An optimal power mean inequality for the complete elliptic integrals

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