On a function involving generalized complete $(p,q)$- elliptic integrals

TL;DR

This paper investigates the monotonicity and convexity of a specific function involving generalized complete (p,q)- elliptic integrals, extending previous work by analyzing its mathematical properties.

Contribution

The authors analyze the monotonicity and convexity of a new function involving generalized (p,q)- elliptic integrals, providing insights into their mathematical behavior.

Findings

Established monotonicity properties of the function elta_{p,q}(r).

Proved convexity characteristics under certain parameter conditions.

Extended previous results on elliptic integrals to the generalized (p,q) case.

Abstract

Motivated by the work of Alzer and Richards \cite{ar}, here authors study the monotonicity and convexity properties of the function $$\Delta_{p,q} (r) = \frac{{E_{p,q}(r) - \left( {r'} \right)^p K_{p,q}(r) }}{{r^p }} - \frac{{E'_{p,q}(r) - r^p K'_{p,q}(r) }}{{\left( {r'} \right)^p }},$$ where $K_{p,q}$ and $E_{p,q}$ denote the complete $(p,q)$- elliptic integrals of the first and the second kind, respectively.