On exterior moduli of quadrilaterals and special functions

TL;DR

This paper proves identities and inequalities involving elliptic integral-based functions, explores their properties, and applies these findings to analyze the growth of the exterior modulus of rectangles.

Contribution

It introduces new identities and inequalities for elliptic integral functions and applies them to geometric modulus problems.

Findings

Established identities involving elliptic integrals

Derived inequalities and estimates for the function

Analyzed the growth of the exterior modulus of rectangles

Abstract

In this paper two identities involving a function defined by the complete elliptic integrals of the first and second kinds are proved. Some functional inequalities and elementary estimates for this function are also derived from the properties of monotonicity and convexity of this function. As applications, some functional inequalities and the growth of the exterior modulus of a rectangle are studied.