Let f be an entire transcendental function and let (λn) be a sequence of positive numbers increasing to +∞. Suppose that the series \(A\left(Z\right)={\sum }_{n=1}^{\infty }{a}_{n}f\left({\lambda }_{n}z\right)\) is regularly convergent in ℂ, i.e., 𝔐(r, A) := \({\sum }_{n=1}^{\infty }\left|{a}_{n}\right|{M}_{f}\left(r{\lambda }_{n}\right)\) < + ∞ for all r ∈ [0,+ ∞). For a positive function l continuous on [0, + ∞), the function A is called a function of bounded l-𝔐-index if there exists N ∈ ℤ+ such that \(\frac{\mathfrak{M}\left(r,{A}^{\left(n\right)}\right)}{n!{l}^{n}\left(r\right)}\le \text{max}\left\{\frac{\mathfrak{M}\left(r,{A}^{\left(k\right)}\right)}{k!{l}^{k}\left(r\right)}:0\le k\le N\right\}\) for all n ∈ ℤ+ and all r ∈ [0,+ ∞). We study the properties of growth of the functions of bounded l- 𝔐-index and formulate some unsolved problems.
