On the lower order (R) of an entire Dirichlet series
Annales de l'Institut Fourier, Tome 24 (1974) no. 1, pp. 123-129.

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

f ( s ) = n = 1 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)= n=1 a n e sλn , 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 ).

@article{AIF_1974__24_1_123_0,
     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://archive.numdam.org/articles/10.5802/aif.494/}
}
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
SP  - 123
EP  - 129
VL  - 24
IS  - 1
PB  - Institut Fourier
PP  - Grenoble
UR  - http://archive.numdam.org/articles/10.5802/aif.494/
DO  - 10.5802/aif.494
LA  - en
ID  - AIF_1974__24_1_123_0
ER  - 
%0 Journal Article
%A Jain, P. K.
%A Jain, D. R.
%T On the lower order $(R)$ of an entire Dirichlet series
%J Annales de l'Institut Fourier
%D 1974
%P 123-129
%V 24
%N 1
%I Institut Fourier
%C Grenoble
%U http://archive.numdam.org/articles/10.5802/aif.494/
%R 10.5802/aif.494
%G en
%F AIF_1974__24_1_123_0
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://archive.numdam.org/articles/10.5802/aif.494/

[1] Alfred Gray and S.M. Shah, Asymptotic values of Holomorphic Functions of irregular growth, Bull. Amer. Math. Soc., 5, 71 (1965), 747-749. | MR | Zbl

[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

Cité par Sources :