广义椭圆积分的若干性质

Some Properties of Generalized Elliptic Integrals

揭示了一类重要的特殊函数———广义椭圆积分的一些诸如单调性、凹凸性等分析性质。利用单调性l’H^opital法则等分析工具来获得由广义椭圆积分定义的函数的单调性与凹凸性,进而获得几个不等式。这些结果有助于对广义Gr tzsch环函数和Ramanujan模方程及其解的研究,这些函数在拟共形理论、数论、几何学等领域中具有非常广泛的应用;同时,从这些结果中可以得到关于完全椭圆积分的一些新的性质。

In this paper, some monotoneity and convexity are obtained for the generalized elliptic integrals , which are very important special functions. We prove the monotoneity and convexity of functions that defined by the generalized elliptic integrals by monotone 1＇ Hopital＇ s rule , and obtain same inequlities of them . These results will be used to study the generalized Grotzsch ring function, Ramanujan＇ s modular equations and the solutions of them, which have been many applications in other mathematical fields, such as number theory , quaciconformal theory and geometry. Some new properties of complete elliptic integrals came be obtained from this paper.