  Let $D$ be a domain in the complex plane, $M$ be an extended real function on
$D$. If $f$ is a non-zero holomorphic function on $D$ with an upper constraint
$|f|\leq \exp M$ on this domain $D$, then it is natural to expect that there
must be some upper constraints on the distribution of zeros of this holomorphic
function exclusively in terms of the function $M$ and the geometry of the
domain $D$. We have investigated this question in detail in our previous works
in the case when $M$ is a subharmonic function and the domain $D$ is arbitrary
or with a non-polar boundary. The answer was given in terms of limiting the
distribution of zeros of $f$ from above via the Riesz measure of the
subharmonic function $M$. In this article, the function $M$ is the difference
of subharmonic functions, or a $\delta$-subharmonic function, and the upper
constraints are given in terms of the Riesz charge of this $\delta$-subharmonic
function $M$. These results are also new to a certain extent for the
subharmonic function $M$. The case when the domain D is the complex plane is
considered separately. For the complex plane, it is possible to reach the
criterion level.
