The logarithm of the modulus of a holomorphic function as a minorant for a subharmonic function. II. The complex plane

Abstract

Let u ≢ −∞be a subharmonic function in the complex plane. We establish necessary and/or sufficient conditions for the existence of a nonzero entire function f for which the modulus of the product of each of its kth derivative k = 0, 1,..., by any polynomial p is not greater than the function Ce^u in the entire complex plane, where C is a constant depending on k and p. The results obtained significantly strengthen and develop a number of results of Lars Hörmander (1997).