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

Abstract

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

Keywords

• relative growth
• entire function
• regularly converging series
• Mittag–Leffler function