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

In the article, we prove that the inequalities

$${H_p}({\cal K}(r),{\cal E}(r)) > {\pi \over 2},\,\,\,\,\,\,{L_q}({\cal K}(r),{\cal E}(r)) > {\pi \over 2}$$

hold for all r ∈ (0, 1) if and only if p ≥ −3/4 and q ≥ −3/4, where H_p(a, b) and L_q(a, b) are respectively the p-th power-type Heronian mean and q-th Lehmer mean of a and b, and \({\cal K}(r)\) and \({\cal E}(r)\) are respectively the complete elliptic integrals of the first and second kinds.