A Sharp Lower Bound for Toader-Qi Mean with Applications - Zhen-Hang Yang; Yu-Ming Chu (Journal of Function Spaces)
We prove that the inequality TQ(a,b)>Lp(a,b) holds for all a,b>0 with a≠b if and only if p≤3/2, where TQ(a,b)=2/π∫0π/2acos2θbsin2θdθ, Lp(a,b)=[(bp-ap)/(p(b-a))]1/p  (p≠0), and L0(a,b)=ab 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.