Schwarz’s Lemma. Conformal Maps of the Unit Disk and the Upper Half-Plane. (Pre)-Compact Subsets of a Metric Space. Continuous Linear Functionals on H(D). Arzelà-Ascoli’s Theorem. Montel’s Theorem. Hurwitz’s Theorem

2017 ◽  
pp. 157-165
Author(s):  
Alexander Isaev
Keyword(s):  
Unit Disk ◽  
Half Plane ◽  
Metric Space ◽  
Continuous Linear ◽  
Linear Functionals ◽  
Conformal Maps ◽  
Montel’S Theorem ◽  
Upper Half Plane ◽  
Schwarz’S Lemma ◽  
Hurwitz’S Theorem

scholarly journals Differential Subordination Results for Analytic Functions in the Upper Half-Plane

2014 ◽  
Vol 2014 ◽  
pp. 1-6
Author(s):  
Huo Tang ◽  
M. K. Aouf ◽  
Guan-Tie Deng ◽  
Shu-Hai Li
Keyword(s):  
Unit Disk ◽  
Half Plane ◽  
Analytic Functions ◽  
Differential Subordination ◽  
Upper Half Plane ◽  
Admissible Functions

There are many articles in the literature dealing with differential subordination problems for analytic functions in the unit disk, and only a few articles deal with the above problems in the upper half-plane. In this paper, we aim to derive several differential subordination results for analytic functions in the upper half-plane by investigating certain suitable classes of admissible functions. Some useful consequences of our main results are also pointed out.

H(ϕ) Spaces

1986 ◽  
Vol 29 (3) ◽  
pp. 295-301 ◽  
Author(s):  
W. Deeb ◽  
M. Marzuq
Keyword(s):  
Unit Disk ◽  
Analytic Functions ◽  
Continuous Linear ◽  
Linear Functionals ◽  
Almost Everywhere ◽  
Subadditive Function

AbstractLet ψ be a non-decreasing continuous subadditive function defined on [0, ∞) and satisfy ψ(x) = 0 if and only if x = 0. The space H(ψ) is defined as the set of analytic functions in the unit disk which satisfyand the space H+ (ψ) is the space of a f ∊ H(ψ) for whichwhere almost everywhere.In this paper we study the H(ψ) spaces and characterize the continuous linear functionals on H+ (ψ).

scholarly journals Some properties of Glisson distances in the upper half-plane

2021 ◽  
Vol 2131 (3) ◽  
pp. 032039
Author(s):  
M Ovchintsev
Keyword(s):  
Unit Ball ◽  
Unit Disk ◽  
Half Plane ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Ordinate Axis ◽  
Upper Half Plane ◽  
Necessary And Sufficient

Abstract The author compares the Gleason distance with the distance of Euclid in the unit disk in the upper half plane. The concept of “the Gleason distance” was formulated in the work of H.S. Bear [1] The Gleason distance is defined as follows (see [1]): d = sup |f(z2)-f(z1)|, f(Z)εB1(K) where B 1 (K) is the unit ball in the space of bounded analytic in K functions. The author of the article proves that in the circle K the distances of Gleason and Euclid are equal only when the points are opposite. He found necessary and sufficient conditions, when the distances are equal for the two given points which are symmetrical about the ordinate axis.

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.

The Upper Half Plane Model for Hyperbolic Geometry

1980 ◽  
Vol 87 (1) ◽  
pp. 48 ◽  
Author(s):  
Richard S. Millman
Keyword(s):  
Half Plane ◽  
Hyperbolic Geometry ◽  
Plane Model ◽  
Upper Half Plane

scholarly journals Extending Edgar's ordering to locally convex spaces

1992 ◽  
Vol 34 (2) ◽  
pp. 175-188
Author(s):  
Neill Robertson
Keyword(s):  
Convex Space ◽  
Dual Pair ◽  
Locally Convex Space ◽  
Vector Spaces ◽  
Locally Convex Spaces ◽  
Continuous Linear ◽  
Linear Functionals ◽  
Hausdorff Topological Vector Space ◽  
Locally Convex ◽  
Mackey Topologies

By the term “locally convex space”, we mean a locally convex Hausdorff topological vector space (see [17]). We shall denote the algebraic dual of a locally convex space E by E*, and its topological dual by E′. It is convenient to think of the elements of E as being linear functionals on E′, so that E can be identified with a subspace of E′*. The adjoint of a continuous linear map T:E→F will be denoted by T′:F′→E′. If 〈E, F〈 is a dual pair of vector spaces, then we shall denote the corresponding weak, strong and Mackey topologies on E by α(E, F), β(E, F) and μ(E, F) respectively.

scholarly journals On Complex Explicit Formulae Connected with the Möbius Function of an Elliptic Curve

2014 ◽  
Vol 57 (2) ◽  
pp. 381-389
Author(s):  
Adrian Łydka
Keyword(s):  
Functional Equation ◽  
Meromorphic Function ◽  
Elliptic Curve ◽  
Explicit Formula ◽  
Half Plane ◽  
Complex Plane ◽  
Möbius Function ◽  
Mobius Function ◽  
Upper Half Plane ◽  
Explicit Formulae

AbstractWe study analytic properties function m(z, E), which is defined on the upper half-plane as an integral from the shifted L-function of an elliptic curve. We show that m(z, E) analytically continues to a meromorphic function on the whole complex plane and satisfies certain functional equation. Moreover, we give explicit formula for m(z, E) in the strip |ℑz| < 2π.

Exact determination of gravity stresses in finite elastic slopes: Part I. Theoretical considerations

1983 ◽  
Vol 20 (1) ◽  
pp. 47-54 ◽  
Author(s):  
V. Silvestri ◽  
C. Tabib
Keyword(s):  
Plane Strain ◽  
Half Plane ◽  
Inclination Angle ◽  
Complex Variable ◽  
Earth Pressure ◽  
Exact Distributions ◽  
Exact Determination ◽  
Upper Half Plane ◽  
Theoretical Considerations

The exact distributions of gravity stresses are obtained within slopes of finite height inclined at various angles, −β (β = π/2, π/3, π/4, π/6, and π/8), to the horizontal. The solutions are obtained by application of the theory of a complex variable. In homogeneous, isotropic, and linearly elastic slopes under plane strain conditions, the gravity stresses are independent of Young's modulus and are a function of (a) the coordinates, (b) the height, (c) the inclination angle, (d) Poisson's ratio or the coefficient of earth pressure at rest, and (e) the volumetric weight. Conformal applications that transform the planes of the various slopes studied onto the upper half-plane are analytically obtained. These solutions are also represented graphically.

scholarly journals Nonlinear potentials in function spaces

2002 ◽  
Vol 165 ◽  
pp. 91-116 ◽  
Author(s):  
Murali Rao ◽  
Zoran Vondraćek
Keyword(s):  
Banach Space ◽  
Potential Theory ◽  
Finite Energy ◽  
Duality Mapping ◽  
Quadratic Functional ◽  
Continuous Linear ◽  
Linear Functionals ◽  
Nonlinear Potential Theory ◽  
Nonlinear Potential ◽  
Nonlinear Potentials

We introduce a framework for a nonlinear potential theory without a kernel on a reflexive, strictly convex and smooth Banach space of functions. Nonlinear potentials are defined as images of nonnegative continuous linear functionals on that space under the duality mapping. We study potentials and reduced functions by using a variant of the Gauss-Frostman quadratic functional. The framework allows a development of other main concepts of nonlinear potential theory such as capacities, equilibrium potentials and measures of finite energy.

Sign in / Sign up

Export Citation Format

Share Document