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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality 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 1.43.

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On entire functions of fast growth
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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.
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Additional Information
  • © Copyright 1975 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 203 (1975), 275-297
  • MSC: Primary 30A64
  • DOI:
  • MathSciNet review: 0372200