Endpoint Absolute Monotonicity for the Complete Elliptic Integral of the First Kind
Abstract
Let fp(x)=𝒦(√x)/(p−ln√(1−x)), where 𝒦 denotes the complete elliptic integral of the first kind, and set gp=1/fp. Tian and Yang conjectured that, at p=ln4, both −fln4′′′ and gln4 are absolutely monotonic on (0,1). Writing 2/πfln4(x)=∑n=0∞an(ln4)xn, π/2gln4(x)=∑n=0∞bn(ln4)xn, we prove an(ln4)<0 (n≥3), bn(ln4)>0 (n≥0), thereby settling both conjectures. The proof of the first inequality is based on the representation an(ln4)=((−1)nC)/(15n+1)−Mn, where C>0 and (Mn) is a positive moment sequence. Its strict log-convexity, together with estimates for the initial coefficients, determines the sign of every an. The second inequality follows from a factorization of the quadratic truncation of the normalized Taylor series and a reciprocal-series argument.
Related
- Maximal Absolute Monotonicity of Balanced Rational Functions with a Single Repeated Pole
- Viscosity solution of systems of integral-partial differential equations with interconnected obstacles without Monotonicity Conditions and infinite Lévy measure
- Symmetry, monotonicity, and asymptotics of singular solutions to semilinear elliptic equations
- Multilinear rough singular integral operators may not be of the weak type at the endpoint
- Deciding monotonicity of simple drawings of the complete graph
BibTeX
@misc{deng2026endpoint,
title = {{Endpoint Absolute Monotonicity for the Complete Elliptic Integral of the First Kind}},
author = {Qintao Deng and Yunjia Kou and Fulin Zhong},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.32366},
eprint = {2609.32366},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.32366},
}
RIS
TY - UNPB
TI - Endpoint Absolute Monotonicity for the Complete Elliptic Integral of the First Kind
AU - Qintao Deng
AU - Yunjia Kou
AU - Fulin Zhong
PY - 2026
DA - 2026/09/26
PB - arXiv
DO - 10.48550/arxiv.2609.32366
UR - https://arxiv.org/abs/2609.32366
AB - Let f_p(x)=𝒦(√x)/(p−ln√(1−x)), where 𝒦 denotes the complete elliptic integral of the first kind, and set g_p=1/f_p. Tian and Yang conjectured that, at p=ln4, both −f_ln4^′′′ and g_ln4 are absolutely monotonic on (0,1). Writing 2/πf_ln4(x)=∑_n=0^∞a_n(ln4)x^n, π/2g_ln4(x)=∑_n=0^∞b_n(ln4)x^n, we prove a_n(ln4)<0 (n≥3), b_n(ln4)>0 (n≥0), thereby settling both conjectures. The proof of the first inequality is based on the representation a_n(ln4)=((−1)^nC)/(15^n+1)−M_n, where C>0 and (M_n) is a positive moment sequence. Its strict log-convexity, together with estimates for the initial coefficients, determines the sign of every a_n. The second inequality follows from a factorization of the quadratic truncation of the normalized Taylor series and a reciprocal-series argument.
ER -