Sharp double inequality for complete elliptic integral of the first kind

2021/03/17 by Bao, Qi
#33C75 #33E05 #Classical Analysis and ODEs (math.CA) #FOS: Mathematics

paper · doi:10.48550/arxiv.2104.11630

Abstract

For r∈(0,1), the function \K(r)=∫₀^π/2(1-r²sin²t)^-1/2dt is known as the complete elliptic integral of the first kind. In this paper, we prove the absolute monotonicity of two functions involving \K(r). As a consequence, we improve Alzer and Richards' result.

Related

• Absolutely monotonic functions related to the asymptotic formula for the complete elliptic integral of the first kind
• Sharp bounds for generalized elliptic integrals of the first kind
• On a function involving generalized complete (p,q)- elliptic integrals
• Convexity and monotonicity for the elliptic integrals of the first kind and applications
• Asymptotical Bounds for Complete Elliptic Integrals of the Second Kind