Monotonicity and sharp inequalities related to complete $(p,q)$-elliptic integrals of the first kind

Abstract

With the aid of the monotone L’Hôpital rule, the authors verify monotonicity of some functions involving complete $(p,q)$-elliptic integrals of the first kind and the inverse of generalized hyperbolic tangent function, derive several sharp inequalities of complete $(p,q)$-elliptic integrals of the first kind, and generalize some known sharp approximation of complete elliptic integrals of the first kind.