Optimal Inequalities between Harmonic, Geometric, Logarithmic, and Arithmetic-Geometric Means

[摘要] We find the least valuesp,q, andsin (0, 1/2) such that theinequalitiesH(pa+(1 − p)b,pb+(1 − p)a)>AG(a,b),G(qa+(1−q)b,qb+(1−q)a)>AG(a,b), andL(sa+(1−s)b,sb+(1−s)a)>AG(a,b)hold for alla,b>0witha≠b, respectively. HereAG(a,b),H(a,b),G(a,b), andL(a,b)denote the arithmetic-geometric, harmonic, geometric, and logarithmic meansof two positive numbersaandb, respectively.

[学科分类] 应用数学