scholarly journals On holomorphic functions with certain extremal properties of its absolute values

1981 ◽  
Vol 4 (4) ◽  
pp. 641-647
Author(s):  
Dieter Schmersau
Keyword(s):  
Complex Plane ◽  
Special Class ◽  
Holomorphic Functions ◽  
Geometrical Characterization ◽  
Line Segments ◽  
Alpha Beta ◽  
Closed Line ◽  
Extremal Properties

This paper is concerned with a special class of holomorphic functions with extremal properties of its absolute values on arbitrary closed line segments in the complex plane. The main result is a geometrical characterization of the functionsz→eaz+b,z→(az+b)nandz→(az+b)α+iβwitha,b,∈ℂ,α,β∈ℝ,n∈ℤ.

scholarly journals Co-convexial reflector curves with applications

1987 ◽  
Vol 10 (2) ◽  
pp. 233-240
Author(s):  
Abdallah M. Al-Rashed ◽  
Neyamat Zaheer
Keyword(s):  
Complex Plane ◽  
Convex Compact ◽  
Compact Sets ◽  
Full Generality ◽  
Line Segments ◽  
General Family ◽  
Family Of Curves ◽  
Closed Line ◽  
Van Vleck ◽  
Convex Compact Sets

The concept of reflector curves for convex compact sets of reflecting type in the complex plane was introduced by the authors in a recent paper (to appear in J. Math. Anal. and Appln.) in their attempt to solve a problem related to Stieltjes and Van Vleck polynomials. Though, in the said paper, certain convex compact sets (e.g. closed discs, closed line segments and the ones with polygonal boundaries) were shown to be of reflecting type, it was only conjectured that all convex compact sets are likewise. The present study not only proves this conjecture and establishes the corresponding results on Stleltjes and Van Vleck polynomials in its full generality as proposed earlier by the authors, but it also furnishes a more general family of curves sharing the properties of confocal ellipses.

scholarly journals OnF-AlgebrasMp  (1

2014 ◽  
Vol 2014 ◽  
pp. 1-10 ◽  
Author(s):  
Romeo Meštrović
Keyword(s):  
Unit Disk ◽  
Complex Plane ◽  
Topological Structure ◽  
Holomorphic Functions ◽  
Open Unit ◽  
Open Unit Disk ◽  
Continuous Linear ◽  
Linear Functionals

We consider the classesMp (1<p<∞)of holomorphic functions on the open unit disk𝔻in the complex plane. These classes are in fact generalizations of the classMintroduced by Kim (1986). The spaceMpequipped with the topology given by the metricρpdefined byρp(f,g)=f-gp=∫02π‍logp1+Mf-gθdθ/2π1/p, withf,g∈MpandMfθ=sup0⩽r<1⁡f(reiθ), becomes anF-space. By a result of Stoll (1977), the Privalov spaceNp (1<p<∞)with the topology given by the Stoll metricdpis anF-algebra. By using these two facts, we prove that the spacesMpandNpcoincide and have the same topological structure. Consequently, we describe a general form of continuous linear functionals onMp(with respect to the metricρp). Furthermore, we give a characterization of bounded subsets of the spacesMp. Moreover, we give the examples of bounded subsets ofMpthat are not relatively compact.

GEOMETRICAL CHARACTERIZATION OF KELVIN-LIKE METAL FOAMS FOR DIFFERENT STRUT SHAPES AND POROSITY

2015 ◽  
Vol 18 (6) ◽  
pp. 637-652 ◽  
Author(s):  
Prashant Kumar ◽  
Frederic Topin ◽  
Lounes Tadrist
Keyword(s):  
Metal Foams ◽  
Geometrical Characterization

scholarly journals The Uniformisation of the Equation $$z^w=w^z$$

2021 ◽  
Author(s):  
A. F. Beardon
Keyword(s):  
Positive Solutions ◽  
Complex Plane ◽  
Complex Variable ◽  
Holomorphic Functions ◽  
Parametric Form ◽  
Lambert Function ◽  
The Real ◽  
Complex Equation ◽  
Real Equation ◽  
Expository Paper

