2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/62786We 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.Complex VariablesDynamical Systems34C07;46E15text