scholarly journals Optimal version of the Picard–Lindelöf theorem

2021 ◽  
pp. 1-8
Author(s):  
Jan-Christoph Schlage-Puchta
Keyword(s):  
Lindelöf Theorem

scholarly journals On a Counterexample in Connection with the Picard-Lindelöf Theorem

2021 ◽  
Vol 52 (3) ◽  
pp. 221-223
Author(s):  
Georgios Passias ◽  
Sven-Ake Wegner
Keyword(s):  
Lindelöf Theorem

The generalized Laguerre inequalities and functions in the Laguerre-Pólya class

2013 ◽  
Vol 11 (9) ◽  
Author(s):  
George Csordas ◽  
Anna Vishnyakova
Keyword(s):  
Entire Function ◽  
Open Problem ◽  
A Priori ◽  
Sufficient Conditions ◽  
Class Definition ◽  
Lindelöf Theorem ◽  
Order And Type ◽  
Geometric Result ◽  
Real Entire Function ◽  
Inequality Theorem

AbstractThe principal goal of this paper is to show that the various sufficient conditions for a real entire function, φ(x), to belong to the Laguerre-Pólya class (Definition 1.1), expressed in terms of Laguerre-type inequalities, do not require the a priori assumptions about the order and type of φ(x). The proof of the main theorem (Theorem 2.3) involving the generalized real Laguerre inequalities, is based on a beautiful geometric result, the Borel-Carathédodory Inequality (Theorem 2.1), and on a deep theorem of Lindelöf (Theorem 2.2). In case of the complex Laguerre inequalities (Theorem 3.2), the proof is sketched for it requires a slightly more delicate analysis. Section 3 concludes with some other cognate results, an open problem and a conjecture which is based on Cardon’s recent, ingenious extension of the Laguerre-type inequalities.

scholarly journals Phragmèn–Lindelöf theorem for minimal surface equations in higher dimensions

2002 ◽  
Vol 207 (1) ◽  
pp. 183-198
Author(s):  
Chun-Chung Hsieh ◽  
Jenn-Fang Hwang ◽  
Fei-Tsen Liang
Keyword(s):  
Minimal Surface ◽  
Higher Dimensions ◽  
Lindelöf Theorem

scholarly journals A Phragmén–Lindelöf Theorem via Proximate Orders, and the Propagation of Asymptotics

2019 ◽  
Vol 30 (4) ◽  
pp. 3458-3483
Author(s):  
Javier Jiménez-Garrido ◽  
Javier Sanz ◽  
Gerhard Schindl
Keyword(s):  
Asymptotic Expansion ◽  
Asymptotic Expansions ◽  
Holomorphic Functions ◽  
Proximate Order ◽  
Convex Sequence ◽  
Lindelöf Theorem ◽  
Bounded Holomorphic ◽  
Refined Version

Abstract We prove that, for asymptotically bounded holomorphic functions in a sector in $$\mathbb {C},$$ C , an asymptotic expansion in a single direction towards the vertex with constraints in terms of a logarithmically convex sequence admitting a nonzero proximate order entails asymptotic expansion in the whole sector with control in terms of the same sequence. This generalizes a result by Fruchard and Zhang for Gevrey asymptotic expansions, and the proof strongly rests on a suitably refined version of the classical Phragmén–Lindelöf theorem, here obtained for functions whose growth in a sector is specified by a nonzero proximate order in the sense of Lindelöf and Valiron.

On the Growth of Entire Functions Bounded on Large Sets

1977 ◽  
Vol 29 (6) ◽  
pp. 1287-1291
Author(s):  
Lowell J. Hansen
Keyword(s):  
Entire Function ◽  
Entire Functions ◽  
Maximum Modulus ◽  
Modulus Function ◽  
Classical Theorem ◽  
Minimum Modulus ◽  
Rate Of Growth ◽  
Large Sets ◽  
Growth Of Entire Functions ◽  
Lindelöf Theorem

There have been many indications of a relationship between the rate of growth of an entire function and the “size” of the set, E(c), where the modulus of the function is larger than the constant, c. Theorems of this type include the classical theorem of Wiman on functions of bounded minimum modulus, the Phragmén-Lindelöf Theorem, the Denjoy-Carleman-Ahlfors Theorem, and its many subsequent improvements. These theorems can all be understood as quantitative versions of the statement that if ƒ is an entire function such that, for some c > 0, the set E(c) is ‘'small”, then the maximum modulus function M(R, f) is forced to grow rapidly with R.

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