Fixed point formula for holomorphic functions

In [Classes of generalized Weingarten surfaces in the Euclidean 3-space, Adv. Geom. 16(1) (2016) 45–55], the authors study a class of generalized special Weingarten surfaces, where coefficients are functions that depend on the support function and the distance function from a fixed point (in short EDSGW-surfaces), this class of surfaces has the geometric property that all the middle spheres pass through a fixed point. In this paper, we present a Weierstrass type representation for EDSGW-surfaces with prescribed Gauss map which depends on two holomorphic functions. Also, we classify isothermic EDSGW-surfaces with respect to the third fundamental form parametrized by planar lines of curvature. Moreover, we give explicit examples of EDSGW-surfaces and isothermic EDSGW-surfaces.

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Integral representations of holomorphic functions with a fixed point

Doklady Mathematics ◽

10.1134/s1064562414070096 ◽

2014 ◽

Vol 90 (3) ◽

pp. 683-684

Author(s):

I. I. Bavrin

Keyword(s):

Fixed Point ◽

Holomorphic Functions ◽

Integral Representations

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Fixed points of holomorphic mappings for domains in Banach spaces

Abstract and Applied Analysis ◽

10.1155/s1085337503205042 ◽

2003 ◽

Vol 2003 (5) ◽

pp. 261-274 ◽

Cited By ~ 12

Author(s):

Lawrence A. Harris

Keyword(s):

Fixed Point ◽

Fixed Point Theorem ◽

Holomorphic Function ◽

Fixed Points ◽

Banach Spaces ◽

Point Theorem ◽

Numerical Range ◽

Holomorphic Functions ◽

Holomorphic Mappings

We discuss the Earle-Hamilton fixed-point theorem and show how it can be applied when restrictions are known on the numerical range of a holomorphic function. In particular, we extend the Earle-Hamilton theorem to holomorphic functions with numerical range having real part strictly less than 1. We also extend the Lumer-Phillips theorem estimating resolvents to dissipative holomorphic functions.

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Fixed points of composite meromorphic functions and normal families

Proceedings of the Royal Society of Edinburgh Section A Mathematics ◽

10.1017/s0308210500003401 ◽

2004 ◽

Vol 134 (4) ◽

pp. 653-660 ◽

Cited By ~ 4

Author(s):

Walter Bergweiler

Keyword(s):

Fixed Point ◽

Meromorphic Function ◽

Fixed Points ◽

Unit Disc ◽

Meromorphic Functions ◽

Holomorphic Functions ◽

Normal Families ◽

The Family

We show that there exists a function f, meromorphic in the plane C, such that the family of all functions g holomorphic in the unit disc D for which f ∘ g has no fixed point in D is not normal. This answers a question of Hinchliffe, who had shown that this family is normal if Ĉ\f(C) does not consist of exactly one point in D. We also investigate the normality of the family of all holomorphic functions g such that f(g(z)) ≠ h(z) for some non-constant meromorphic function h.

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FIXED POINT THEOREMS FOR INFINITE DIMENSIONAL HOLOMORPHIC FUNCTIONS

Journal of the Korean Mathematical Society ◽

10.4134/jkms.2004.41.1.175 ◽

2004 ◽

Vol 41 (1) ◽

pp. 175-192 ◽

Cited By ~ 3

Author(s):

Lwarence-A. Harris

Keyword(s):

Fixed Point ◽

Fixed Point Theorems ◽

Holomorphic Functions ◽

Infinite Dimensional

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Variations on the Brouwer Fixed Point Theorem: A Survey

Mathematics ◽

10.3390/math8040501 ◽

2020 ◽

Vol 8 (4) ◽

pp. 501

Author(s):

Jean Mawhin

Keyword(s):

Fixed Point ◽

Existence Theorems ◽

Holomorphic Functions ◽

Cauchy Integral ◽

Integral Theorem ◽

Exterior Calculus ◽

Continuous Mappings ◽

Cauchy Integral Theorem ◽

Basic Facts ◽

Special Case

This paper surveys some recent simple proofs of various fixed point and existence theorems for continuous mappings in R n . The main tools are basic facts of the exterior calculus and the use of retractions. The special case of holomorphic functions is considered, based only on the Cauchy integral theorem.

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Normality, Quasinormality and Periodic Points

Nagoya Mathematical Journal ◽

10.1017/s0027763000009715 ◽

2009 ◽

Vol 195 ◽

pp. 77-95 ◽

Cited By ~ 1

Author(s):

Jianming Chang

Keyword(s):

Fixed Point ◽

Periodic Point ◽

Holomorphic Functions ◽

Periodic Points

AbstractLet M > 1 be a positive number. Let be a family of holomorphic functions f in some domain D ⊂ ℂ for which there exists an integer k = k(f) > 2 such that |(fk)′(ζ)| ≤ Mk for every periodic point ζ of period k of f in D. We show first that is quasinormal of order at most one in D. This strengthens a result of W. Bergweiler. Secondly, for the case M = 1, we prove that is normal in D if there exists a positive number K < 3 such that | f(η)| ≤ K for each f ∈ and every fixed point η of f in D. This improves a result of M. Esséen and S. J. Wu. We also construct an example which shows that the condition |f’(η)| ≤ K < 3 can not be replaced by | f′(η) | < 3.

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