Monotonicity properties of generalized elliptic integrals with respect to the parameter

For a is an element of (0, 1/2] and r is an element of (0, 1), let K-a(r) and E-a(r) (K(r) and E(r)) denote the generalized elliptic integrals (the complete elliptic integrals, respectively) of the first and second kinds, respectively. In this paper, the authors present the necessary and sufficient conditions under which certain familiar combinations defined in terms of the generalized elliptic integrals and elementary functions are monotone in a is an element of (0, 1/2]. By these results, sharp bounds expressed in terms of K(r), E(r) and elementary functions are obtained for K-a(r) and E-a(r), thus showing the dependence on the parameter a of K-a(r) and E-a(r), and substantially improving the known related results. (C) 2020 Elsevier Inc. All rights reserved.