AbstractThe positive solutions of the equation $$x^y = y^x$$ x y = y x have been discussed for over two centuries. Goldbach found a parametric form for the solutions, and later a connection was made with the classical Lambert function, which was also studied by Euler. Despite the attention given to the real equation $$x^y=y^x$$ x y = y x , the complex equation $$z^w = w^z$$ z w = w z has virtually been ignored in the literature. In this expository paper, we suggest that the problem should not be simply to parametrise the solutions of the equation, but to uniformize it. Explicitly, we construct a pair z(t) and w(t) of functions of a complex variable t that are holomorphic functions of t lying in some region D of the complex plane that satisfy the equation $$z(t)^{w(t)} = w(t)^{z(t)}$$ z ( t ) w ( t ) = w ( t ) z ( t ) for t in D. Moreover, when t is positive these solutions agree with those of $$x^y=y^x$$ x y = y x .

scholarly journals A Characterization of the Dynamics of Schröder’s Method for Polynomials with Two Roots

2021 ◽  
Vol 5 (1) ◽  
pp. 25
Author(s):  
Víctor Galilea ◽  
José M. Gutiérrez
Keyword(s):  
Nonlinear Equations ◽  
Complex Plane ◽  
Iterative Process ◽  
Dynamical Behavior ◽  
Basins Of Attraction ◽  
Schröder’S Method

The purpose of this work is to give a first approach to the dynamical behavior of Schröder’s method, a well-known iterative process for solving nonlinear equations. In this context, we consider equations defined in the complex plane. By using topological conjugations, we characterize the basins of attraction of Schröder’s method applied to polynomials with two roots and different multiplicities. Actually, we show that these basins are half-planes or circles, depending on the multiplicities of the roots. We conclude our study with a graphical gallery that allow us to compare the basins of attraction of Newton’s and Schröder’s method applied to some given polynomials.

A new approach to the bienergy and biangle of a moving particle lying in a surface of lorentzian space

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Talat Körpınar ◽  
Yasin Ünlütürk
Keyword(s):  
Vector Fields ◽  
Elastic Surface ◽  
Geometrical Characterization ◽  
Lorentzian Space ◽  
New Approach ◽  
Darboux Vector ◽  
Moving Particle ◽  
The Relationship ◽  
Surface Curve

AbstractIn this research, we study bienergy and biangles of moving particles lying on the surface of Lorentzian 3-space by using their energy and angle values. We present the geometrical characterization of bienergy of the particle in Darboux vector fields depending on surface. We also give the relationship between bienergy of the surface curve and bienergy of the elastic surface curve. We conclude the paper by providing bienergy-curve graphics for different cases.

scholarly journals A note on a space Hp, a of holomorphic functions

1987 ◽  
Vol 35 (3) ◽  
pp. 471-479
Author(s):  
H. O. Kim ◽  
S. M. Kim ◽  
E. G. Kwon
Keyword(s):  
Hardy Space ◽  
Complex Plane ◽  
Unit Disc ◽  
Weak Type ◽  
Holomorphic Functions ◽  
Embedding Theorem ◽  
Type Operator ◽  
Taylor Coefficients ◽  
Bounded Holomorphic

For 0 < p < ∞ and 0 ≤a; ≤ 1, we define a space Hp, a of holomorphic functions on the unit disc of the complex plane, for which Hp, 0 = H∞, the space of all bounded holomorphic functions, and Hp, 1 = Hp, the usual Hardy space. We introduce a weak type operator whose boundedness extends the well-known Hardy-Littlewood embedding theorem to Hp, a, give some results on the Taylor coefficients of the functions of Hp, a and show by an example that the inner factor cannot be divisible in Hp, a.

scholarly journals Geometrical characterization of fluorescently labelled surfaces from noisy 3D microscopy data

2017 ◽  
Vol 269 (3) ◽  
pp. 259-268 ◽  
Author(s):  
ELIJAH SHELTON ◽  
FRIEDHELM SERWANE ◽  
OTGER CAMPÀS
Keyword(s):  
Geometrical Characterization ◽  
3D Microscopy ◽  
Microscopy Data
Sign in / Sign up

Export Citation Format

Share Document