  Let $f$ be a meromorphic function on the complex plane $\mathbb C$ with the
maximum function of its modulus $M(r,f)$ on circles centered at zero of radius
$r$. A number of classical, well-known and widely used results allow us to
estimate from above the integrals of the positive part of the logarithm
$\ln^+M(t,f)$ over subsets of $E\subset [0,r]$ via the Nevanlinna
characteristic $T(r,f)$ and the linear Lebesgue measure of the set $E$. The
paper gives similar estimates for the Lebesgue-Stieltjes integrals of $\ln^+M
(t,f)$ over the increasing integration function of $m$. The main part of the
presentation is conducted immediately for the differences of subharmonic
functions on closed discs with the center at zero, i.e, $\delta$-subharmonic
functions. The only condition in the main theorem is the Dini condition for the
modulus of continuity of the integration function $m$. This condition is, in a
sense, necessary. Thus, the first part of the work to a certain extent
completes in a general form the research on the upper estimates of the
integrals of the radial maximum growth characteristics of arbitrary meromorphic
and $\delta$-subharmonic functions through the Nevanlinna characteristic with
its versions and through the characteristics of the integration function $m$.
