2020
DOI: 10.48550/arxiv.2011.09532
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On subharmonic and entire functions of small order: after Kjellberg

Abstract: We give a general method for constructing examples of transcendental entire functions of given small order, which allows precise control over the size and shape of the set where the minimum modulus of the function is relatively large. Our method involves developing a new technique to give an upper bound for the growth of a positive harmonic function defined in a certain type of multiply connected domain, giving a sharp estimate for the growth in many cases.n∈Z [β n , αβ n ], where 1 < α < β, whereas for us it … Show more

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Cited by 1 publication
(5 citation statements)
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“…The structure of the paper is as follows. In Section 2 we prove our positive results, including Theorem 1.2, and in Section 3 we recall some results from [13] needed for the proofs of Examples 1.3 and 1.4, which are given in Section 4.…”
Section: Introductionmentioning
confidence: 93%
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“…The structure of the paper is as follows. In Section 2 we prove our positive results, including Theorem 1.2, and in Section 3 we recall some results from [13] needed for the proofs of Examples 1.3 and 1.4, which are given in Section 4.…”
Section: Introductionmentioning
confidence: 93%
“…Our method of proving Examples 1.4 and 1.3 uses a recent generalisation [13] of Kjellberg's method [6, Chapter 2] for constructing transcendental entire functions of order less than 1/2 by approximating certain continuous subharmonic functions by functions of the form log |f | where f is a transcendental entire function. In this section we summarise the results from [13] which are needed to construct our examples.…”
Section: Constructing Entire Functions Of Small Ordermentioning
confidence: 99%
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