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.

中文翻译：

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关于广义完全（p，q）椭圆积分

在本文中，我们研究了第一类和第二类广义完全（p，q）椭圆积分作为具有两个参数的广义三角函数的应用，并建立了这些函数的单调性，广义凸性和凹性。特别是，给出了一些图兰类型不等式。最后，我们还通过应用Alzer和Richard的方法展示了这些功能的一些新系列表示。