Abstract
In this paper, we prove that the inequalities H p ( K ( r ) , E ( r ) ) < π 2 , L q ( K ( r ) , E ( r ) ) < π 2 hold for all r ∈ ( 0 , 1 ) if and only if p ≤ − log 3 / log ( π / 2 ) and q<−1, 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 K ( r ) and E ( r ) are respectively the complete elliptic integrals of the first and second kinds.
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