关键词: 广义椭圆积分, 广义Gr\"otzsch 环函数, 单调性, 不等式

Abstract: For $a\in(0,1/2]$ and $r\in(0,1)$,let $\mu_a(r)$ be the so-called generalized Gr\"otzschring function which is defined by $\mu_a(r)=[\pi\K'_a(r)]/[2\sin(\pi a)\K_a(r)]$,appearing in Ramanujan's generalized modular equations.In this paper, the authors study the relations of the functions$\mu_a(r)$ and $\mu(r)$ and the monotonicity and convexity of combinations of the function $\mu_a(r)$ and some elementary functions, and obtain several sharp
inequalities for $\mu_a(r)$. Some properties of the function $\mu(r)$ are generalized to the function $\mu_a(r)$.

Key words: Generalized elliptic integrals, Generalized Gr"otzsch ring function, Monotoniceity, Inequalities