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Transactions of the American Mathematical Society

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

 

 

On entire functions of fast growth


Authors: S. K. Bajpai, G. P. Kapoor and O. P. Juneja
Journal: Trans. Amer. Math. Soc. 203 (1975), 275-297
MSC: Primary 30A64
DOI: https://doi.org/10.1090/S0002-9947-1975-0372200-0
MathSciNet review: 0372200
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Abstract: Let

$\displaystyle (\ast )\quad f(z) = \sum\limits_{n = 0}^\infty {{a_n}{z^{{\lambda _n}}}} $

be a transcendental entire function. Set

$\displaystyle M(r) = \mathop {max}\limits_{\vert z\vert = r} \vert f(z)\vert,\q... ...(r) = \mathop {\max }\limits_{n \geq 0} \{ \vert{a_n}\vert{r^{{\lambda _n}}}\} $

and

$\displaystyle N(r) = \mathop {\max }\limits_{n \geq 0} \{ {\lambda _n}\vert m(r) = \vert{a_n}\vert{r^{{\lambda _n}}}\} .$

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

$\displaystyle (\ast \ast )\quad \lambda = \mathop {\lim }\limits_{r \to \infty } \sup ({\log ^{[q]}}M(r)\vert\log r)$

and if $ 0 < \lambda < \infty $, then

$\displaystyle (\ast \ast \ast )\quad T = \mathop {\lim }\limits_{r \to \infty } \sup ({\log ^{[q - 1]}}M(r)\vert{r^\lambda })$

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

$\displaystyle v = \mathop {\lim }\limits_{r \to \infty } \inf ({\log ^{[q]}}M(r)\vert\log r)$

and if $ 0 < \lambda < \infty $ then

$\displaystyle t = \mathop {\lim }\limits_{r \to \infty } \inf ({\log ^{[q - 1]}}M(r)\vert{r^\lambda }),\quad q = 2,3,4, \cdots .$

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{displaymath}\begin{array}{*{20}{c}} \delta \\ \mu \\ \end{array} = \matho... ...\inf } \\ \end{array} ({\log ^{[q - 1]}}N(r)\vert{r^\lambda }),\end{displaymath}

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

DOI: https://doi.org/10.1090/S0002-9947-1975-0372200-0
Keywords: Entire functions, growth estimates, order, lower order, type, lower type, decomposition theorems, Sato $ q$th iterate order, Sato $ q$th iterate lower order, growth numbers, lower growth numbers, coefficient characterizations
Article copyright: © Copyright 1975 American Mathematical Society