Growth Of Subharmonic Functions Research Articles

Harmonic measure of the outer boundary of colander sets

We present two companion results: Phragmén–Lindelöf type tight bounds on the minimal possible growth of subharmonic functions with a recurrent zero set, and tight bounds on the maximal possible decay of the harmonic measure of the outer boundary of coland

Growth of subharmonic functions along line and distribution of their Riesz measures

Growth of subharmonic functions along line and distribution of their Riesz measures

The generalized Matsaev theorem on growth of subharmonic functions admitting a lower bound in ℝn

We generalize Matsaev’s theorem for subharmonic functions from two to higher dimension. The proofs are nontrivial and constructive.

Growth and Asymptotic Sets of Subharmonic Functions

We study the relation between the growth of a subharmonic function in the half space and the size of its asymptotic set. A function f defined on a domain D has an asymptotic value b∈[−∞, ∞] at a∈∂D if there exists a path γ in D ending at a such that u(p) tends to b as p tends to a along γ. The set of all points on ∂D at which f has an asymptotic value b is denoted by A(f, b). G. R. MacLane [10, 11] studied the class of analytic functions in the unit disk having asymptotic values at a dense subset of the unit circle. Hornblower [8, 9] studied the analogous class for subharmonic functions. Many theorems have since been proved having the following character: for a function f of a given growth, if A(f, +∞) is a small set then f has nice boundary behavior on a large set. See [1, 3–7] and the references therein. For α>0, let Mα be the class of subharmonic functions u in R + n + 1 ≡{(x, y):x∈ℝn, y>0} satisfying the growth condition u(x, y) ⩽ C(u)y−α for 0 < y < 1 for some constant C(u) depending on u. Denote by F(u) the Fatou set of u, which consists of points on ∂ R + n + 1 where u has finite vertical limits. For β>0, denote by Hβ the β-dimensional Hausdorff content. The following theorem is due to Barth and Rippon [1], Fernández, Heinonen and Llorente [5], and Gardiner [6].

Growth and asymptotic sets of subharmonic functions II

We study the relation between the growth of a subharmonic function in the half space $\Bbb R_+^{n+1}$ and the size of its asymptotic set. In particular, we prove that for any $n \ge 1$ and $0 < \alpha \le n$, there exists a subharmonic function $u$ in the $\Bbb R^{n+1}_+$ satisfying the growth condition of order $\alpha: u(x) \le x_{n+1}^{-\alpha}$ for $0 < x_{n+1} < 1$, such that the Hausdorff dimension of the asymptotic set $\bigcup_{\lambda\ne -\infty} A(\lambda)$ is exactly $n - \alpha$. Here $A (\lambda)$ is the set of boundary points at which $f$ tends to $\lambda$ along some curve. This proves the sharpness of a theorem due to Berman, Barth, Rippon, Sons, Fernandez, Heinonen, Llorente and Gardiner cumulatively

On the growth of subharmonic functions in space

Consider a function u = u1 - u2, where u1 and u2 are subharmonic in BY’ (m 2 3), and harmonic in a neighborhood of the origin. Let v, and v2 be the Riesz-masses of U, and u2, respectively. We assume that vi and v2 are mutually singular and that they are equal respectively to the positive and negative parts of the Riesz-measure [ 11, p. 3531 of U. We use the notation

Convexity of means and growth of subharmonic functions in strips

On demontre les theoremes de convexite et de croissance pour certaines moyennes ponderees de fonctions sousharmoniques dans une bande

Asymptotic regularity of growth of subharmonic functions of finite order

Asymptotic regularity of growth of subharmonic functions of finite order

On the growth of subharmonic functions along paths

On the growth of subharmonic functions along paths

Representation and Growth of Subharmonic Functions in Half-Spaces

Representation and Growth of Subharmonic Functions in Half-Spaces

On the growth of subharmonic functions in $\{{\rm Re}z&gt;0\}$

On the growth of subharmonic functions in $\{{\rm Re}z&gt;0\}$

On the growth of subharmonic functions

On the growth of subharmonic functions

Eigenvalue inequalities for the dirichlet problem on spheres and the growth of subharmonic functions

Eigenvalue inequalities for the dirichlet problem on spheres and the growth of subharmonic functions

On the Growth of Subharmonic Functions on Asymptotic Paths

On the Growth of Subharmonic Functions on Asymptotic Paths