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Let $f$ be a meromorphic function on the complex plane $\mathbb C$ with themaximum function of its modulus $M(r,f)$ on circles centered at zero of radius$r$. A number of classical, well-known and widely used results allow us toestimate from above the integrals of the positive part of the logarithm$\ln^+M(t,f)$ over subsets of $E\subset [0,r]$ via the Nevanlinnacharacteristic $T(r,f)$ and the linear Lebesgue measure of the set $E$. Thepaper gives similar estimates for the Lebesgue-Stieltjes integrals of $\ln^+M(t,f)$ over the increasing integration function of $m$. The main part of thepresentation is conducted immediately for the differences of subharmonicfunctions on closed discs with the center at zero, i.e, $\delta$-subharmonicfunctions. The only condition in the main theorem is the Dini condition for themodulus of continuity of the integration function $m$. This condition is, in asense, necessary. Thus, the first part of the work to a certain extentcompletes in a general form the research on the upper estimates of theintegrals of the radial maximum growth characteristics of arbitrary meromorphicand $\delta$-subharmonic functions through the Nevanlinna characteristic withits versions and through the characteristics of the integration function $m$.

Meromorphic functions and differences of subharmonic functions in  integrals and the difference characteristic of Nevanlinna. I. Radial maximum  growth characteristics