Let
f
f
be a meromorphic function on the complex plane with Nevanlinna characteristic
T
(
r
,
f
)
T(r,f)
and maximal radial characteristic
ln
M
(
t
,
f
)
\ln M(t,f)
, where
M
(
t
,
f
)
M(t,f)
is the maximum of the modulus
|
f
|
|f|
on circles centered at zero and of radius
t
t
. A number of well-known and widely used results make it possible to estimate from above the integrals of
ln
M
(
t
,
f
)
\ln M (t,f)
over subsets
E
E
on segments
[
0
,
r
]
[0,r]
in terms of
T
(
r
,
f
)
T(r,f)
and the linear Lebesgue measure of
E
E
. In the paper, such estimates are obtained for the Lebesgue–Stieltjes integrals of
ln
M
(
t
,
f
)
\ln M(t,f)
with respect to an increasing integration function
m
m
, and the sets
E
E
on which the function
m
m
is not constant can have fractal nature. At the same time, it is possible to obtain nontrivial estimates in terms of the
h
h
-content and
h
h
-Hausdorff measure of the set
E
E
, as well as their partial
d
d
-dimensional power versions with
d
∈
(
0
,
1
]
d\in (0,1]
. All preceding similar estimates known to the author correspond to the extreme case of
d
=
1
d=1
and an absolutely continuous integration function
m
m
with density of class
L
p
L^p
for
p
>
1
p>1
. The main part of the presentation is carried out immediately for the differences of subharmonic functions, or
δ
\delta
-subharmonic functions, on circles centered at zero with explicit constants in the estimates. The only restriction in the main theorem is that the modulus of continuity of the function
m
m
should satisfy the Dini condition at zero, and this condition, as a counterexample shows, is essential.