Motivated by the work of Alzer and Richards (Anal Math 41:133–139, 2015), here authors study the monotonicity and convexity properties of the function$$\begin{aligned} \Delta _{p,q} (r) = \frac{{E_{p,q}(r) - \left( {r'} \right) ^p K_{p,q}(r) }}{{r^p }} - \frac{{E'_{p,q}(r) - r^p K'_{p,q}(r) }}{{\left( {r'} \right) ^p }}, \end{aligned}$$where \(K_{p,q}\) and \(E_{p,q}\) denote the complete (p, q)-elliptic integrals of the first and second kind, respectively.

中文翻译：

---

关于涉及广义完整（p，q）椭圆积分的函数

受Alzer和Richards的研究（2015年肛门数学41：133–139）的激励，作者研究了函数$$ \ begin {aligned} \ Delta _ {p，q}（r）=的单调性和凸性。\ frac {{E_ {p，q}（r）-\ left（{r'} \ right）^ p K_ {p，q}（r）}} {{r ^ p}}-\ frac {{E '_ {p，q}（r）-r ^ p K'_ {p，q}（r）}} {{\ left（{r'} \ right）^ p}}，\ end {aligned} $ $其中\（K_ {p，q} \）和\（E_ {p，q} \）分别表示第一类和第二类的完整（p，  q）椭圆积分。