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The maximum of the modulus of a meromorphic function cannot be restrictedfrom above by the Nevanlinna characteristic of this meromorphic function. Butintegrals from the logarithm of the module of a meromorphic function allowsimilar restrictions from above. This is illustrated by one of the importanttheorems of Rolf Nevanlinna in the classical monograph by A.A. Goldberg andI.V. Ostrovskii on meromorphic functions, as well as by the Edrei-Fuchs Lemmaon small arcs and its versions for small intervals in articles by A.F. Grishin,M.L. Sodin, T.I. Malyutina. Similar results for integrals of differences ofsubharmonic functions even with weights were recently obtained by B.N.Khabiblullin, L.A. Gabdrakhmanova. All these results are on integrals oversubsets on a ray. In this article, we establish such results for integrals ofthe logarithm of the modulus of a meromorphic function and the difference ofsubharmonic functions over discs and planar small sets. Our estimates areuniform in the sense that the constants in these estimates are explicitlywritten out and do not depend on meromorphic functions and the difference ofsubharmonic functions provided that these functions has an integralnormalization near zero.

Integrals of the difference of subharmonic functions over discs and planar small sets