An optimal power mean inequality for the complete elliptic integrals

摘要

In this work, we prove that Mp(K(r),?(r))>π2 for all r∈(0,1) if and only if p≥-1/2, where Mp(x,y) denotes the power mean of order p of two positive numbers x and y, and K(r) and ? (r) denote the complete elliptic integrals of the first and second kinds, respectively.