Optimal Convex Combination Bounds For Centroidal Mean

Various means of two positive numbers a and b and their inequalities have wideapplications in the fields of statistics, mathematics and engineering. In this paper,we consider optimal convex combination bounds for the centroidal mean in terms ofcontraharmonic, geometric, logarithmic, Seifert and golden section means. We find theoptimal values for α1, β1, α2, β2, α3, β3, α4and β4such that the inequalitiesα1C(a, b)+(1α1)Mq(a, b)<E(a, b)<β1C(a, b)+(1β1)Mq(a, b),α2C(a, b)+(1α2)G(a, b)<E(a, b)<β2C(a, b)+(1β2)G(a, b),α3C(a, b)+(1α3)L(a, b)<E(a, b)<β3C(a, b)+(1β3)L(a, b),α4C(a, b)+(1α4)P (a, b)<E(a, b)<β4C(a, b)+(1β4)P (a, b)hold for all a, b>0with a=b, where C(a, b), Mq(a, b), G(a, b), L(a, b), P (a, b) andE(a, b) be the contraharmonic mean, golden section mean, geometric mean, logarithmicmean, Seifert’s mean and centroidal mean of two positive numbers a and b, respectively.