  Let $E$ be a measurable subset in a segment $[0,r]$ in the positive part of
the real axis in the complex plane, and $U=u-v$ be the difference of
subharmonic functions $u\not\equiv -\infty$ and $v\not\equiv-\infty$ on the
complex plane. An integral of the maximum on circles centered at zero of
$U^+:=\sup\{0,U\} $ or $|u|$ over $E$ with a function-multiplier $g\in L^p(E)$
in the integrand is estimated, respectively, in terms of the characteristic
function $T_U$ of $U$ or the maximum of $u$ on circles centered at zero, and
also in terms of the linear Lebesgue measure of $E$ and the $ L^p$-norm of $g$.
Our main theorem develops the proof of one of the classical theorems of Rolf
Nevanlinna in the case $E=[0,R]$, given in the classical monograph by Anatolii
A. Gol'dberg and Iosif V. Ostrovskii, and also generalizes analogs of the
Edrei-Fuchs Lemma on small arcs for small intervals from the works of A.F.
Grishin, M.L. Sodin, T.I. Malyutina. Our estimates are uniform in the sense
that the constants in these estimates do not depend on $U$ or $u$, provided
that $U$ has an integral normalization near zero or $u(0)\geq 0$, respectively.
