  The maximum of the modulus of a meromorphic function cannot be restricted
from above by the Nevanlinna characteristic of this meromorphic function. But
integrals from the logarithm of the module of a meromorphic function allow
similar restrictions from above. This is illustrated by one of the important
theorems of Rolf Nevanlinna in the classical monograph by A.A. Goldberg and
I.V. Ostrovskii on meromorphic functions, as well as by the Edrei-Fuchs Lemma
on 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 of
subharmonic functions even with weights were recently obtained by B.N.
Khabiblullin, L.A. Gabdrakhmanova. All these results are on integrals over
subsets on a ray. In this article, we establish such results for integrals of
the logarithm of the modulus of a meromorphic function and the difference of
subharmonic functions over discs and planar small sets. Our estimates are
uniform in the sense that the constants in these estimates are explicitly
written out and do not depend on meromorphic functions and the difference of
subharmonic functions provided that these functions has an integral
normalization near zero.
