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A Flexible Proximate Order

A Flexible Proximate Order DAVID DRASINj 1. Introduction A continuous function p(r) (r > 0) which is continuously differentiate off a discrete set D and such that 0 < p(r) -• p (0 < p < oo)l (r -• oo) (1.1) p'(r)r logr^ O (r->oo, r£D ) (1.2) is called a proximate order. An auxiliary function /j(r) which satisfies /,(,.) ~ p w (r-oo), (1.3) with p(r) a proximate order, is useful in the study of entire and meromorphic functions, both for obtaining positive results and examples. Useful references are [1; Ch. 4] and [6; Ch. 1]. Recently, A. Edrei [3] has developed a theory which does not require a globally-defined function h{r), as in (1.3), but instead one defined only in intervals (A ~ r , A r ) where {r }, {A } are sequences which independently tend to infinity. n n n n n n Edrei's comparison function is then given by h{r) ~ {rlr y»h{r ) {A " r < r < A r ) (1.4) n n H n n n where k is a nonnegative constant, and he showed that this simple function h(r) was adequate to establish the most important results which heretofore had depended on http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Bulletin of the London Mathematical Society Wiley

A Flexible Proximate Order

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Publisher
Wiley
Copyright
© London Mathematical Society
ISSN
0024-6093
eISSN
1469-2120
DOI
10.1112/blms/6.2.129
Publisher site
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Abstract

DAVID DRASINj 1. Introduction A continuous function p(r) (r > 0) which is continuously differentiate off a discrete set D and such that 0 < p(r) -• p (0 < p < oo)l (r -• oo) (1.1) p'(r)r logr^ O (r->oo, r£D ) (1.2) is called a proximate order. An auxiliary function /j(r) which satisfies /,(,.) ~ p w (r-oo), (1.3) with p(r) a proximate order, is useful in the study of entire and meromorphic functions, both for obtaining positive results and examples. Useful references are [1; Ch. 4] and [6; Ch. 1]. Recently, A. Edrei [3] has developed a theory which does not require a globally-defined function h{r), as in (1.3), but instead one defined only in intervals (A ~ r , A r ) where {r }, {A } are sequences which independently tend to infinity. n n n n n n Edrei's comparison function is then given by h{r) ~ {rlr y»h{r ) {A " r < r < A r ) (1.4) n n H n n n where k is a nonnegative constant, and he showed that this simple function h(r) was adequate to establish the most important results which heretofore had depended on

Journal

Bulletin of the London Mathematical Society – Wiley

Published: Jul 1, 1974

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