The logarithm of the modulus of a holomorphic function as a minorant for a subharmonic function

Abstract

For an arbitrary subharmonic function not identically equal to −∞ in a domain D of the complex plane C, we prove the existence of a nonzero holomorphic function in D the logarithm of whose modulus is majorized by locally averaging a subharmonic function with logarithmic additions or even without them in the case D = C.