A representation formula for reproducing subharmonic functions in the unit disc

2006 ◽  
pp. 165-173 ◽  
Author(s):  
Anders Olofsson
Keyword(s):  
Unit Disc ◽  
Representation Formula ◽  
Subharmonic Functions

scholarly journals Riesz-Martin representation for positive super-polyharmonic functions in A Riemannian manifold

2006 ◽  
Vol 2006 ◽  
pp. 1-9
Author(s):  
V. Anandam ◽  
S. I. Othman
Keyword(s):  
Riemannian Manifold ◽  
Complex Plane ◽  
Unit Disc ◽  
Biharmonic Function ◽  
Subharmonic Functions ◽  
Polyharmonic Functions ◽  
Superharmonic Functions ◽  
Martin Representation

Letube a super-biharmonic function, that is,Δ2u≥0, on the unit discDin the complex plane, satisfying certain conditions. Then it has been shown thatuhas a representation analogous to the Poisson-Jensen representation for subharmonic functions onD. In the same vein, it is shown here that a functionuon any Green domainΩin a Riemannian manifold satisfying the conditions(−Δ)iu≥0for0≤i≤mhas a representation analogous to the Riesz-Martin representation for positive superharmonic functions onΩ.

scholarly journals ROTARU ALPHA - CONVEX FUNCTIONS

1992 ◽  
Vol 23 (4) ◽  
pp. 355-362
Author(s):  
SUBHAS S. TIHOOSNURMATH ◽  
S. R. SWAMY
Keyword(s):  
Real Number ◽  
Convex Functions ◽  
Analytic Functions ◽  
Unit Disc ◽  
Representation Formula ◽  
Negative Real Number

Let $S^*(a,b)$ denote the class of analytic functions $f$ in the unit disc $E$, with $f(0) =f'(0) - 1 =0$, satisfying the condition $|(zf'(z)/f(z))- a|<b$, $a\in C$, $|a- 1|<b\le Re(a)$, $z\in E$. In this paper the class $S^*(\alpha, a, b)$ of functions $f$ analytic in $E$, with $f(0) = f'(0)- 1 =0$, $f(z)f'(z)/z\neq 0$ for $z$ in $E$ and satisfying in $E$ the condition $|J(\alpha,f)- a|<b$, $a \in C$, $|a-1|<b\le Re(a)$, where $J(\alpha, f) =(1- \alpha)(zf'(z)/f(z)) +\alpha((zf'(z))'/f'(z))$, $\alpha$ a non-negative real number is introduced. It is proved that $S^*(\alpha, a,b)\subset S^*(a,b)$, if $a> (4b/c)|Im(a)|$, $c=(b^2- |a- 1|^2)/b$. Further a representation formula for $f \in S^*(\alpha, a, b)$ and an inequality relating the coefficients of functions in $S^*(\alpha, a, b)$ are obtained.

The Growth and Borel Points of Random Algebroid Functions in the Unit Disc

2021 ◽  
Vol 41 (4) ◽  
pp. 1119-1129
Author(s):  
Daochun Sun ◽  
Yingying Huo ◽  
Fujie Chai
Keyword(s):  
Unit Disc ◽  
Algebroid Functions

scholarly journals Connected perimeter of planar sets

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
François Dayrens ◽  
Simon Masnou ◽  
Matteo Novaga ◽  
Marco Pozzetta
Keyword(s):  
Minimization Problem ◽  
Existence Of Solutions ◽  
Lower Semicontinuous ◽  
Representation Formula ◽  
Steiner Trees ◽  
Lower Semicontinuous Envelope ◽  
Simply Connected

AbstractWe introduce a notion of connected perimeter for planar sets defined as the lower semicontinuous envelope of perimeters of approximating sets which are measure-theoretically connected. A companion notion of simply connected perimeter is also studied. We prove a representation formula which links the connected perimeter, the classical perimeter, and the length of suitable Steiner trees. We also discuss the application of this notion to the existence of solutions to a nonlocal minimization problem with connectedness constraint.

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