scholarly journals Bohr radius and its asymptotic value for holomorphic functions in higher dimensions

2021 ◽  
Vol 359 (7) ◽  
pp. 911-918
Author(s):  
Bappaditya Bhowmik ◽  
Nilanjan Das
Keyword(s):  
Holomorphic Functions ◽  
Higher Dimensions ◽  
Bohr Radius ◽  
Asymptotic Value

On a Question of Seidel Concerning Holomorphic Functions Bounded on a Spiral

1969 ◽  
Vol 21 ◽  
pp. 1255-1262 ◽  
Author(s):  
K. F. Barth ◽  
W. J. Schneider
Keyword(s):  
Oral Communication ◽  
Holomorphic Functions ◽  
Asymptotic Value ◽  
Asymptotic Values

Let S be a spiral contained in D = {|z| < 1} such that S tends to C = {|z| = 1}. For the sake of brevity, by “f is bounded on S” we shall mean that f is holomorphic in D, unbounded, and bounded on S. The existence of such functions was first discussed by Valiron (9; 10); see also (1; 3; 8). Valiron also proved that any function that is “bounded on a spiral” must have the asymptotic value ∞ (10, p. 432). Functions that are bounded on a spiral may also have finite asymptotic values (1, p. 1254). In view of the above, Seidel has raised the question (oral communication): “Does there exist a function bounded on a spiral that has only the asymptotic value ∞?”. The following theorem answers this question affirmatively.

Some results of Lindelöf type involving the segmental behaviour of holomorphic functions

1993 ◽  
Vol 114 (1) ◽  
pp. 57-65 ◽  
Author(s):  
Francisca Bravo ◽  
Daniel Girela
Keyword(s):  
Analytic Function ◽  
Unit Disc ◽  
Slow Growth ◽  
Holomorphic Functions ◽  
Maximum Modulus ◽  
Classical Theorem ◽  
Asymptotic Value

AbstractA classical theorem of Lindelöf asserts that if ƒ is a function analytic and bounded in the unit disc δ which has the asymptotic value L at a point ξ ε ∂ δ then it has the non-tangential limit L at ξ. This result does not remain true for functions f analytic in δ whose maximum modulus grows to infinity arbitrarily slowly. However, the second author has recently obtained some results of Lindelöf type valid for these functions. In this paper we obtain new results of this kind. We prove that if f is an analytic function of slow growth in δ and ξ ε ∂ δ, then certain restrictions on the growth of ƒ′ along a segment which ends at ξ do imply that ƒ has a non-tangential limit at ξ.

Application of Holomorphic Functions in Two and Higher Dimensions

2016 ◽  
Author(s):  
Klaus Gürlebeck ◽  
Klaus Habetha ◽  
Wolfgang Sprößig
Keyword(s):  
Holomorphic Functions ◽  
Higher Dimensions

Hedgehogs in higher dimensions and their applications

Astérisque ◽  
2020 ◽  
Vol 416 ◽  
pp. 213-251
Author(s):  
Mikhail LYUBICH ◽  
Remus RADU ◽  
Raluca TANASE
Keyword(s):  
Higher Dimensions
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