ABSOLUTE MONOTONICITY INVOLVING THE COMPLETE ELLIPTIC INTEGRALS OF THE FIRST KIND WITH APPLICATIONS

Let K(r) be the complete elliptic integrals of the first kind for r 2 (0, 1) and f(p) (x) = [(1 - x)(p) K(root x)]. Using the recurrence method, we find the necessary and sufficient conditions for the functions -f(p)', ln f(p), - (ln f(p))((i)) (i = 1, 2, 3) to be absolutely monotonic on (0, 1). As applications, we establish some new bounds for the ratios and the product of two complete integrals of the first kind, including the double inequalities exp [r(2) (1 - r(2))/64]/(1 + r)(1/4) < K(r)/K(root r) < exp < [- r(1 - r)/4], pi/2 exp [theta(0) (1 - 2r(2))] < pi K(r')/2 K(r) < pi/2 (r'/r)(p) exp [theta(p)(1 - 2r(2))], K-2 (1/root 2) <= K(r)K(r') <= 1/root 2rr' K-2 (1/root 2) for r is an element of (0, 1) and p >= 13/32, where r' = root 1 - r(2) and theta(p) = 2 Gamma (3/4)(4) /pi(2) - p.