Relative Growth of Series in Systems of Functions and Laplace—Stieltjes-Type Integrals

For a regularly converging-in-C series A(z)=?n=1?anf(?nz), where f is an entire transcendental function, the asymptotic behavior of function Mf?1(MA(r)), Mf(r)=max{|f(z)|:|z|=r}, investigated. It proven that, under certain conditions on functions f, ?, and coefficients an, equality limr?+??(Mf?1(MA(r)))?(r)=1 correct. A similar result obtained for Laplace–Stiltjes-type integral I(r)=?0?a(x)f(rx)dF(x). Unresolved problems are formulated.