The Logarithm of the Modulus of a Holomorphic Function as a Minorant for a Subharmonic Function

For an arbitrary subharmonic function not identically equal to $-\infty$ in a domain $D$ of the complex plane $\mathbb C$, 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 $D=\mathbb C$.