Density functions with respect to a model growth function
M. V. Kabanko,
K. G. Malyutin,
T. I. Malyutina Faculty of Physics, Mathematics, Computer Science, Kursk State University, Kursk, Russia
Abstract:
The properties of general density functions with respect to a model growth function
$M$ and related semiadditive functions are discussed. The concept of function of slow growth with respect to the model growth function
$M$ 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
$\rho$-semiadditive functions with respect to the model function
$M$ are presented. One of the central results is a theorem that can be viewed as an extension of Pólya's theorem on the existence of minimal and maximal densities to a wider class of functions, whose growth is bounded by an arbitrary model growth function
$M$. Examples of functions
$f$ and their density functions are presented.
Bibliography: 17 titles.
Keywords:
model growth function, semiadditive function, density function, minimal and maximal densities, Pólya's theorem.
MSC: 26A12 Received: 11.09.2024 and 17.01.2025
DOI:
10.4213/sm10188
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