球面导数，高阶导数与正现族

本文应用陈怀惠和顾永兴关于Ｚａｌｃｍａｎ不正规性条件的改进结果，推广和加强了Ｌａｐｐａｎ的一个正规定则．Ｌａｐｐａｎ证明：若亚纯函数族中的所有函数的球面导数的幂方（大于２）在紧集上的积分一致有界，则该族是正则的．本文证明，把积分限制在函数值的模小于给定常数的子集上，结论仍然成立．同时，用高阶导数的积分替代球面导数的积分，得到十分一般的结果．另外对幂方为２的情形也进行了讨论．

Spherical Derivative,Higher Derivatives and Normal Families

This paper extends and improves a criterion due to Aulaskari and Lappan about normal families by applying an improvemed,due to Chen and Gu,of the Zalcman's condition for nonnormality.Aulaskari and Lappan showed that a family of meromophic functions is a normal family if,for all functions in the family,the integrals of a power(greater than 2)of the spherical derivatives are uniformly bounded on each compact set.The present paper proves the same conclusion under the weaker condition that the integration is performed only on sets on which the functions are bounded by a given constant.A general result is obtained when the spherical derivative is replaced by a higher order derivative in the integral under considerations.Also,it is proved that the result keeeps true for a power equal to 2 if the functions in the family omit two complex values.