A Sharp Lower Bound for Toader-Qi Mean with Applications. (17th January 2016) - Yang, Zhen-Hang; Chu, Yu-Ming
Abstract : We prove that the inequality T Q ( a, b ) > L p ( a, b ) holds for all a, b > 0 with a ≠ b if and only if p ≤ 3 / 2, where T Q ( a, b ) = 2 / π ∫ 0 π / 2 a c o s 2 θ b s i n 2 θ d θ, L p ( a, b ) = [ ( b p - a p ) / ( p ( b - a ) ) ] 1 / p     ( p ≠ 0 ), and L 0 ( a, b ) = a b are, respectively, the Toader-Qi and p -order logarithmic means of a and b . As applications, we find two fine inequalities chains for certain bivariate means.