Skip to Main Content

Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48.

What is MCQ? The Mathematical Citation Quotient (MCQ) measures journal impact by looking at citations over a five-year period. Subscribers to MathSciNet may click through for more detailed information.


On entire functions of fast growth
HTML articles powered by AMS MathViewer

by S. K. Bajpai, G. P. Kapoor and O. P. Juneja PDF
Trans. Amer. Math. Soc. 203 (1975), 275-297 Request permission


Let \[ (\ast )\quad f(z) = \sum \limits _{n = 0}^\infty {{a_n}{z^{{\lambda _n}}}} \] be a transcendental entire function. Set \[ M(r) = \max _{|z| = r} |f(z)|,\; m(r) = \max _{n \geq 0} \{ |{a_n}|{r^{{\lambda _n}}}\} \] and \[ N(r) = \max _{n \geq 0} \{ {\lambda _n}|m(r) = |{a_n}|{r^{{\lambda _n}}}\} .\] Sato introduced the notion of growth constants, referred in the present paper as ${S_q}$-order $\lambda$ and ${S_q}$-type $T$, which are generalizations of concepts of classical order and type by defining \[ (\ast \ast )\quad \lambda = \lim _{r \to \infty } \sup ({\log ^{[q]}}M(r)|\log r)\] and if $0 < \lambda < \infty$, then \[ (\ast \ast \ast )\quad T = \lim _{r \to \infty } \sup ({\log ^{[q - 1]}}M(r)|{r^\lambda })\] for $q = 2,3,4, \cdots$ where ${\log ^{[0]}}x = x$ and ${\log ^{[q]}}x = \log ({\log ^{[q - 1]}}x)$. Sato has also obtained the coefficient equivalents of $(\ast \ast )$ and $(\ast \ast \ast )$ for the entire function $(\ast )$ when ${\lambda _n} = n$. It is noted that Sato’s coefficient equivalents of $\lambda$, and $T$ also hold true for $(\ast )$ if $n$’s are replaced by ${\lambda _n}$’s in his coefficient equivalents. Analogous to $(\ast \ast )$ and $(\ast \ast \ast )$ lower ${S_q}$-order $v$ and lower ${S_q}$-type $t$ for entire function $f(z)$ are introduced here by defining \[ v = \lim _{r \to \infty } \inf ({\log ^{[q]}}M(r)|\log r)\] and if $0 < \lambda < \infty$ then \[ t = \lim _{r \to \infty } \inf ({\log ^{[q - 1]}}M(r)|{r^\lambda }),\quad q = 2,3,4, \cdots .\] For the case $q = 2$, these notions are due to Whittakar and Shah respectively. For the constant $v$, two complete coefficient characterizations have been found which generalize the earlier known results. For $t$ coefficient characterization only for those entire functions for which the consecutive principal indices are asymptotic is obtained. Determination of a complete coefficient characterization of $t$ remains an open problem. Further ${S_q}$-growth and lower ${S_q}$-growth numbers for entire function $f(z)$ we defined \[ \begin {array}{*{20}{c}} \delta \\ \mu \\ \end {array} = \lim _{r \to \infty } \begin {array}{*{20}{c}} {\sup } \\ {\inf } \\ \end {array} ({\log ^{[q - 1]}}N(r)|{r^\lambda }),\] for $q = 2,3,4, \cdots$ and $0 < \lambda < \infty$. Earlier results of Juneja giving the coefficients characterization of $\delta$ and $\mu$ are extended and generalized. A new decomposition theorem for entire functions of ${S_q}$-regular growth but not of perfectly ${S_q}$-regular growth has been found.
Similar Articles
  • Retrieve articles in Transactions of the American Mathematical Society with MSC: 30A64
  • Retrieve articles in all journals with MSC: 30A64
Additional Information
  • © Copyright 1975 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 203 (1975), 275-297
  • MSC: Primary 30A64
  • DOI:
  • MathSciNet review: 0372200