Optimal power-type Heronian and Lehmer means inequalities for the complete elliptic integrals

In this paper, we prove that the inequalities Hp(K(r),E(r))<π2,Lq(K(r),E(r))<π2hold for all r∈(0,1) if and only if p≤−log⁡3/log⁡(π/2) and q<−1, where Hp(a,b) and Lq(a,b) are respectively the p-th power-type Heronian mean and q-th Lehmer mean of a and b, and K(r) and E(r) are respectively the complete elliptic integrals of the first and second kinds.