广义Grtzsch环函数的若干性质

Some Properties of Generalized Grtzsch Ring Function

设0<a,r<1,r′=1-r2,Ka(r),Ea(r)分别为第一类和第二类广义椭圆积分,定义广义Grtzsch环函数为μa(r)={π/[2sin(πa)]}K′a(r)/Ka(r),μa(r)出现于广义Ramanujan模方程。文章揭示了μa(r)和广义椭圆积分的若干诸如单调性、凹凸性等性质,这些结果将有助于对广义Ramanujan模方程的研究。

Let Ka（r）, Ea（r） be the generalized elliptic integrals of the first and the second kind, respectively, where a,r∈ （0,1） and r＇=√（1-r^2）, and μa（r）={π/[2sin（πa）]}K＇a（r）/Ka（r）be generalized Groetzsch ring function which appears in generalized Ramanujan＇s modular equations. In this paper, monotoniety and convexity, of functions denoted by the generalized elliptic integrals and the generalized Groetzsch ring function are investigated. These results will be helpful to the study of the generalized Ramanujan＇s modular equations.