Monotonicity properties and inequalities for the generalized elliptic integral of the first kind

The authors obtain some monotonicity and concavity properties, and asymptotically sharp bounds for the generalized elliptic integral Ka(r) of the first kind and its special case K(r), the complete elliptic integral of the first kind, by studying the monotonicity and concavity properties of certain combinations defined in terms of the Gaussian hypergeometric function F(a,1−a;1;x) and elementary functions, thus improving some well-known results.