A Sharp Lower Bound for Toader-Qi Mean with Applications

Abstract EN

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.