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 K a ( 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.