Sharp bounds for the elliptic integral of the first kind in terms of two classes of logarithmic-type functions

For r ? (0, 1), let K(r) be the complete elliptic integral of the first kind. In this paper, by introducing two classes of logarithmic-type functions, and proving the monotonicity and absolutely monotonicity properties of certain functions involving K(r) and the logarithmic-type functions, several new functional inequalities for K(r) will be derived, which improve some previous known results.