Geometric Proofs of some Classical Results on Boundary Values for Analytic Functions

1994 ◽  
Vol 37 (2) ◽  
pp. 263-269 ◽  
Author(s):  
Enrique Villamor
Keyword(s):  
Analytic Function ◽  
Analytic Functions ◽  
Unit Disc ◽  
Logarithmic Capacity ◽  
Geometric Proof ◽  
Boundary Values ◽  
Dirichlet Integral ◽  
Capacity Zero ◽  
Extremal Metric ◽  
Nontangential Limits

AbstractIn this note we are going to give a geometric proof, using the method of the extremal metric, of the following result of Beurling. For any analytic function f(z) in the unit disc Δ of the plane with a bounded Dirichlet integral, the set E on the boundary of the unit disc where the nontangential limits of f(z) do not exist has logarithmic capacity zero. Also, using an unpublished result of Beurling, we will prove different results on boundary values for different classes of functions.

scholarly journals Class of Analytic Function Related with Uniformly Convex and Janowski’s Functions

2018 ◽  
Vol 2018 ◽  
pp. 1-6 ◽  
Author(s):  
Akhter Rasheed ◽  
Saqib Hussain ◽  
Muhammad Asad Zaighum ◽  
Maslina Darus
Keyword(s):  
Analytic Function ◽  
Analytic Functions ◽  
Unit Disc ◽  
Extreme Points ◽  
Distortion Theorem ◽  
Open Unit ◽  
Uniformly Convex ◽  
Coefficient Estimates ◽  
Open Unit Disc

In this paper, we introduce a new subclass of analytic functions in open unit disc. We obtain coefficient estimates, extreme points, and distortion theorem. We also derived the radii of close-to-convexity and starlikeness for this class.

Cauchy and Poisson integral representations for ultradistributions of compact support and distributional boundary values

1981 ◽  
Vol 91 (1-2) ◽  
pp. 49-62 ◽  
Author(s):  
R. S. Pathak
Keyword(s):  
Analytic Function ◽  
Growth Condition ◽  
Compact Support ◽  
Analytic Functions ◽  
Convex Cone ◽  
Integral Representations ◽  
Boundary Values ◽  
Cauchy Integral ◽  
Boundedness Condition ◽  
Poisson Integrals

SynopsisUltradistributions of compact support are represented as the boundary values of Cauchy and Poisson integrals corresponding to tubular radial domains Tc' =ℝn + iC', C'⊂⊂C, where C is an open, connected, convex cone. The Cauchy integral of is shown to be an analytic function in TC' which satisfies a certain boundedness condition. Analytic functions which satisfy a specified growth condition in TC' have a distributional boundary value which can be used to determine an distribution.

scholarly journals Positively Infinite Singularities of a Superharmonic Function

1968 ◽  
Vol 31 ◽  
pp. 89-96
Author(s):  
Kikuji Matsumoto
Keyword(s):  
Complex Plane ◽  
Unit Disc ◽  
Measure Zero ◽  
Logarithmic Potential ◽  
Logarithmic Capacity ◽  
Superharmonic Function ◽  
Compact Set ◽  
Linear Measure ◽  
Capacity Zero ◽  
Positive Capacity

Let E be a compact set of logarithmic capacity zero in the complex plane. Then the following is well-known as Evans-Selberg’s theorem [1] [8]: there is a measure with support contained in E such that its logarithmic potential is positively infinite at each point of E. But such a potential does not exist for E of logarithmic positive capacity. Now suppose that E is contained in the circumference of the unit disc |z| < 1 and is of linear measure zero.

scholarly journals A distributional representation of strip analytic functions

1982 ◽  
Vol 5 (1) ◽  
pp. 1-9 ◽  
Author(s):  
Dragiŝa Mitrović
Keyword(s):  
Analytic Function ◽  
Analytic Functions ◽  
Boundary Values ◽  
Cauchy Integrals ◽  
The Difference

A strip analytic function converging in theD′topology to certain boundary values (from the interior of the strip) is represented as the difference of two generalized Cauchy integrals.

scholarly journals On a subclass of analytic functions defined by Ruscheweyh operator

2012 ◽  
Vol 21 (1) ◽  
pp. 49-56
Author(s):  
ABDUL RAHMAN SALMAN JUMA ◽  
◽  
LUMINITA-IOANA COTIRLA ◽  
Keyword(s):  
Analytic Function ◽  
Analytic Functions ◽  
Unit Disc ◽  
Inclusion Relation ◽  
Distortion Bounds ◽  
Ruscheweyh Derivative ◽  
Coefficient Condition ◽  
Necessary And Sufficient

By using the Ruscheweyh derivative, we have introduced a subclass of analytic functions with negative coefficients in the unit disc. Some properties of analytic function as necessary and sufficient coefficient condition for this class are provided. Distortion bounds, inclusion relation and various properties are also determined.

scholarly journals A CONVOLUTION APPROACH TO CERTAIN SUBCLASSES OF STARLIKE FUNCTIONS

