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Let $E$ be a measurable subset in a segment $[0,r]$ in the positive part ofthe real axis in the complex plane, and $U=u-v$ be the difference ofsubharmonic functions $u\not\equiv -\infty$ and $v\not\equiv-\infty$ on thecomplex 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 characteristicfunction $T_U$ of $U$ or the maximum of $u$ on circles centered at zero, andalso 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 RolfNevanlinna in the case $E=[0,R]$, given in the classical monograph by AnatoliiA. Gol'dberg and Iosif V. Ostrovskii, and also generalizes analogs of theEdrei-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 sensethat the constants in these estimates do not depend on $U$ or $u$, providedthat $U$ has an integral normalization near zero or $u(0)\geq 0$, respectively.

Integrals of subharmonic functions and their differences with weight over small sets on a ray