On the lower order ( R ) of an entire Dirichlet series

P. K. Jain; D. R. Jain

Annales de l'institut Fourier (1974)

  • Volume: 24, Issue: 1, page 123-129
  • ISSN: 0373-0956

Abstract

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The estimations of lower order ( R ) λ in terms of the sequences { a n } and { λ n } for an entire Dirichlet series f ( s ) = n = 1 a n e s λ n , have been obtained, namely : λ = max { λ n p } lim inf p λ n p log λ n p - 1 log | a n p | - 1 = max { λ n p } lim inf p ( λ n p - λ n p - 1 ) log λ n p - 1 log | a n p - 1 | a n p | . One of these estimations improves considerably the estimations earlier obtained by Rahman (Quart. J. Math. Oxford, (2), 7, 96-99 (1956)) and Juneja and Singh (Math. Ann., 184(1969), 25-29 ).

How to cite

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Jain, P. K., and Jain, D. R.. "On the lower order $(R)$ of an entire Dirichlet series." Annales de l'institut Fourier 24.1 (1974): 123-129. <http://eudml.org/doc/74157>.

@article{Jain1974,
abstract = {The estimations of lower order $(R)$$\lambda $ in terms of the sequences $\lbrace a_n\rbrace $ and $\lbrace \lambda _n\rbrace $ for an entire Dirichlet series $f(s)=\sum ^\infty _\{n=1\}a_ne^\{s\lambda n\}$, have been obtained, namely :\begin\{align\} \lambda & =\max \_\{ \lbrace \lambda \_\{n\_p\}\rbrace \}\lim \inf \_\{p\rightarrow \infty \} \{ \lambda \_\{n\_p\}\log \lambda \_\{n\_\{p-1\}\} \over \log \vert a\_\{n\_p\}\vert ^\{-1\}\}\\ & =\max \_\{ \lbrace \lambda \_\{n\_p\}\rbrace \}\lim \inf \_\{p\rightarrow \infty \} \{(\lambda \_\{n\_p\}-\lambda \_\{n\_\{p-1\}\}) \log \lambda \_\{n\_\{p-1\}\}\over \log \vert a\_\{n\_\{p-1\}\}\vert a\_\{n\_p\}\vert \}.\end\{align\}One of these estimations improves considerably the estimations earlier obtained by Rahman (Quart. J. Math. Oxford, (2), 7, 96-99 (1956)) and Juneja and Singh (Math. Ann., 184(1969), 25-29 ).},
author = {Jain, P. K., Jain, D. R.},
journal = {Annales de l'institut Fourier},
language = {eng},
number = {1},
pages = {123-129},
publisher = {Association des Annales de l'Institut Fourier},
title = {On the lower order $(R)$ of an entire Dirichlet series},
url = {http://eudml.org/doc/74157},
volume = {24},
year = {1974},
}

TY - JOUR
AU - Jain, P. K.
AU - Jain, D. R.
TI - On the lower order $(R)$ of an entire Dirichlet series
JO - Annales de l'institut Fourier
PY - 1974
PB - Association des Annales de l'Institut Fourier
VL - 24
IS - 1
SP - 123
EP - 129
AB - The estimations of lower order $(R)$$\lambda $ in terms of the sequences $\lbrace a_n\rbrace $ and $\lbrace \lambda _n\rbrace $ for an entire Dirichlet series $f(s)=\sum ^\infty _{n=1}a_ne^{s\lambda n}$, have been obtained, namely :\begin{align} \lambda & =\max _{ \lbrace \lambda _{n_p}\rbrace }\lim \inf _{p\rightarrow \infty } { \lambda _{n_p}\log \lambda _{n_{p-1}} \over \log \vert a_{n_p}\vert ^{-1}}\\ & =\max _{ \lbrace \lambda _{n_p}\rbrace }\lim \inf _{p\rightarrow \infty } {(\lambda _{n_p}-\lambda _{n_{p-1}}) \log \lambda _{n_{p-1}}\over \log \vert a_{n_{p-1}}\vert a_{n_p}\vert }.\end{align}One of these estimations improves considerably the estimations earlier obtained by Rahman (Quart. J. Math. Oxford, (2), 7, 96-99 (1956)) and Juneja and Singh (Math. Ann., 184(1969), 25-29 ).
LA - eng
UR - http://eudml.org/doc/74157
ER -

References

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  1. [1] Alfred GRAY and S.M. SHAH, Asymptotic values of Holomorphic Functions of irregular growth, Bull. Amer. Math. Soc., 5, 71 (1965), 747-749. Zbl0147.06703MR31 #3614
  2. [2] Alfred GRAY and S.M. SHAH, Holomorphic Functions with Gap Power Series, Math. Zeit, 86 (1965), 375-394. Zbl0133.03501MR33 #7501
  3. [3] Alfred GRAY and S.M. SHAH, Holomorphic Functions with Gap Power Series, (II), Math. Anal. and Appl. 2, 13, (1966). Zbl0163.08802
  4. [4] O.P. JUNEJA and Prem SINGH, On the Lower order of an entire function defined by Dirichlet series, Math. Ann. 184 (1969), 25-29. Zbl0182.09702MR40 #7448
  5. [5] P.K. KAMTHAN, On entire functions represented by Dirichlet series (IV), Ann. Inst. Fourier, Grenoble, 16, 2 (1966), 209-223. Zbl0145.08103MR37 #1606
  6. [6] Q.I. RAHMAN, On the lower order of entire functions defined by Dirichlet series. Quart. J. Math. Oxford (2), 7 (1956), 96-99. Zbl0074.29901MR20 #5282

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