Functions of density with respect to a model function of growth

The properties of general density functions with respect to a model function of growth $M$ and related semiadditive functions are discussed. The concept of function of slow growth with respect to the model function of growth $M$ is introduced; it is shown that the function $L(r)= M^{-\rho}(r)v()r)$ has a slow growth with respect to $M$. The concept of $\rho$-semiadditive function with respect to $M$ is also introduced and the main properties of such functions are established. Density functions are studied; a criterion of the continuity of the density $N_M(\alpha)$ and lower density $\underline N_M(\alpha)$ of a function $f$ is obtained. A uniformity theorem is proved. The main properties of $\rho$-additive and -semiadditive functions with respect to the model function $M$ are presented. POn of the central results is a theorem that can be viewed as an extension of Polya's theorem on the existence of minimal and maximal densities to a wider class of functions, whose growth is bounded by an arbitrary model function of growth $M$. Examples of function $f$ and their density functions are presented.

model function of growth, semiadditive function, density function, minimal and maximal densities, Polya's theorem