SHARP BOUNDS FOR TOADER-TYPE MEANS IN TERMS OF TWO-PARAMETER MEANS  

作　　者：Yueying YANG Weimao QIAN Hongwei ZHANG Yuming CHU 杨月英;钱伟茂;张宏伟;褚玉明(School of Mechanical and Electrical Engineering,Huzhou Vocational&Technical College,Huzhou 313000,China;School of Continuing Education,Huzhou Vocational&Technical College,Huzhou 313000,China;School of Mathematics and Statistics,Changsha University of Science&Technology,Changsha 410014,China;Department of Mathematics,Huzhou University,Huzhou 313000,China)

机构地区：[1]School of Mechanical and Electrical Engineering,Huzhou Vocational&Technical College,Huzhou 313000,China [2]School of Continuing Education,Huzhou Vocational&Technical College,Huzhou 313000,China [3]School of Mathematics and Statistics,Changsha University of Science&Technology,Changsha 410014,China [4]Department of Mathematics,Huzhou University,Huzhou 313000,China

出　　处：《Acta Mathematica Scientia》2021年第3期719-728,共10页数学物理学报（B辑英文版）

基　　金：supported by the Natural Science Foundation of China(61673169,11301127,11701176,11626101,11601485)。

摘　　要：In the article,we prove that the double inequalities Gp[λ1a+(1-λ1)b,λ1 b+(1-λ1)a]A1-p(a,b)<T[A(a,b),G(a,b)]<Gp[μ1 a+(1-μ1)b,μ1b+(1-μ1)a]A1-p(a,b),Cs[λ^(2) a+(1-λ2)b,λ2 b+(1-λ2)a]A1-s(a,b)<T[A(a,b),Q(a,b)]<Cs[μ2 a+(1-μ2)b,μ2 b+(1-μ2)a]A1-p(a,b)hold for all a,b>0 with a≠b if and only ifλ1≤1/2-(1-(2/π)2/p)1/2/2,μ1≥1/2-(2p)1/2/(4 p),λ2≤1/2+(2(3/(2 s)(E(21/2/2)/π)1/s)-1)1/2/2 andμ2≥1/2+s1/2/(4 s)ifλ1,μ1∈(0,1/2),λ2,μ2∈(1/2,1),p≥1 and s≥1/2,where G(a,b)=(ab)1/2,A(a,b)=(a+b)/2,T(a,b)=∫0π/2(a2 cos2 t+b2 sin2)1/2 tdt/π,Q(a,b)=((a2+b2)/2)1/2,C(a,b)=(a2+b2)/(a+b)and E(r)=∫0π/2(1-r^(2) sin^(2))1/2 tdt.

关 键 词：Geometric mean arithmetic mean Toader mean ontraharmonic mean complete elliptic integral 

分 类 号：O172.2[理学—数学]