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On the proximate type of an entire function. (English) Zbl 0535.30029

Let \(f(z)=\sum^{\infty}_{n=0}a_ nz^ n\) be an entire function. The proximate type T and lower proximate type t of f(z) with respect to the proximate order \(\rho\) (r) are defined as \[ \lim_{r\to \infty}\left\{ \begin{matrix} \sup \\ \inf \end{matrix} \right\}\frac{\log M(r)}{r^{\rho(r)}}=\left\{ \begin{matrix} T\\ t\end{matrix} \right\},\quad 0\leq t\leq T\leq \infty. \] In this paper the authors have studied some of the growth properties of f(z) with the help of proximate order \(\rho\) (r) and proximate type T and have also proved that the proximate type of the derivative of f(z) is the same as that of f(z) with respect to the proximate order \(\rho\) (r).
Reviewer: S.N.Srivastava

MSC:

30D15 Special classes of entire functions of one complex variable and growth estimates
30D10 Representations of entire functions of one complex variable by series and integrals
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