Sharing values and normality

Abstract

According to Bloch's principle every condition, which reduces a meromorphic function in the plane to a constant, makes a family of meromorphic functions in a domain G normal. Although the principle is false in general ([6]), many authors proved normality criteria for families of meromorphic functions by starting from picard type theorems ([1], [7], [8]). Yang ([911 expanded such considerations to singular direcnons. It seems. however, that an attempt has not yet been made to prove normality criteria using conditions known from sharing value theorems (two meromorphic functions f and g share a e I~ iff f -1 ({a}) = 0 -1 ({a})). Nevanlinna's famous five point theorem says that two meromorphic functions in the plane which share five distinct values are identical. But if each pair f and g of meromorphic functions of a family shares five fixed values a t, the sets f-1 ({as}) are independent from f and the normality follows immediately from Montel's theorem. Therefore it is more interesting to discuss conditions, where a function f e F is not compared with all other functions in the family, but with specia! functions related to f. Mues and Steinmetz ([5]) proved