these results, several well-known results for the classical complete elliptic integrals including its bounds and logarithmic inequalities are remarkably improved.

Keywords

• complete p-elliptic integrals
• monotonicity
• inequality
• bounds inequalities
• logarithmic inequalities
• 33c75
• 33e05
• 33f05