On Boundedness of the l-ð"-Index of Entire Functions Represented by Series in a System of Functions

Let f be an entire transcendental function and let (λn) be a sequence of positive numbers increasing to +∞. Suppose that the series AZ=∑n=1∞anfλnz is regularly convergent in ℂ, i.e., ð"(r, A) := ∑n=1∞anMfrλn< + ∞ for all r ∈ [0,+ ∞). For a positive function l continuous on [0, + ∞), the function A is called a function of bounded l-ð"-index if there exists N ∈ ℤ+ such that Mr,Ann!lnr≤maxMr,Akk!lkr:0≤k≤N for all n ∈ ℤ+ and all r ∈ [0,+ ∞). We study the properties of growth of the functions of bounded l- ð"-index and formulate some unsolved problems.