Let \(\mathbb{C}\) be the complex plane, E be a measurable subset of a segment \([0, R]\) of the positive semiaxis \(\mathbb{R}^+\), and \(u\not\equiv - \infty\) 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 the linear Lebesgue measure of subset E. Our result develops one of the classical theorems of R. Nevanlinna, established 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, and T.I. Malyutina. Our estimate is uniform in the sense that the constants in the inequality are absolute and do not depend on the subharmonic function u, under the semi-normalization \(u(0)\geq 0\).

令\（\ mathbb {C} \）为复平面，E为正半轴\（\ mathbb {R} ^ + \）的线段\（[0，R] \）的 可测量子集，并且\ （u \ not \ equiv-\ infty \）是\（\ mathbb {C} \）上的子谐波函数 。本文的主要结果是通过以半径为零的半径R的圆上的函数u的最大值和E的线性Lebesgue测度对E的子集上的模块\（| u | \）的积分进行较高估计。子集Ë。我们的结果建立了R. Nevanlinna的经典定理之一，该定理在\（E = [0，R] \）的情况下建立，以及AF Grishin，ML Sodin和TI Malyutina等人的著作（\ mathbb {R} ^ + \）上以小间隔出现的Edrei–Fuchs所谓的小弧引理版本 。在半正规化\（u（0）\ geq 0 \）下，在不等式中的常数是绝对的且不依赖于次谐波函数u的意义上，我们的估计是均匀的。