Set of asymptotic values of a meromorphic function of finite order

Abstract

It is well known that an entire function of finite order has an at most countable number of asymptotic values. Gross [I] constructed an example of an entire function of infinite order whose set of asymptotic values coincides with the extended complex plane. Dreisin and Weizmann [2] raised the question of whether there are any restrictions from above on the size of the set of asymptotic values of a meromorphic function of finite order. In this direction only the result of Valiron [3] is known; he constructed a meromorphic functlon of finite order with a set of asymptotic values having the cardinality of the continuum 9 In this paper we construct a meromorphic function of finite order whose set of asymptotic values coincides with the extended complex plane, thereby giving a negative answer to the question of Dreisin and Weizmann.