An optimal power mean inequality for the complete elliptic integrals

In this work, we prove that M p ( K ( r ) , E ( r ) ) > π / 2 for all r ∈ ( 0 , 1 ) if and only if p ≥ − 1 / 2 , where M p ( x , y ) denotes the power mean of order p of two positive numbers x and y , and K ( r ) and E ( r ) denote the complete elliptic integrals of the first and second kinds, respectively.