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In this paper, we study the generalized complete ( p, q )-elliptic integrals of the first and second kind as an application of generalized trigonometric functions with two parameters, and establish the monotonicity, generalized convexity and concavity of these functions. In particular, some Turán type inequalities are given. Finally, we also show some new series representations of these functions by applying Alzer and Richard’s methods.

On Generalized Complete (p, q)-Elliptic Integrals