A generalization of completely regular growth. (Q2782536)

The paper deals with entire functions of a single complex variable of order \(\rho\) and normal type, the set of which is denoted by \(A(\rho)\). For \(f\in A(\rho)\), its \textit{Phragmén-Lindelöf indicator} is defined by NEWLINE\[NEWLINE h_f (\phi) = \limsup_{r\rightarrow \infty} r^{-\rho}\log| f(re^{i\phi})| . NEWLINE\]NEWLINE Such a function is called \textit{a function of completely regular growth} (a \(CRG\)-function) in the sense of Levin-Pfluger [see \textit{B. Ya. Levin}, Distribution of zeros of entire functions, Providence, R.I.: AMS (1964; Zbl 0152.06703)], if NEWLINE\[NEWLINE | z| ^{-\rho}\log| f(z)| - h_f (\arg z) \rightarrow 0, NEWLINE\]NEWLINE as \(z\rightarrow \infty\) outside of a small set (a \(C_0\)-set). For a fixed \(T>1\), let \(t_n = T^n\), \(n\in {\mathbb N}\). The authors introduce the following generalization of the above indicator function NEWLINE\[NEWLINE h_f (z, T) = \limsup_{n\rightarrow \infty} t^{-\rho}_n \log| f(t_n z)| . NEWLINE\]NEWLINE By means of it, they define a \(T-CRG\)-function as an element of \(A(\rho)\), for which NEWLINE\[NEWLINE | z| ^{-\rho}\log| f(z)| - h_f ( z, T) \rightarrow 0, NEWLINE\]NEWLINE as \(z\rightarrow \infty\) outside of a \(C_0\)-set. A number of properties of such a new indicator function and of \(T-CRG\)-functions are described.NEWLINENEWLINEFor the entire collection see [Zbl 0980.00031].