Abstract
We prove that the double inequality(π/2)(arthr/r)3/4+α*r<K(r)<(π/2)(arthr/r)3/4+β*rholds for allr∈(0,1)with the best possible constantsα*=0andβ*=1/4, which answer to an open problem proposed by Alzer and Qiu. Here,K(r)is the complete elliptic integrals of the first kind, and arth is the inverse hyperbolic tangent function.
Highlights
For r ∈ 0, 1, Lengedre’s complete elliptic integrals of the first and second kind 1 are defined byK Kr π/2 1 − r2sin2θ −1/2dθ,K K r Kr, K0 π 2 K 1 ∞,E Er π/2 1 − r2sin2θ 1/2dθ,E E r Er, E0 π 2E 1 1, respectively
In order to establish our main result, we need several formulas and lemmas, which we present
From 2.3 and Lemma 2.1 together with l’Hopital’s rule, we know that f1 r is strictly increasing in 0, 1, f1 0 2/3 and f1 1− 1
Summary
On Alzer and Qiu’s Conjecture for Complete Elliptic Integral and Inverse Hyperbolic Tangent Function. We prove that the double inequality π/2 arthr/r 3/4 α∗r < K r < π/2 arthr/r 3/4 β∗r holds for all r ∈ 0, 1 with the best possible constants α∗ 0 and β∗ 1/4, which answer to an open problem proposed by Alzer and Qiu. Here, K r is the complete elliptic integrals of the first kind, and arth is the inverse hyperbolic tangent function
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