函数的增长性质及其推广

【作者】 张艳慧；

【导师】 邓冠铁；

【作者基本信息】 北京师范大学 ， 基础数学， 2005， 博士

【摘要】 本文得到右半平面中几类解析函数与调和函数的积分表示,利用Cartan估计和Hayman定理的方法,研究了它们的增长性质,并且把这些增长性质推广到了n维欧式半空间中.在右半平面中,作者主要考虑了级小于2的调和函数当它在边界上连续时的积分表示和级小于2的解析函数当它在边界上奇异时的积分表达式,并得到了它们的增长性质,这一性质直接推广了Hayman定理。作者应用这种方法还得到了右半平面中一类级小于3的解析函数的积分表示,同时作者还得到右半平面中一类由积分表示的次调和函数的增长性质。在n维欧式半空间中,作者研究了一类次调和函数的增长性质,一类调和函数(由Piosson积分表示的Dirichlet问题的解)的渐进性质和另一类调和函数(由修正的Piosson积分表示的Dirichlet问题的解)的渐进性质。在n维欧式半空间中,作者得到了调和函数的Carleman公式和具体的半球上调和函数的清楚表达式,并且由一个调和函数的上界得到了该调和函数的下界,当n=2时就是复平面中的结果。这些结果推广了经典的复变函数理论,深刻揭示了复变函数和调和分析之间的联系与区别。更多还原

【Abstract】 In these thesis, the author studied the integral representations of several classes of analytic and harmonic functions with new conditions, and gave the growth estimates by the ways of Cartan’s estimate and Hayman’s theorem, then the author generalized those estimate properties to n-dimension Euclidean half space.On the right half plane, the author focused on the integral representions of harmonic functions that countious to the boundary and analytic functions that singular to the boundary with order less than 2, and proved their estimate properties, which generalized the Hayman theorem. The author gave the integral repressentation of a class of analytic functions with order less than 3, at the same time the author gave the growth properties of a class of subharmonic functions and functions whose order is finite in right half plane.On the n dimensional Euclidean half space, the author studied the growth properties of subharmonic functions, the asymptotic properties of a class of harmonic functions(the solutions of half space Dirichlet problem represented by Poisson integral) and another class of harmonic functions (the modified solutions of half space Dirichlet problem represented by the modified Poisson integral).On the n dimensional Euclidean half space, the author derived the Carleman formula of harmonic functions and the precise representation of harmonic function in the half sphere by using Hormander’s theorem, the author also derivd a lower bound of harmonic function in the half space from the upper bound by using the representation of harmonic function in the half sphere and the carleman formula for the half space, which is the results of complex plane when n = 2.These results generalize the theory of classic complex variables functions, illustrate the links and the difference between the complex variables functions and the harmonic analysis.更多还原