  Let $\mathbb C$ be the complex plane, $E$ be a measurable subset in a segment
$[0, R]$ of the positive semiaxis $\mathbb R^+$, $u\not\equiv -\infty$ be a
subharmonic function on $\mathbb C$. The main result of this article is an
upper estimate of the integral of the module $|u|$ over a subset of $E$ through
the maximum of the function $u$ on a circle of radius $R$ centered at zero and
a linear Lebesgue measure of subset $E$. Our result develops one of the
classical theorems of R. Nevanlinna in the case of $E=[0, R]$ and versions of
so-called Small Arcs Lemma by Edrei-Fuchs for small intervals on $\mathbb R^+$
from the works of A.F. Grishin, M.L. Sodin, T.I. Malyutina. Our obtained
estimate is uniform in the sense that the constants in the estimates are
absolute and do not depend on the subharmonic function under the
semi-normalization $u(0)\geq 0$.
