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

Jain, P. K.; Jain, D. R.

doi:10.5802/aif.494

On the lower order

(R)

of an entire Dirichlet series

Jain, P. K. ; Jain, D. R.

Annales de l'Institut Fourier, Tome 24 (1974) no. 1, pp. 123-129.

Résumé
Abstract

Des estimations $λ$ de l’ordre inférieur $(R)$ d’une série de Dirichlet

f (s) = \sum_{n = 1}^{\infty} a_{n} e^{s λ n}

ont été obtenues en fonction des suites ${a_{n}}$ et ${λ_{n}}$ .

Ces estimations améliorent considérablement celles obtenues précédemment par Rahman (Quart. J. Math. Oxford, (2), 7, 96-99 (1956)), Juneja et Singh (Math. Ann., 184 (1969), 25-29 ).

The estimations of lower order $(R)$ $λ$ in terms of the sequences ${a_{n}}$ and ${λ_{n}}$ for an entire Dirichlet series $f (s) = \sum_{n = 1}^{\infty} a_{n} e^{s λ n}$ , have been obtained, namely :

\begin{matrix} λ & = max_{{λ_{n_{p}}}} lim inf_{p \to \infty} \frac{λ_{n_{p}} log λ_{n_{p - 1}}}{log | a_{n_{p}} |^{- 1}} \\ = max_{{λ_{n_{p}}}} lim inf_{p \to \infty} \frac{(λ_{n_{p}} - λ_{n_{p - 1}}) log λ_{n_{p - 1}}}{log | a_{n_{p - 1}} | a_{n_{p}} |} . \end{matrix}

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 ).

MR Zbl

DOI : 10.5802/aif.494

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     author = {Jain, P. K. and Jain, D. R.},
     title = {On the lower order $(R)$ of an entire {Dirichlet} series},
     journal = {Annales de l'Institut Fourier},
     pages = {123--129},
     publisher = {Institut Fourier},
     address = {Grenoble},
     volume = {24},
     number = {1},
     year = {1974},
     doi = {10.5802/aif.494},
     mrnumber = {50 #7520},
     zbl = {0273.30021},
     language = {en},
     url = {http://www.numdam.org/articles/10.5802/aif.494/}
}

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Jain, P. K.; Jain, D. R. On the lower order $(R)$ of an entire Dirichlet series. Annales de l'Institut Fourier, Tome 24 (1974) no. 1, pp. 123-129. doi : 10.5802/aif.494. http://www.numdam.org/articles/10.5802/aif.494/

Bibliographie
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[2] Alfred Gray and S.M. Shah, Holomorphic Functions with Gap Power Series, Math. Zeit, 86 (1965), 375-394. | MR | Zbl

[3] Alfred Gray and S.M. Shah, Holomorphic Functions with Gap Power Series, (II), Math. Anal. and Appl. 2, 13, (1966). | Zbl

[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. | MR | Zbl

[5] P.K. Kamthan, On entire functions represented by Dirichlet series (IV), Ann. Inst. Fourier, Grenoble, 16, 2 (1966), 209-223. | Numdam | MR | Zbl

[6] Q.I. Rahman, On the lower order of entire functions defined by Dirichlet series. Quart. J. Math. Oxford (2), 7 (1956), 96-99. | MR | Zbl

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