On Properties of Karamata Slowly Varying Functions with Remainder and Their Applications

In this paper, we study the asymptotic properties of slowly varying functions of one real variable in the sense of Karamata. We establish analogs of fundamental theorems on uniform convergence and integral representation for slowly varying functions with a remainder depending on the types of remainder. We also prove several important theorems on the asymptotic representation of integrals of Karamata functions. Under certain conditions, we observe a "narrowing" of classes of slowly varying functions concerning the types of remainder. At the end of the paper, we discuss the possibilities of the application of slowly varying functions in the theory of stochastic branching systems. In particular, under the condition of the finiteness of the moment of the type Exlnx for the particle transformation intensity, it is established that the property of slow variation with a remainder is implicitly present in the asymptotic structure of a non-critical Markov branching random system.