一类解析函数的Jensen不等式
A Jensen Inequality for a Family of Analytic Functions
摘要：
我们改进了一个估计（在“A.Brudnyi，Small amplitude极限环and the distribution of zeroes of families of解析函数，Ann. of Math.154（2）（2001），227-243”）对于位于原点邻域中的平面多项式向量场的极限环的平均数，假设在较大邻域中的场足够接近线性中心。这个结果是由一个新的分布不等式得出的，该不等式关于一个单变量全纯函数族的零点个数，全纯地依赖于一个参数。
英文摘要：
We improve an estimate (obtained in "A.Brudnyi, Small amplitude limit cycles and the distribution of zeros of families of analytic functions, Ann. of Math. 154 (2) (2001), 227-243") for the average number of limit cycles of a planar polynomial vector field situated in a neighbourhood of the origin provided that the field in a larger neighbourhood is close enough to a linear center. The result follows from a new distributional inequality for the number of zeros of a family of univariate holomorphic functions depending holomorphically on a parameter.
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