The logarithm of the modulus of a holomorphic function as a minorant for a subharmonic function. II. The complex plane

2017 ◽  
Vol 101 (3-4) ◽  
pp. 590-607 ◽  
Author(s):  
T. Yu. Baiguskarov ◽  
B. N. Khabibullin ◽  
A. V. Khasanova
Keyword(s):  
Holomorphic Function ◽  
Complex Plane ◽  
Subharmonic Function

scholarly journals Extension of holomorphic L2-functions with weighted growth conditions

1992 ◽  
Vol 126 ◽  
pp. 141-157 ◽  
Author(s):  
Klas Diederich ◽  
Gregor Herbort
Keyword(s):  
Fixed Point ◽  
Holomorphic Function ◽  
Lebesgue Measure ◽  
Complex Plane ◽  
Pseudoconvex Domain ◽  
Smooth Boundary ◽  
Small Neighborhood ◽  
Growth Conditions

In this article a new contribution to the following question is given: Let Ω ⊂ ⊂ Cn be a bounded pseudoconvex domain with C∞-smooth boundary, q ∈ ∂Ω a fixed point and H a k-dimensional affine complex plane such that q ∈ H and H intersects ∂Ω at q transversally. Let U be a suitably small neighborhood of q, and denote by r a C∞-defining function of Ω on U. Under which conditions on ∂Ω near q is it possible to find an exponent η>0 > 0 such that every holomorphic function f on Ω′ = H ∩Ω∩ U withwhere dλ′ denotes the Lebesgue-measure on H, can be extended to a holomorphic function ^f on Ω ∩ U such that even

Complex Uniform Convexity and Riesz Measures

2004 ◽  
Vol 56 (2) ◽  
pp. 225-245 ◽  
Author(s):  
Gordon Blower ◽  
Thomas Ransford
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Mass Distribution ◽  
Complex Plane ◽  
Subharmonic Function ◽  
Uniform Convexity ◽  
Von Neumann ◽  
Riesz Measure

AbstractThe norm on a Banach space gives rise to a subharmonic function on the complex plane for which the distributional Laplacian gives a Riesz measure. This measure is calculated explicitly here for LebesgueLpspaces and the von Neumann-Schatten trace ideals. Banach spaces that areq-uniformly PL-convex in the sense of Davis, Garling and Tomczak-Jaegermann are characterized in terms of the mass distribution of this measure. This gives a new proof that the trace idealscpare 2-uniformly PL-convex for 1 ≤p≤ 2.

scholarly journals Functional equation of a special Dirichlet series

1987 ◽  
Vol 10 (2) ◽  
pp. 395-403
Author(s):  
Ibrahim A. Abou-Tair
Keyword(s):  
Functional Equation ◽  
Holomorphic Function ◽  
Dirichlet Series ◽  
Half Plane ◽  
Complex Plane ◽  
Positive Integers

In this paper we study the special Dirichlet seriesL(s)=23∑n=1∞sin(2πn3)n−s,  s∈CThis series converges uniformly in the half-planeRe(s)>1and thus represents a holomorphic function there. We show that the functionLcan be extended to a holomorphic function in the whole complex-plane. The values of the functionLat the points0,±1,−2,±3,−4,±5,…are obtained. The values at the positive integers1,3,5,…are determined by means of a functional equation satisfied byL.

On a non-regular parametric integral

1979 ◽  
Vol 83 (3-4) ◽  
pp. 185-188 ◽  
Author(s):  
W. K. Hayman
Keyword(s):  
Holomorphic Function ◽  
Complex Plane ◽  
Open Interval ◽  
Image Position

Suppose that f(t, λ) is, for fixed t in an open interval (a, b), a regular (or analytic or holomorphic) function of λ, when λ lies in a domain D of the complex plane. We assume that f is integrable with respect to t for fixed λ ∈ D, and consider the parametric integralBy hypothesis F(λ) exists for λ ∈ D and Professor W. N. Everitt has asked whether F(λ) is necessarily regular as a function of λ.

scholarly journals A remark on boundedness of Bloch functions

1991 ◽  
Vol 44 (3) ◽  
pp. 527-528 ◽  
Author(s):  
Krzysztof Samotij
Keyword(s):  
Holomorphic Function ◽  
Unit Circle ◽  
Lebesgue Measure ◽  
Unit Disk ◽  
Complex Plane ◽  
Hausdorff Measure ◽  
Bloch Space ◽  
Bloch Functions ◽  
Zero Measure

Two consequences of a theorem of Dahlberg are derived. Let f be a holomorphic function in the unit disk D of the complex plane, and let E be an Fσ subset of the unit circle T. Suppose that |f(rw)| ≤ M, ω ∈ T/E, for some constant M.Then f is bounded in either of the two cases:(i) if f is in the Bloch space and E is of zero measure with respect to the Hausdorff measure associated with the function ψ(t) = t log log (2πee/t),(ii) if f is integrable with respect to the planar Lebesgue measure on D and E is of zero measure with respect to the Hausdorff measure associated with the function ψ(t) = t log(2πee/t).

scholarly journals Qualitative analysis of approximations of functions of a complex variable by Taylor series

2019 ◽  
Author(s):  
Lucas José Muñoz Dentello ◽  
Denis Rafael Nacbar
Keyword(s):  
Power Series ◽  
Holomorphic Function ◽  
Qualitative Analysis ◽  
Complex Plane ◽  
Taylor Series ◽  
Complex Variable ◽  
Complex Function ◽  
Holomorphic Functions ◽  
Vectorial Functions ◽  
Series Of Functions

The present work has been analysed graphically the approximations as Taylor series of functions of a complex variable, using the domain coloring. In this method each point is coloring according to the value of range. Functions of a complex variable have a property to associate one point of the plane other point of the plane, what describe them as vectorial functions from the complex plane to the complex plane. A complex function is called a holomorphic function in a region if is differentiable in all points of that region. Holomorphic functions can be described as power series.

A system of functions biorthogonal with the derivatives of Chebyshev second-kind polynomials of a complex variable

2020 ◽  
Vol 17 (2) ◽  
pp. 256-277
Author(s):  
Ol'ga Veselovska ◽  
Veronika Dostoina
Keyword(s):  
Complex Plane ◽  
Complex Variable ◽  
Analytic Functions ◽  
Combinatorial Identities ◽  
Independent Interest ◽  
Closed Curves ◽  
In Series ◽  
Derivatives Of

For the derivatives of Chebyshev second-kind polynomials of a complex vafiable, a system of functions biorthogonal with them on closed curves of the complex plane is constructed. Properties of these functions and the conditions of expansion of analytic functions in series in polynomials under consideration are established. The examples of such expansions are given. In addition, we obtain some combinatorial identities of independent interest.

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