The authors consider the symmetric elliptic integrals and the Legendre’s incomplete integral of the first kind. These integrals are written as particular cases of the more general
-hypergeometric function and then the authors show a log-convex property of
function. From this property, analogous properties for the elliptic integrals are deduced. Using these log-convex properties, the authors derive several lower and upper bounds for the symmetric elliptic integrals and the Legendre’s incomplete integral of the first kind, as well as some inequalities between those integrals.