Sharp bounds for the lemniscatic mean by the one-parameter geometric and quadratic means

In the article, we present the best possible parameters alpha(1), alpha(2), alpha(3), alpha(4), beta(1), beta(2), beta(3) and beta(4) on the interval (0, 1) such that the double inequalities G(alpha 1) (a, b) < LMGA (a, b) < G(beta 1) (a, b), G(alpha 2) (a, b) < LMAG (a, b) < G(beta 2) (a, b), Q(alpha 3) (a, b) < LMAQ (a, b) < Q(beta 3) (a, b) and Q(alpha 4) (a, b) < LMQA (a, b) < Q(beta 4) (a, b) hold for a, b > 0 with a not equal b, where G(p)(a, b) and Q(p)(a, b) are respectively the one-parameter geometric and quadratic means, LMGA(a, b), LMAG(a, b), LMAQ(a, b) and LMQA(a, b) are four lemniscatic means of a and b. As applications, some new bounds for the arc lemniscate functions are given.