On the coefficients of an entire series

Juneja, O. P.

doi:10.1007/bf02790381

Cited by 8 publications

(6 citation statements)

References 2 publications

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“…For all c > 0, the function f clearly has Fabry gaps, and it is not difficult to check that it has order and lower order equal to c 2 . These can be calculated either directly or by using the following formulae (see [32]) for the order and lower order of an entire function of the form f (z) = ∞ n=0 a n z n : There are also functions of infinite order with gap series for which Corollary 8.3(a) is satisfied for each m > 1. Hayman [25] showed that Fuch's result above, for functions of finite order with Fabry gaps, holds for functions of any order provided that a stronger gap series condition is satisfied, which can be stated as follows:…”

Section: Gap Seriesmentioning

confidence: 99%

Fast escaping points of entire functions

Rippon

Stallard

2012

Proceedings of the London Mathematical Society

100

274

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Abstract. Let f be a transcendental entire function and let A(f ) denote the set of points that escape to infinity 'as fast as possible' under iteration. By writing A(f ) as a countable union of closed sets, called 'levels' of A(f ), we obtain a new understanding of the structure of this set. For example, we show that if U is a Fatou component in A(f ), then ∂U ⊂ A(f ) and this leads to significant new results and considerable improvements to existing results about A(f ). In particular, we study functions for which A(f ), and each of its levels, has the structure of an 'infinite spider's web'. We show that there are many such functions and that they have a number of strong dynamical properties. This new structure provides an unexpected connection between a conjecture of Baker concerning the components of the Fatou set and a conjecture of Eremenko concerning the components of the escaping set.

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Section: Gap Seriesmentioning

confidence: 99%

Fast escaping points of entire functions

Rippon

Stallard

2012

Proceedings of the London Mathematical Society

100

274

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“…Then there exists a set E ⊂ (0, ∞) with E 1/t dt < ∞ such that, for r / ∈ E and z ∈ D(r), (15) g(z) = z z(r)…”

Section: New Results On Regularitymentioning

confidence: 99%

“…We define the order ρ(f ) and lower order λ(f ) of a transcendental entire function f by We note from, for example, [15] that if f (z) = ∞ n=0 a n z n , then We use the following three lemmas, all discussed in [19, Corollary 8.3 and the following remarks]. The first is from [14, p.205], and gives a sufficient condition for a transcendental entire function to be log-regular.…”

Section: Functions Of Finite Order For Which a R (F ) Is A Spider's Webmentioning

confidence: 99%

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Entire functions for which the escaping set is a spider's web

Sixsmith¹

2011

Math. Proc. Camb. Phil. Soc.

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Abstract. We construct several new classes of transcendental entire functions, f , such that both the escaping set, I(f ), and the fast escaping set, A(f ), have a structure known as a spider's web. We show that some of these classes have a degree of stability under changes in the function. We show that new examples of functions for which I(f ) and A(f ) are spiders' webs can be constructed by composition, by differentiation, and by integration of existing examples. We use a property of spiders' webs to give new results concerning functions with no unbounded Fatou components.

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“…11/12; cf. also [19], [20], [6], [7], [8]) and this has been extended to iterated orders by Schonhage [17], Sato [16], Reddy [14], Juneja, Kapoor, and Bajpai [9] (also [22], [13]), and to generalized orders by Seremeta [18], Bajpai, Gautam, and Bajpai [1] as well as Kapoor and Nautiyal [10]. Combining the two kinds of characterizations (as done, e.g., by Reddy [15], p. 105) approximation theorems in terms of the sequence {/ (k) (0)} teN are obtained.…”

Section: Introduction and Resultsmentioning

confidence: 99%

Polynomial approximation of an entire function and rate of growth of Taylor coefficients

Freund¹,

Görlich

1985

Proceedings of the Edinburgh Mathematical Society

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The best uniform approximation of a functionfon [-1,1] by real algebraic polynomials satisfiesif and only if ƒ is the restriction to [- 1,1] of an entire function (Bernstein [2], p. 113, see also [12], pp. 83–85). For such functions ƒ the rate of best approximation has been characterized by Varga [24], Reddy [24], Shah [21], and Kapoor and Nautiyal [10] in terms of order and type of ƒ , lower order and type, and in terms of more general concepts of order. On the other hand, order and type of ƒ are connected with the Taylor coefficients, i.e. with the rate of growth of the sequence(see [23], p. 41 or [3], pp. 11/12; cf. also [19], [20], [6], [7], [8]) and this has been extended to iterated orders by Schonhage [17], Sato [16], Reddy [14], Juneja, Kapoor, and Bajpai [9] (also [22], [13]), and to generalized orders by Seremeta [18], Bajpai, Gautam, and Bajpai [1] as well as Kapoor and Nautiyal [10]. Combining the two kinds of characterizations (as done, e.g., by Reddy [15], p. 105) approximation theorems in terms of the sequenceare obtained. But in such results the rate ofbest approximation is always described by a limit relation, e.g. of the form, and this causes a considerable loss of precision, as will be discussed in more detail in Section 3 (in this respect cf. also the remark by Bernstein [2], pp. 114/115).

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On the coefficients of an entire series

Cited by 8 publications

References 2 publications

Fast escaping points of entire functions

Fast escaping points of entire functions

Entire functions for which the escaping set is a spider's web

Polynomial approximation of an entire function and rate of growth of Taylor coefficients

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