Complete $(p,q)$-elliptic integrals with application to a family of means

TL;DR

This paper generalizes complete elliptic integrals using two-parameter generalized trigonometric functions, establishes a key relation, and applies these to provide an alternative proof for a family of means involving the logarithmic and arithmetic-geometric means.

Contribution

It introduces a two-parameter generalization of elliptic integrals and applies this to derive an alternative proof for a known family of means.

Findings

A relation for the generalized elliptic integrals is established.

An alternative proof for a family of means involving logarithmic and arithmetic-geometric means is provided.

The generalized integrals extend classical elliptic integral properties.

Abstract

The complete elliptic integrals are generalized by using the generalized trigonometric functions with two parameters. It is shown that a particular relation holds for the generalized integrals. Moreover, as an application of the integrals, an alternative proof of a result for a family of means by Bhatia and Li, which involves the logarithmic mean and the arithmetic-geometric mean, is given.