1992 ◽  
Vol 23 (4) ◽  
pp. 311-320
Author(s):  
T . RAM REDDY ◽  
O. P. JUNEJA ◽  
K. SATHYANARAYANA
Keyword(s):  
Analytic Function ◽  
Analytic Functions ◽  
Unit Disc ◽  
Starlike Functions ◽  
Integral Transforms ◽  
Open Unit ◽  
Sufficient Condition ◽  
Open Unit Disc ◽  
Containment Property ◽  
Convolution Approach

The class $R_\gamma(A,B)$ for $-1\le B < A\le 1$ and $\gamma> (A- 1)/(1- B)$ consisting of normalised analytic functions in the open unit disc is defined with the help of Convolution technique. It consists of univalent starlike functions for $\gamma\ge 0$. We establish containment property, integral transforms and a sufficient condition for an analytic function to be in $R\gamma(A,B)$. Using the concept of dual spaces we find a convolution condition for a function in this class.

scholarly journals Characterisations for analytic functions of bounded mean oscillation

1992 ◽  
Vol 46 (1) ◽  
pp. 115-125 ◽  
Author(s):  
Jie Miao
Keyword(s):  
Analytic Function ◽  
Fractional Derivative ◽  
Lebesgue Measure ◽  
Analytic Functions ◽  
Unit Disc ◽  
Carleson Measure ◽  
Bounded Mean Oscillation ◽  
Mean Oscillation ◽  
Significant Extension

Let α > 0 and let f[α](z) be the αth fractional derivative of an analytic function f on the unit disc D. In this paper we show that f ∈ BMOA if and only if |f[α](z)|2 (l - |z|2)2α−1dA(z) is a Carleson measure and f ∈ VMOA if and only if |f[α](z)|2 (1 − |z|2)2α−1dA(z) is a vanishing Carleson measure, where A denotes the normalised Lebesgue measure on D. Hence a significant extension of familiar characterisations for analytic functions of bounded and vanishing mean oscillation is obtained.

scholarly journals On Boundary Values of an Analytic Transformation of a Circle into a Riemann Surface

1956 ◽  
Vol 10 ◽  
pp. 171-175
Author(s):  
Makoto Ohtsuka
Keyword(s):  
Riemann Surface ◽  
Universal Covering ◽  
Logarithmic Capacity ◽  
Hyperbolic Type ◽  
Linear Measure ◽  
Classical Theorem ◽  
Boundary Values ◽  
Covering Surface ◽  
Analytic Transformation ◽  
Capacity Zero

Let f(z) be a nonconstant analytic transformation of the unit circle U : ∣z∣ < 1 into a Riemann surface ℜ. As an extension of a classical theorem of F. and M. Riesz, the author proved in Theorem 3.4 of [2] that if the image of U is relatively compact in ℜ and has universal covering surface of hyperbolic type, and if, at every point of a set on ∣z∣ = 1 of positive inner linear measure, there terminates a curve along which f(z) has limit, then the set of such limits has positive inner logarithmic capacity. This theorem was followed by the first proposition in Kuramochi [1], which asserts that, if ℜ has a null boundary and the image of U excludes a set of positive logarithmic capacity on ℜ and if, at every point of a set E on ∣z∣ = 1, there terminates a curve along which f(z) has limit in the union of a set of inner logarithmic capacity zero on ℜ and the boundary components of ℜ then the inner linear measure of E is zero.

INTEGRAL MEANS AND DIRICHLET INTEGRAL FOR CERTAIN CLASSES OF ANALYTIC FUNCTIONS

2015 ◽  
Vol 99 (3) ◽  
pp. 315-333
Author(s):  
MD FIROZ ALI ◽  
A. VASUDEVARAO
Keyword(s):  
Fluid Dynamics ◽  
Analytic Function ◽  
Unit Disk ◽  
Analytic Functions ◽  
Starlike Functions ◽  
Extremal Problems ◽  
Dirichlet Integral ◽  
Integral Means ◽  
Maximum Value ◽  
Image Position

For a normalized analytic function$f(z)=z+\sum _{n=2}^{\infty }a_{n}z^{n}$in the unit disk$\mathbb{D}:=\{z\in \mathbb{C}:|z|<1\}$, the estimate of the integral means$$\begin{eqnarray}L_{1}(r,f):=\frac{r^{2}}{2{\it\pi}}\int _{-{\it\pi}}^{{\it\pi}}\frac{d{\it\theta}}{|f(re^{i{\it\theta}})|^{2}}\end{eqnarray}$$is an important quantity for certain problems in fluid dynamics, especially when the functions$f(z)$are nonvanishing in the punctured unit disk$\mathbb{D}\setminus \{0\}$. Let${\rm\Delta}(r,f)$denote the area of the image of the subdisk$\mathbb{D}_{r}:=\{z\in \mathbb{C}:|z|<r\}$under$f$, where$0<r\leq 1$. In this paper, we solve two extremal problems of finding the maximum value of$L_{1}(r,f)$and${\rm\Delta}(r,z/f)$as a function of$r$when$f$belongs to the class of$m$-fold symmetric starlike functions of complex order defined by a subordination relation. One of the particular cases of the latter problem includes the solution to a conjecture of Yamashita, which was settled recently by Obradovićet al.[‘A proof of Yamashita’s conjecture on area integral’,Comput. Methods Funct. Theory13(2013), 479–492].

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