Optimal Convex Combination Bounds for Toader Mean

In this paper, the authors prove that the double inequalitiesα1T(a, b) + (1−α1)H(a, b)< T[A(a, b), G(a, b)]< β1T(a, b) + (1−β1)H(a, b),α2T(a, b) + (1−α2)G(a, b)< T[A(a, b), G(a, b)]< β2T(a, b) + (1−β2)G(a, b)hold for alla, b >0 witha̸=bif and only ifα1≤1/2,β1≥3/5,α2≤1/3 andβ2≥1/2.HereT(a, b),T[A(a, b), G(a, b)],H(a, b) andG(a, b) are the Toader, Toader-type, harmonicand geometric means ofaandb, respectively.