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Takeuchi's generalized complete elliptic integrals related to generalized trigonometric functions are of importance in the computation of the generalized pi and in the elementary proof of Ramanujan's cubic transformation. In this paper we generalize some well-known results of the classical complete elliptic integrals to the case of Takeuchi's generalized complete elliptic integrals. We obtain sharp monotonicity and convexity results for combinations of these functions, as well as functional inequalities.