广义Grotzsch环函数的几个精确不等式

对$a\in(0,1/2]$和$r\in(0,1)$, Ramanujan广义模方程中的广义Gr\"otzsch环函数$\mu_a(r)$定义如下:$\mu_a(r)=\pi\K'_a(r)]/2\sin(\pi a)\K_a(r)]$. 该文通过研究$\mu_a(r)$和$\mu(r)$的关系,以及$\mu_a(r)$和一些初等函数的组合的单调性和凹凸性,获得了$\mu_a(r)$的几个精确不等式,从而把$\mu(r)$的一些熟知的性质推广到$\mu_a(r)$上.

Some Sharp Inequalities for Generalized Grotzsch Ring Function

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 sharpinequalities for $\mu_a(r)$. Some properties of the function $\mu(r)$ are generalized to the function $\mu_a(r)$.