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In this paper we rigorously compute the average multifractal spectrum of harmonic measure on the boundary of SLE clusters.
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We give several new characterizations of Caratheodory convergence of simply connected domains. We then investigate how different definitions of convergence generalize to the multiply-connected case.
Let $E$ be a continuum in the closed unit disk $|z|le 1$ of the complex $z$-plane which divides the open disk $|z| < 1$ into $nge 2$ pairwise non-intersecting simply connected domains $D_k,$ such that each of the domains $D_k$ contains some point $a_k$ on a prescribed circle $|z| = rho, 0 <rho <1, k=1,...,n,. $ It is shown that for some increasing function $Psi,$ independent of $E$ and the choice of the points $a_k,$ the mean value of the harmonic measures $$ Psi^{-1}[ frac{1}{n} sum_{k=1}^{k} Psi(omega(a_k,E, D_k))] $$ is greater than or equal to the harmonic measure $omega(rho, E^*, D^*),,$ where $E^* = {z: z^n in [-1,0] }$ and $D^* ={z: |z|<1, |{rm arg} z| < pi/n} ,.$ This implies, for instance, a solution to a problem of R.W. Barnard, L. Cole, and A. Yu. Solynin concerning a lower estimate of the quantity $inf_{E} max_{k=1,...,n} omega(a_k,E, D_k),$ for arbitrary points of the circle $|z| = rho ,.$ These authors stated this hypothesis in the particular case when the points are equally distributed on the circle $|z| = rho ,.$
We study a family of real rational functions with prescribed critical points and the evolution of its poles and critical points under particular Loewner flows. The recent work by Peltola and Wang shows that the real locus of these rational functions contains the multiple SLE$(0)$ curves, the deterministic $kappa to 0$ limit of the multiple SLE$(kappa)$ system. Our main results highlight the importance of the poles of the rational function in determining properties of the SLE$(0)$ curves. We show that solutions to the classical limit of the the null vector equations, which are used in Loewner evolution of the multiple SLE$(0)$ curves, have simple expressions in terms of the critical points and the poles of the rational function. We also show that the evolution of the poles and critical points under the Loewner flow is a particular Calogero-Moser integrable system. A key step in our analysis is a new integral of motion for the deterministic Loewner flow.
Harmonic functions are natural generalizations of conformal mappings. In recent years, a lot of work have been done by some researchers who focus on harmonic starlike functions. In this paper, we aim to introduce two classes of harmonic univalent functions of the unit disk, called hereditarily $lambda$-spirallike functions and hereditarily strongly starlike functions, which are the generalizations of $lambda$-spirallike functions and strongly starlike functions, respectively. We note that a relation can be obtained between this two classes. We also investigate analytic characterization of hereditarily spirallike functions and uniform boundedness of hereditarily strongly starlike functions. Some coefficient conditions are given for hereditary strong starlikeness and hereditary spirallikeness. As a simple application, we consider a special form of harmonic functions.
In the present paper, we will study geometric properties of harmonic mappings whose analytic and co-analytic parts are (shifted) generated functions of completely monotone sequences.
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