ABSOLUTE MONOTONICITY INVOLVING THE COMPLETE ELLIPTIC INTEGRALS OF THE FIRST KIND WITH APPLICATIONS  

作　　者：Zhenhang YANG 田景峰 杨镇杭;田景峰(Engineering Research Center of Intelligent Computing for Complex Energy Systems of Ministry of Education,North China Electric Power University,Baoding 071003,China;Zhejiang Society for Electric Power,Hangzhou 310014,China;Department of Mathematics and Physics,North China Electric Power University,Baoding 071003,China)

机构地区：[1]Engineering Research Center of Intelligent Computing for Complex Energy Systems of Ministry of Education,North China Electric Power University,Baoding 071003,China [2]Zhejiang Society for Electric Power,Hangzhou 310014,China [3]Department of Mathematics and Physics,North China Electric Power University,Baoding 071003,China

出　　处：《Acta Mathematica Scientia》2022年第3期847-864,共18页数学物理学报（B辑英文版）

摘　　要：Let K(r)be the complete elliptic integrals of the first kind for r∈(0,1)and f_(p)(x)=[(1−x)^(p)K(√x)].Using the recurrence method,we find the necessary and sufficient conditions for the functions−f′_(p),ln f_(p),−(ln f_(p))^((i))(i=1,2,3)to be absolutely monotonic on(0,1).As applications,we establish some new bounds for the ratios and the product of two complete integrals of the first kind,including the double inequalities exp[r^(2)(1−r^(2))/^(64)]/(1+r)^(1/4)<K(r)/K(√r)<exp[−r(1−r)/4],π/2 exp[θ0(1−2r^(2))]<π/2 K(r′)/K(r)<π/2(r′/r)^(p)exp[θ_(p)(1−2r^(2))],K^(2)(1/√2)≤K(r)K(r′)≤1/√2rr′K^(2)(1/√2)for r∈2(0,1)and p≥13/32,where r′=√1−r^(2) and θ_(p)=2Γ(3/4)^(4)/π^(2)−p.

关 键 词：Complete elliptic integrals of the first kind absolute monotonicity hypergeometric series recurrence method INEQUALITY 

分 类 号：O172.2[理学—数学]