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 | References | Similar Articles | Additional Information

Abstract: 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 Satoβs coefficient equivalents of $\lambda$, and $T$ also hold true for $(\ast )$ if $n$βs are replaced by ${\lambda _n}$βs 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

Keywords:
Entire functions,
growth estimates,
order,
lower order,
type,
lower type,
decomposition theorems,
Sato <IMG WIDTH="15" HEIGHT="37" ALIGN="MIDDLE" BORDER="0" SRC="images/img1.gif" ALT="$q$">th iterate order,
Sato <IMG WIDTH="15" HEIGHT="37" ALIGN="MIDDLE" BORDER="0" SRC="images/img49.gif" ALT="$q$">th iterate lower order,
growth numbers,
lower growth numbers,
coefficient characterizations

Article copyright:
© Copyright 1975
American Mathematical Society